FINITE ELEMENT SOLUTION OF SCATTERING by …math.ucdenver.edu/graduate/thesis/popa_thesis.pdfFINITE ELEMENT SOLUTION OF SCATTERING IN COUPLED FLUID-SOLID SYSTEMS by Mirela O. Popa

FINITE ELEMENT SOLUTION OF SCATTERING

IN COUPLED FLUID-SOLID SYSTEMS

by

Mirela O. Popa

B.S., Harvey Mudd College, Claremont CA, 1995

M.S., University of Colorado at Denver, 1997

A thesis submitted to the

University of Colorado at Denver

in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Applied Mathematics

2002

This thesis for the Doctor of Philosophy

degree by

Mirela O. Popa

has been approved

by

Jan Mandel

Leopoldo P. Franca

William L. Briggs

Lynn S. Bennethum

Charbel Farhat

Date

Popa, Mirela O. (Ph.D., Applied Mathematics)

Finite element solution of scattering in coupled fluid-solid systems

Thesis directed by Professor Jan Mandel

ABSTRACT

In this thesis we investigate the mathematical theory of wave scatter-

ing by an obstacle. The obstacle considered is a bounded elastic body in a fluid

domain. We analyse finite element methods and show existence and uniqueness

of the solution for the coupled fluid-solid interaction problem in more than one

dimension. We study the stability and regularity properties of acoustic wave

scattering by introducing interpolation of spaces and scaled norms. This type

of analysis, to the author’s knowledge, has not been investigated. Then we con-

sider a multigrid method for the coupled fluid-solid interaction model problem

in higher dimension.

We use the Garding Inequality to obtain coercivity, then we use the

Fredholm Alternative to analyze spectral properties and show uniqueness and

existence of solutions. We need only weak regularity assumptions, and give

a rigorous treatment of scale of spaces with constants independent of wave

number k. A new approach is used to show stability of the coupled problem

by using intermediate spaces and norms.

iii

Finally, we present a multigrid algorithm to solve a coupled solid-

fluid interface problem. To the author’s knowledge, multigrid methods for the

coupled acoustic-elastic problem have not been investigated. In this thesis, we

formulate such a method and present numerical experiments from a prototype

implementation in MATLAB.

This abstract accurately represents the content of the candidate’s thesis. I

recommend its publication.

SignedJan Mandel

iv

DEDICATION

To my family.

ACKNOWLEDGMENTS

I would like to thank my advisor, Prof. Jan Mandel, for his support,

guidance, and generosity.

This research was supported by the National Science Foundation un-

der grants ECS-9725504 and DMS-0074278, and by the Office of Naval Research

under grant N-00014-95-1-0663.

CONTENTS

Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Existing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3. Theoretical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Finite Element Approximation . . . . . . . . . . . . . . . . . . . . 16

3.3 Garding Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Generalized Korn Inequality . . . . . . . . . . . . . . . . . . . . . 20

3.5 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Trace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.7 Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . 22

3.8 Lax-Milgram Theorem . . . . . . . . . . . . . . . . . . . . . . . . 22

3.9 Hilbert Interpolation Spaces . . . . . . . . . . . . . . . . . . . . . 22

4. Statement of Coupled Problem . . . . . . . . . . . . . . . . . . . . . 24

4.1 Derivation of the Coupled Problem . . . . . . . . . . . . . . . . . 24

4.1.1 Elastic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2 Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.4 Solid-Fluid Interface Conditions . . . . . . . . . . . . . . . . . . 32

vii

4.2 Variational Form of Coupled Problem . . . . . . . . . . . . . . . . 34

4.3 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.1 Hilbert scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1 Garding Inequality for the Coupled Problem . . . . . . . . . . . . 42

5.2 Existence of Solution . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3 Intermediate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3.1 Intermediate norms . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.2 Regularity of Solution for Coupled Problem. . . . . . . . . . . . 63

5.4 Discretization and Error Bound . . . . . . . . . . . . . . . . . . . 65

6. Multigrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1 Multigrid for the Coupled Problem . . . . . . . . . . . . . . . . . 72

6.1.1 GMRES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.1.2 BICG-STAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.1.3 Gauss-Seidel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.1 Numerical Verification of the Discretization . . . . . . . . . . . . 78

7.2 Computational Results with Multigrid . . . . . . . . . . . . . . . 89

7.3 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . 114

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

viii

FIGURES

Figure

4.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.1 Configuration of sample points for Table 7.1. Sample points

1,2,3,4 are fluid pressure values, and sample points 5 and 6 are

ux and uy displacements, respectively. . . . . . . . . . . . . . . 80

7.2 Log log plot of difference of solution values xh at sample points

for mesh size 10×10 to 320×320 and extrapolated exact solution

x∗, k=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.3 Solution for a 40 × 40 mesh, k = 5, right-hand side modified for

the Dirichlet boundary condition p = p0 on Γd . . . . . . . . . . 83

7.4 Exact solution for a 64 × 64 mesh, k = 15, right-hand side is

modified for the Dirichlet boundary condition p = p0 on Γd . . . 84

7.5 Three multigrid iterations, 20 smoothing steps, smoother GM-

RES, 2 levels to solve a 64 × 64 mesh, k = 15, right-hand side

is modified for the Dirichlet boundary condition p = p0 on Γd. . 85

7.6 Contour of an obstacle (0.4 m) and (0.2 m) in the x and the

y direction, respectively. The gap is on the y axis. The size of

the gap is 0.5 or 50% in the x direction and 0.4 or 40% in the y

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

ix

7.7 Contour of an obstacle (0.2 m) and (0.4 m) in the x and the y

direction, respectively. The gap is on x axis. The size of the gap

is 0.4 or 40% in the x and 0.5 or 50% in the y direction. . . . . 87

7.8 Solution for a 64 × 64 mesh with ushaped obstacle (0.2 m) by

(0.4 m). The obstacle has a gap of size 0.4 by 0.5 on the y axis.

The wave number is k = 15, and the right-hand side is modified

for the Dirichlet boundary condition p = p0 on Γd . . . . . . . . 88

7.9 Solution for a 64 × 64 mesh ushaped obstacle (0.2 m) by (0.4

m). The obstacle has a gap of 0.4 by 0.5 on the x axis. The

wave number is k = 15, and the right-hand side is modified for

the Dirichlet boundary condition p = p0 on Γd . . . . . . . . . . 90

7.10 Residual reduction as a function of adding coarse meshes, de-

creasing mesh size h while keeping k3h2 constant. . . . . . . . . 98

7.11 Residual reduction as a function of adding coarse meshes, de-


7.12 Residual reduction as a function of adding smoothing steps, de-


7.13 Residual reduction as a function of adding smoothing steps, de-


7.14 Relative residual varies as mesh size h is decreased for the case

in which k3h2 is constant and in the case of resonance . . . . . . 112

7.15 Solution for a 64 × 64 mesh, k = 4π, and right-hand side mod-

ified for the Dirichlet boundary condition p = p0 on Γd. A gap

of half wavelength in x and y direction . . . . . . . . . . . . . . 116

x

TABLES

Table

7.1 Values of the solution at the 6 sample points as explained in

(Fig.7.1), mesh size h is halved at each run and wave number is

kept constant k = 5. . . . . . . . . . . . . . . . . . . . . . . . . 79

7.2 Exact solution is x∗ and solution at mesh size h is xh. . . . . . . 81

7.3 h = step size, k = wave number, sm = number of smoothing

steps, lv = number of levels, mth = iterative method used for the

smoothing algorithm, it = number of multigrid iterations, res red

= residual reduction, rel rs = relative residual, res fl = residual

in the fluid part, and res el = residual in the elastic part. BC

is BICG-STAB, GM is GMRES, GD is GMRES preconditioned

by D−1 (as explained in (7.5)), GA is Gauss-Seidel . . . . . . . 92







by D−1 (as explained in (7.5)), GA is Gauss-Seidel . . . . . . . 93

xi

7.5 Iteration counts for multigrid V cycle to achieve a relative resid-

ual of order 1e-6, for smoothers GMS=GMRES; BGS=BICG-

STAB; GMD=GMRES preconditioned by the inverse of the

lower triangular part of A. . . . . . . . . . . . . . . . . . . . . 95







by D−1 (as explained in (7.5)). . . . . . . . . . . . . . . . . . . 96


steps, lv = number of levels, it = number of multigrid iterations,

res red = residual reduction, rel rs = relative residual, res fl =

residual in the fluid part, and res el = residual in the elastic part. 103





in the fluid part, and res el = residual in the elastic part. BC is

BICG-STAB, GM is GMRES, GT is GMRES preconditioned by

the inverse of the lower triangular part of A, GA=Gauss-Seidel. 104

xii

7.9 h = the step size, k = the wave number, sm = the number

of smoothing steps, lv = the number of levels, sm =smoother,

mth = the iterative method used for the smoothing algorithm,

it = the number of multigrid iterations, res red = the residual

reduction, rl res = the relative residual, rs fl = the residual in

the fluid part, and rs el = the residual in the elastic part. BC

is BICG-STAB, GT is GMRES preconditioned by the inverse of

the lower triangular part of A . . . . . . . . . . . . . . . . . . . 106

7.10 h = the step size, k = the wave number, sm = the number of

smoothing steps, lv = the number of levels, mth = the iterative

method used for the smoothing algorithm, it = the number of

multigrid iterations, res red = the residual reduction, rl res =

the relative residual, rs fl = the residual in the fluid part, and

rs el = the residual in the elastic part. BC is BICG-STAB, GT

is GMRES preconditioned by the inverse of the lower triangular

part of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.11 h = the step size, x is size of obstacle in x direction, y size of

obstacle in y direction, gap x is size of the gap on the x axis as

a percentage, gap y is size of gap on the y axis, k = the wave

number, rs red = the residual reduction, rl rs = the relative

residual, rs fl = the residual in the fluid part, and rs el = the

residual in the elastic part. . . . . . . . . . . . . . . . . . . . . 108

xiii

7.12 h = the step size, x is size of obstacle in x direction, y size of

obstacle in y direction, gp x is size of the gap on the x axis as

a percentage, gp y is size of gap on the y axis, k = the wave



residual in the elastic part. . . . . . . . . . . . . . . . . . . . . . 109

7.13 h = the step size, k = the wave number, rel res = the relative

residual, rel res fl = the relative residual in the fluid part, and

rel res el = the relative residual in the elastic part. . . . . . . . 110

7.14 GMRES preconditioned by ILU h = the step size, k = the wave

number, rel res = the relative residual, res fl = the residual in

the fluid part, and res el = the residual in the elastic part. . . . 111

7.15 h = the step size, k = the wave number, conv fctr = convergence

factor or the residual reduction, rel res = the relative residual,

res fl = the residual in the fluid part, and res el = the residual

in the elastic part. . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.16 h = the step size, k = the wave number, conv fctr = convergence

factor or the residual reduction, rel res = the relative residual,

res fl = the residual in the fluid part, and res el = the residual

in the elastic part. . . . . . . . . . . . . . . . . . . . . . . . . . 113

xiv

7.17 Residual Reduction per unit of work for one Multigrid cycle. h

= the step size, x is size of obstacle in x direction, y size of

obstacle in y direction, gp x is size of the gap on the x axis as

a percentage, gp y is size of gap on the y axis, k = the wave



residual in the elastic part. . . . . . . . . . . . . . . . . . . . . . 115

xv

1. Introduction

Time harmonic acoustics of coupled fluid-solid systems in fluid pres-

sure and solid displacement formulation has many wide ranging applications.

For this reason it has been extensively studied in recent years in an effort

to predict the dynamic response of elastic objects immersed in fluids. Wave

propagation in fluid, assuming time-harmonic behavior, is described by the

Helmholtz equation,

4p+ k2p = 0, (1.1)

where the the critical parameter is the wave number k. In elastic media, waves

propagate in the form of oscillations of the stress field, and are described by

the elastodynamic Helmholtz equation,

∇ · τ + ω2ρu = 0, (1.2)

where u is displacement, ρ is density of the elastic medium, and τ is the stress

tensor. Wave propagation in a composite medium, consisting of a fluid part

and an elastic part, is described by (1.1) and (1.2), in the acoustic and in

the solid part, respectively, together with boundary conditions imposed on the

fluid solid interface.

Important applications of elastic wave scattering are found in geo-

physical exploration, seismology, the response of structures to seismic waves,

1

acoustic response of elastic objects immersed in fluids, response of aircraft parts

to dynamic loads, and applications of ultrasound to biological systems for di-

agnostic of therapeutic purposes [64, 69]. A significant growth in the literature

employing scattering problems can be seen in recent years; [14, 48, 50, 34, 60,

59, 21]. The question of existence and uniqueness of solutions of Helmholtz

problems was addressed by the end of the 1950s; [46, 13, 58].

In this thesis, we analyze a time-harmonic solution of coupled fluid-

solid interaction model problem in more than one dimension. We analyse finite

element methods, show existence and uniqueness of the solution, and study its

stability and regularity properties by introducing interpolation of spaces and

scaled norms. We give a rigorous treatment of scale of spaces with constants

independent of the wave number k. We present a multigrid method for the

solution of linear systems arising from the finite element discretization of the

coupled fluid-solid system. The obstacle considered is a bounded elastic body

embedded in fluid.

It is known, that numerical solutions to the Helmholtz equation de-

teriorate for increasing wave numbers k, [37, 36], which is called the pollution

effect. The effect of pollution is that the wave number of the finite element

solution is different from the wave number of the exact solution, and pollution

means that the local error has a global effect, [4, 20]. In one dimension the

pollution effect has been extensively studied, and different approaches have

been proposed that lead to solutions that do not suffer the pollution effect,

2

[3, 4]. In two and three dimensions it has been shown that pollution cannot be

avoided, [4]. In this thesis we will not be concerned with the pollution effect

or dispersion, and we will keep in mind that for large wave numbers, numer-

ical pollution in the error dominates the error of interpolation. Completely

different methods are required to address the pollution effect.

Iterative methods consisting of alternating solution in the fluid and

the solid region are known [14, 50]. Numerical solution of elliptic partial differ-

ential equations, in two or three dimensions, is a typical application for iterative

solvers based on the multilevel paradigm. In contrast to other methods, multi-

grid does not depend on the separability of the equations; thus using multigrid

methods for solving the Helmholtz equation in modeling acoustic scattering is

classical [25, 31, 63, 62]; for more recent developments, see [22, 44, 45, 62].

The remainder of the thesis is organized as follows. In Chapter 2,

we give an overview of some existing methods and approaches to solving the

problem of scattering by an elastic obstacle in an acoustic fluid. In particular,

we describe the analysis of a fluid-solid interaction problem in one dimension

studied by Makridakis et. al. [48]. In Chapter 3, we present some theoretical

preliminaries, and describe notation we use later in this thesis. In Chapter 4,

we present the relations of linear wave physics, derive the differential equations

with an emphasis on the boundary conditions that describe the solid-fluid inter-

action, and then we define the spaces necessary to derive the weak formulation

of the coupled problem. Chapter 5 is concerned with analyzing the existence

3

and uniqueness of the solution of scattering by an elastic obstacle in an acoustic

fluid in a bounded region. We use the Garding Inequality to obtain coercivity,

then we use the Fredholm Alternative to analyze the spectral properties and

show existence of solution. We show that if a solution exists, then the solution

is unique. By using intermediate spaces, we show stability of the solution.

In Chapter 6, we present a multigrid algorithm to solve a coupled solid-fluid

interface problem. In Chapter 7, we present numerical experiments from a

prototype implementation in MATLAB.

4

2. Existing Methods

The scattering and propagation of waves is a classical area, which

has engaged the interest of several generations of mathematicians, physicists,

and engineers. Extensive research in the area of acoustics, electro-magnetics,

and elastodynamics has resulted in many mathematical and computational

techniques [5, 8, 47, 64, 68, 69, 70].

Bielak, MacCamy, and Zeng [8] studied acoustic scattering in an un-

bounded domain, representing the solution in the unbounded domain by a

combination of potentials. Demkowicz [15, 16] considered elastic scattering in

unbounded domains for spherical shells submerged in fluid, and showed that

an inf-sup condition holds for the reduced problem on the boundary of the

scatterer. In [21], Djellouli, Farhat, Tezaur, and Macedo apply a finite element

method coupled with the Bayliss-Turkel-like non reflecting boundary condition

to solve direct acoustic problems. Domain decomposition approaches to solv-

ing the Helmholtz equation are in papers by Hetmaniuk and Farhat [34], and

Tezaur, Macedo and Farhat [59]. Analysis and domain decomposition methods

for the time dependent coupled fluid-solid interaction problem are found in pa-

pers by Cummings and Feng [14, 24]. Mandel in [50] presents a domain decom-

position method for solving the time harmonic coupled fluid-solid interaction

5

problem. Ohayon and Valid [55] present symmetric variational formulations for

the problem of transient and modal analysis of bounded coupled fluid-structure

linear systems, taking into account gravity and compressibility effects. Gener-

alized added mass operators, independent of time and circular frequency, are

introduced. Numerical results are presented for an incompressible hydroelas-

tic modal analysis of a liquid propelled launch vehicle, elasto-acoustic modal

analysis of an incompressible structure containing a compressible gas, and an

elastic cylinder partially filled with liquid under gravity effects. In [22], Elman

et. al. present a multigrid method enhanced by Krylov subspace iterations

for the discrete Helmholtz equation. In this thesis, we extend the approach

of [22] to the coupled problem. Makridakis et. al. [48] proved existence and

regularity of the solution of elastic scattering in bounded domains in one di-

mension, using the Garding inequality and an inf-sup condition, which leads

to asymptotically optimal estimates. Our results extend [48] to more than one

dimension; however, some tools which work in one dimension are not available

here, so we do not obtain asymptotically optimal estimates. More details on

the approaches outlined above follow.

Bielak, MacCamy, and Zeng [8] discuss a scattering problem and its

solution by a coupling method. Existence, uniqueness, convergence, as well

as accuracy of the numerical approximations is also presented. This is ac-

complished by using potential theory and three different representations of the

pressure p to solve the coupled fluid-solid problem. The domain is unbounded,

6

and it is separated into a bounded region Ω in R3, with boundary Γ, and exte-

rior Ω+. The domain Ω represents an inhomogeneous elastic obstacle and Ω+

is a compressible, non viscous, homogeneous fluid. Three different coupling

problems are presented, of which, the first two both fail for different sets of

frequencies. The third coupling method is based on potential theory, and is

shown to be stable.

Another approach to the problem of scattering by an elastic obstacle

in an acoustic fluid was presented by Demkowicz [15, 16]. The main difference

between what Demkowicz did and this thesis is that Demkowicz investigates

rigid scattering and vibrations of an elastic submerged shell. From the spectral

decomposition of the operator for an elastic spherical shell in fluid, he com-

putes the LBB constant as a function of the wave number k, and shows its

effects on convergence. This work is motivated by the earlier paper [17, 18],

where asymptotic convergence was studied for both finite and boundary el-

ement methods on acoustic problems, and a 1-D model acoustic interaction

problem is presented. Elastic scattering problems investigated by Demkowicz

in [15, 16] assume an elastic spherical shell freely floating in fluid. The spectral

decomposition for this operator is computed, and finding the LBB constant

reduces to solving a saddle point problem. Pointwise infimum of the spectral

decomposition show dependence of the LBB constant on the wave number k.

Demkowicz is able to prove that the magnitude of the LBB constant depends

upon the distance from the nearest resonant frequency, and that without a

7

strict control of the discrete LBB constant during the solution process, the

results may be unreliable.

Cummings and Feng [14] present a domain decomposition method for

the time dependent system of coupled acoustic and elastic interaction prob-

lems. Two classes of iterative methods are proposed for decoupling the domain

problem into fluid and solid subdomains, and replacing the physical interface

conditions with equivalent relaxation conditions as the transmission conditions.

The nonoverlapping domain decomposition methods developed are regarded as

Jacobi and Gauss-Seidel type algorithms, and they use convex combinations

of the original physical interface conditions to transmit information between

the subdomains. Strong convergence in the energy norm for the fluid-solid

interaction problem is shown for the iterative methods, and their findings are

supported by numerical test results.

Feng [24] analyzes some finite Galerkin approximations for the time

dependent fluid-solid interaction model, and presents a domain decomposition

method for the time dependent system of coupled elastic acoustic problem. An

optimal order apriori error estimates in L∞(H1)-norm and in L∞(L2)-norm for

the semi-discrete and fully discrete Galerkin approximation to the solution of

the model is established. Higher order time derivatives of the errors are used

for the error estimates of the interface conditions describing the interaction

between the fluid and the solid. To handle the terms involving the interface

conditions, the boundary duality argument due to Douglas and Dupont [38]

8

is used. The error estimates for the fully discrete methods are obtained by

averaging error equations at different time steps and choosing some nonstan-

dard test functions. Feng defines a second order discrete-time Galerkin method,

and presents a generalized parallelizable nonoverlapping domain decomposition

method for solving the coupled fluid-solid interaction problem.

Another domain decomposition method for time harmonic coupled

fluid-solid systems that decomposes the fluid and solid domains into nonover-

lapping subdomains is presented by Mandel in [49]. The continuity of the

solution uses Lagrange multipliers to ensure that the values of the degrees of

freedom coincide on the interface between the subdomains. This method is

known as the FETI-H domain decomposition method originally proposed by

Farhat and Roux [23]. Mandel finds that the division into subdomains does

not need to match across the wet interface. The system is augmented by dupli-

cating the degrees of freedom on the wet interface, and the original degrees of

freedom are eliminated. The intersubdomain Lagrange multipliers and the du-

plicates of the degrees of freedom on the wet interface are retained and form the

reduced problem. The resulting system is solved by iterations preconditioned

by a coarse space correction. Numerical results are presented for a bounded

2D region.

Elman et. al. [22] study the exterior Helmholtz problem in an un-

bounded domain. The unbounded domain is truncated to a finite domain

9

by introducing an artificial boundary on which the radiation boundary condi-

tion approximates the outgoing Sommerfeld radiation condition. In this pa-

per multigrid methods are used to solve the discretized Helmholtz equation.

The authors identify difficulties arising in a standard multigrid iteration for

the Helmholtz equation, and analyze and test techniques designed to address

these difficulties. Some difficulties encountered are with the smoothing and

coarse grid corrections. Standard smoothers such as Jacobi and Gauss-Seidel

relaxation become unstable for indefinite problems, since there are error com-

ponents that are amplified by these smoothers. The difficulties with the coarse

grid corrections are due to the poor approximation of the Helmholtz operator

on very coarse meshes. The approach used in [22] for smoothing is to use stan-

dard damped Jacobi relaxation when it works reasonably well, on fine enough

grids, and then, to replace it with a Krylov subspace iteration when standard

damped Jacobi fails as smoother. For the coarse grid correction, the number of

eigenvalues that are handled poorly during the correction is identified, and an

acceleration for multigrid is introduced by using multigrid as a preconditioner

for an outer Krylov subspace iteration. The authors observe that multigrid

does a poor job of eliminating some modes from the error, so it converges

slowly or even diverges in some cases, and an outer Krylov subspace iteration

is needed for the method to be robust. GMRES is used as the Krylov sub-

space method. In this thesis we extended the approach of [22] to the coupled

problem.

10

The analysis for a fluid-solid interaction problem in one dimension

done by Makridakis, Ihlenburg, and Babuska [48], is the closest to the analysis

presented in this thesis. The approach taken in [48] focuses on the stability

of the continuous problem (2.1) and on the stability and convergence of the

discrete problem (2.9) with respect to the wave number k. The problem is a 1D

layered fluid-solid-fluid medium with configuration Ω = Ω1 ∪ Ω2 ∪ Ω3 = [0, L],

L > 0,

pxx + k2p = −g1, in Ω1

(aux)x + k2gu = −f, in Ω2 (2.1)

pxx + k2p = −g2, in Ω3

with radiation boundary conditions at the boundary of Ω,

px(0) + ikp(0) = 0,

px(L)− ikp(L) = 0,

transmission conditions on the interface boundary at x1,

px(x1)− k2u(x1) = 0,

p(x1) + aux(x1) = 0,

and at x2,

px(x2)− k2u(x2) = 0,

p(x2) + aux(x2) = 0.

11

The space considered is H = H1(Ω1) × H1(Ω2) × H1(Ω3). With the weak

formulation, we find U = (p, u, p) ∈ H such that

B(U, V ) = (F, V )0,H ∀V ∈ H (2.2)

where V = (q, v, q) ∈ H,

B(U, V ) =∫Ω1

pxqxdx− k2∫Ω1

pqdx− k2u(x1)q(x1)− ikp(0)q(0)

+k2∫Ω2

auxvxdx− k4∫Ω2

guvdx− k2p(x1)v(x1) + k2p(x2)v(x2) (2.3)

+∫Ω3

px¯qxdx− k2

∫Ω3

p¯qdx+ k2u(x2)¯q(x2)− ikp(L)¯q(L)

and

(F, V )0,H = (g1, q) + k2(f, v) + (g2, q). (2.4)

The standard definition for the inner product is used, (u, v) =∫Ωi

uv dx. With

a suitable choice of norms,

(U, V )0,H = (p, q) + k2(u, v) + (p, q), (2.5)

(U, V )1,H = (Ux, Vx)0,H + k2(U, V )0,H , (2.6)

it was shown using variational techniques, that if the right-hand side F ∈ L2,

then the solution satisfies a regularity estimate of the form

‖U‖1,H ≤ C1‖F‖0,H , (2.7)

where C1 is a constant independent of k. The estimate (2.7) establishes the

uniqueness of the solution of (2.2), and is needed in the analysis of the discrete

12

problem. Uniqueness combined with the fact that the variational form satisfies

a Garding type inequality, yields the existence of the solution U . Using an

estimate of the form (2.7) for a properly chosen auxiliary problem, the Babuska-

Brezzi condition for the bilinear form B(·, ·) is proved:

supV ∈H

ReB(U, V )

‖v‖1,H

≥ γ1

k‖U‖1,H , ∀U ∈ H. (2.8)

This approach follows earlier Babuska work relying on the LBB constant. LBB

condition (2.8) is equivalent to a bound on the norm of the solution operator,

which depends on k linearly. Using (2.7) and (2.8), other similar regularity

estimates are derived. These estimates are useful in the convergence analysis

of the numerical methods.

If Sh is a suitable finite element subspace of H consisting of piecewise

polynomial functions, approximation Uh ∈ Sh to U is defined as the solution

of

B(Uh, φ) = (F, φ)0,H , ∀φ ∈ Sh. (2.9)

In [48], the authors showed the existence of a unique solution of (2.9),

provided the quantity hdk2 is small enough, where h is the maximum mesh size

and d is the polynomial degree of the functions of Sh. It is also shown that the

discrete analog of the LBB condition (2.8) is satisfied on Sh. The authors used

(2.7) to show that an optimal estimate of the form

‖U − Uh‖1,H ≤ C∗ infφ∈Sh

‖U − φ‖1,H , (2.10)

13

holds, where C∗ is a positive constant independent of h and k. The approxi-

mation properties of Sh are known, thus (2.10) implies convergence of optimal

order of the finite element approximations in the ‖ · ‖1,H norm. In 1D, the final

estimate has no dependence on wave number k. In more than 1D, the equiv-

alent of (2.8) does not hold; instead we use estimates and FEM error bounds

that rely on the Garding inequality. We have dependence on wave number

k, because the estimate relies on the boundedness of (I − βk2G)−1, where β

is a constant independent of k and G is a compact operator with finite but

unknown norm.

14

3. Theoretical Preliminaries

In this chapter, we describe notation and summarize some concepts

and algorithms used throughout this thesis. Let Ω be a bounded open con-

nected domain in Rn, n ∈ 1, 2, 3, with Lipschitz-continuous boundary Γ. Let

∂Ω = Γd∪Γn with Γd,Γn disjoint, and meas(Γd) > 0. We will denote by Γd,Γn

the parts of the boundary with Dirichlet and Neumann boundary conditions,

respectively.

3.1 Sobolev Spaces

Let W qp (Ω) be the usual Sobolev space of complex valued functions

W qp (Ω) =

f ∈ L1

loc : ‖f‖W qp (Ω) <∞

,

where q is a non-negative integer, and ‖f‖W qp (Ω) =

(∑|α|≤q ‖Dα

wf‖pLp(Ω)

)1/p.

We use the multi-index notation α for denoting partial derivatives by an n-

tuple with non-negative integer components, α = (α1, ..., αn), with the length

of α given by |α| = ∑ni=1 αi. For p = 2, the space W q

p (Ω) is a Hilbert space,

and we denote W q2 (Ω) by Hq(Ω). For q = 1, we obtain the Hilbert space H1

equipped with the norm ‖f‖H1 =(‖f‖2L2(Ω) + ‖∇f‖2L2(Ω)

)1/2. In addition to

integer-order Sobolev spaces, there are fractional-order Sobolev spaces. For

15

s ∈ R, 0 < s < 1, 1 ≤ p <∞, and q ≥ 0, we recall

‖u‖pW q+s

p (Ω):= ‖u‖pW q

p (Ω) +∑|α|=q

∫Ω

∫Ω

|u(α)(x)− u(α)(y)|p

|x− y|n+spdxdy,

with the seminorm [54]

|u|pW q+s

p (Ω):=

∑|α|=q

∫Ω

∫Ω

|u(α)(x)− u(α)(y)|p

|x− y|n+spdxdy,

where Ω ⊂ Rn. For more details, see [2, 10, 54].

3.2 Finite Element Approximation

The finite element method (FEM) is a general technique for the nu-

merical solution of partial differential equations in structural engineering. From

the engineering point of view, the method was thought of as a generalization

of earlier methods in structural engineering for beams, frames, and plates,

where the structure was subdivided into small parts, called finite elements,

with known simple behavior. The presentation here follows [10], where more

details can be found. To fix ideas, as a simple example, consider the second

order elliptic equation with Dirichlet boundary condition,

Au = f in Ω (3.1)

u = 0 on ∂Ω,

where Ω is a Lipschitz domain in R2 or R3, and

Au(x) := −n∑

i,j=1

∂

∂xj

(aij(x)

∂u

∂xi

(x)

)+

n∑k=1

bk(x)∂u

∂xk

(x) + b0(x)u(x) (3.2)

16

with the matrix [a(i, j)] symmetric, positive definite, and bounded on Ω. Mul-

tiplying by a test function v ∈ V (Ω), where the Sobolev space V (Ω) = H1(Ω),

is a given set of admissible functions, and integrating by parts, the model

problem (3.1) can be written in the variational form,

Find u ∈ V (Ω) such that a(u, v) = F (v) for all v ∈ V (Ω), (3.3)

where F : V → R is a functional with F (v) = (f, v), and the bilinear form

a(·, ·) : V × V → R given by

a(u, v) :=∫Ω

n∑i,j=1

aij∂u

∂xi

∂v

∂xj

+n∑

k=1

bk∂u

∂xk

v + b0uv dx (3.4)

is defined for all u and v in the Sobolev space V (Ω) = H1(Ω).

The functions v ∈ V (Ω) usually represent a continuously varying

quantity, such as a displacement in an elastic body, and F (v) is the total en-

ergy associated with v. In general, the functions in V (Ω) cannot be described

by a finite number of parameters, and so the problem cannot be solved directly.

The standard Galerkin approximation approach is to look for an approximate

solution of (3.3) in a finite dimensional subspace Vh(Ω) of the space V (Ω), in

which the weak form is posed. The space Vh(Ω) consists of simple functions

only depending on finitely many parameters, usually chosen to be piecewise

polynomials. This leads to a finite-dimensional problem. The Galerkin ap-

proximation is the solution to the following problem:

Find uh ∈ Vh(Ω) such that a(uh, vh) = F (vh) for all vh ∈ Vh(Ω). (3.5)

17

When a basis is chosen for Vh, the Galerkin approximation leads to a system

of equations. Let φiNi=1 be a basis for Vh(Ω). Assuming that

uh =N∑

i=1

uiφi,

equation (3.5) becomes

N∑i=1

a(φi, φj)ui = (f, φj), j = 1, ..., N.

The left-hand side matrix K = [a(φi, φj)] is called the stiffness matrix, the

right-hand side vector [(f, φj)] is called the load vector, and the vector of

unknowns [ui] is referred to as the vector of degrees of freedom.

For a wide class of approximation spaces Vh(Ω), uh is a good ap-

proximation for u. The choice of the finite dimensional subspace Vh(Ω) is

influenced by the variational formulation, accuracy requirements, and regular-

ity properties of the exact solution. The construction of suitable spaces Vh uses

triangulation, which splits the domain Ω into small disjoint simple geometries.

In R2, triangular and rectangular shapes are considered, and for R3, tetra-

hedrons and hexahedrons are used. By imposing certain assumptions on the

triangulation, the finer the triangulation, the closer a finite element Galerkin

solution is to the exact solution.

As in the example above, functions in Vh arise from a polynomial

interpolation on the elements of the triangulation, and are generated by basis

functions that are usually polynomial on each element of the triangulation of

the domain. The supports of basis functions have only small overlaps. Every

18

polynomial defined on a given region is uniquely determined by its values and

perhaps the values of its derivatives at some nodal points. Thus, each function

in Vh is determined by a set of values at nodal points, so called degrees of

freedom. Simple examples of finite element spaces include spaces formed by

continuous linear or triangle regions in R2, or tetrahedrons in R3. These spaces

are referred to as P1 and Q1 respectively. In this thesis we use standard Q1

finite element spaces.

For finite element spaces, the standard basis functions chosen are

the continuous piecewise linear functions that take the value 1 at one node

point and the value 0 at other node points. Thus, the unknowns in the linear

system arising from the discretization are the degrees of freedom of the Galerkin

approximation. Since the overlaps of the supports of the basis functions are

small, the stiffness matrix is sparse.

3.3 Garding Inequality

A bilinear form a(·, ·) on a normed linear space H, is said to be

bounded (or continuous), if ∃ c1 <∞ such that

|a(u, v)| ≤ c1‖u‖H‖v‖H ∀u, v ∈ H,

and coercive on V ⊂ H if ∃ c2 > 0 such that

|a(v, v)| ≥ c2‖v‖2H ∀ v ∈ V. (3.6)

There can be well-posed elliptic problems (3.1) for which the corresponding

variational problem (3.3) is not coercive, although a suitably large additive

19

constant can always make it coercive as follows. By Garding inequality there

is a constant, κ <∞, such that

a(v, v) + κ‖v‖2L2(Ω) ≥ α‖v‖2H1(Ω), (3.7)

for some α > 0, where a(·, ·) is defined in equation (3.4).

3.4 Generalized Korn Inequality

Korn’s inequality plays an important role in establishing the existence

and uniqueness of a solution in linearized elasticity, and it is used to establish

coercivity of the operator. Let Ω ∈ Rn be a domain with Lipschitz boundary.

There exists a constant Ck > 0, such that for all u ∈ (H1(Ω))n, [40, 53]

‖e(u)‖2(L2(Ω))n×n + ‖u‖2(L2(Ω))n ≥ Ck‖u‖2(H1(Ω))n , (3.8)

where e(u) = 12(∇u + (∇u)T ) is the strain tensor (see section 4.1).

3.5 Fredholm Alternative

Compactness of a linear operator is essential in Fredholm’s theory. For

X and Y normed spaces, an operator T : X → Y is called a compact linear

operator if T is linear, and if for every bounded subset M of X, the closure

T (M) is compact, i.e. every sequence in T (M) has a convergent subsequence

whose limit is an element of T (M).

A bounded linear operator A : X → X on a normed space X is said

to satisfy the Fredholm alternative [42] if A is such that either (I) or (II) holds:

(I) The nonhomogeneous equations Ax = y, A×f = g , have solutions x and

20

f , respectively, for every given y ∈ X and g ∈ X ′ the dual space of X, the

solutions being unique.

(II) The homogeneous equations Ax = 0, A×f = 0 , have the same number of

linearly independent solutions x1, ..., xn and f1, ..., fn, respectively. The non-

homogeneous equations Ax = y, A×f = g, have a solution if and only if y and

g are such that fk(y) = 0, g(xk) = 0 (k = 1, ..., n), respectively. The particular

important case is that in which both Ax = 0 and Af = 0 admit only the trivial

solution; then there is a solution of Ax = y for any y ∈ Y .

Compact operators and the Fredholm alternative theory are related

by the following result. Let T : X → Y be a compact linear operator on a

normed space X, and let λ 6= 0. Then Tλ = T − λI satisfies the Fredholm

alternative.

3.6 Trace Theorem

Let Ω be a bounded open set of Rn with a Lipshitz boundary ∂Ω and

Γ a subset of the boundary. Consider the Sobolev space Hm(Ω); the trace γu =

uΓ of a function u ∈ Hm(Ω) on Γ is a bounded linear operator, and is defined

as a restriction of the function u to the boundary γ : Hm(Ω) → Hm−1/2(Γ),

m > 12

[27]. There is a constant CΓ(m), such that

‖uΓ‖Hm−1/2(Γ) ≤ CΓ‖u‖Hm(Ω). (3.9)

21

3.7 Riesz Representation Theorem

Any continuous linear functional F on a Hilbert space H, can be

represented uniquely in terms of the inner product, F (x) = 〈x, z〉, where z

depends on F , is uniquely determined by f , and has norm ‖z‖H = ‖F‖H′ .

In general, let H1, H2 be Hilbert spaces and, h : H1 × H2 → K a bounded

sesquilinear form. Then h has a representation (x, y) = 〈Sx, y〉, where S :

H1 → H2 is bounded linear operator, uniquely determined by h, and has norm

‖S‖ = ‖h‖.

3.8 Lax-Milgram Theorem

Given a Hilbert space (V, (·, ·)), a continuous, coercive bilinear form

a(·, ·) and a continuous linear functional F ∈ V ′, there exists a unique u ∈ V

such that a(u, v) = F (v), ∀ v ∈ V. Since the bilinear form a(·, ·) is coercive,

we have the estimate ‖u‖V ≤ 1c2‖F‖V ′ , where c2 is the coercivity constant (3.6).

3.9 Hilbert Interpolation Spaces

There are many ways in which one can define an interpolation space,

with the most common methods being the real and the complex interpolation

methods. We will describe the complex interpolation method, and we will

follow [41], for the real interpolation method [10, 61].

Here and in section 5.2, we use the notation Xa → Xb to denote the

continuous embedding of Xa in Xb (i.e. convergence in Xa strongly implies

convergence in Xb strongly), [58]. Similarly, we let Xac→ Xb denote compact

22

embedding of Xa in Xb (i.e. the embedding operator is compact). We are

interested in the interpolation theory of Hilbert spaces. Given two Hilbert

spaces X0 and X1 → X0 with inner products (·, ·)X0 , (·, ·)X1 and norms ‖ · ‖X0 ,

‖ · ‖X1 , respectively, we will define Hilbert spaces that “interpolate” between

them. The spaces Xθ = [X0, X1]θ, 0 < θ < 1 are called interpolation spaces

between X0 and X1.

Let an unbounded positive definite self-adjoint operator A exist in

X0, with dense domain D(A) = X1, such that ‖u‖X1 = ‖Au‖X0 , u ∈ D(A).

The existence of operator A follows from the Riesz representation theorem.

For 0 < θ < 1, the set

[X0, X1]θ = Xθ :=u ∈ X0 : ‖u‖[X0,X1]θ <∞

(3.10)

forms a Hilbert scale of spaces (Xθ, ‖ · ‖Xθ) given by Xθ = D(Aθ), with norm

‖u‖[X0,X1]θ = ‖Aθu‖X0 .

The following theorem is known as the Convexity Theorem, and will

be used later.

Theorem 3.1 [10] Suppose that Xi and Yi (i = 0, 1) are two pairs of Banach

spaces, and that A is a linear operator that maps Xi to Yi. Then A maps Xθ

to Yθ for 0 < θ < 1. Moreover,

‖A‖Xθ→Yθ≤ ‖A‖1−θ

X0→Y0‖A‖θX1→Y1

. (3.11)

23

4. Statement of Coupled Problem

4.1 Derivation of the Coupled Problem

In this section, we summarize some of the basic relations of linear

wave physics, starting with elastic waves, proceeding to acoustic waves, and

then fluid-solid interaction. Derivation of the coupled problem is standard, and

this section is intended for completeness only. We will derive the differential

equations and the boundary conditions that describe the fluid-solid interaction.

We are interested in the time-harmonic case, and we assume that all waves are

steady-state with circular frequency ω. An acoustic wave that is incident onto

an elastic obstacle is not totally reflected, and part of the incident energy is

transmitted in the form of elastic vibrations. Basic ideas regarding the nature

of the elastic field were established long before the application of elasticity

theory [64]. We will use the word ”field” to refer to any physical quantities

that vary in space and time. For time harmonic fields, the scalar and vector

potentials satisfy the scalar and vector Helmholtz equations.

4.1.1 Elastic Waves In an elastic medium, waves propagate in

the form of small oscillations of the stress field. In a solid elastic material,

the physical quantities that are of interest are the displacement, the stress, the

24

strain tensors and, if any are present, the body forces. The speed of propaga-

tion is usually denoted by c. The governing equations for the elastic medium

are obtained from the basic relations of continuum mechanics. Consider the

momentum equation

∂(ρv)

∂t+ ∇ · (ρvv)−∇ · τ = f.

Assuming no external forces, the momentum equation becomes

∂(ρv)

∂t+ ∇ · (ρvv)−∇ · τ = 0.

Assume that the density, ρ(x, t), and the velocity, v(x, t), undergo small fluc-

tuations about equilibrium values, ρ0 and v0, respectively. The domain as a

whole is assumed to be at rest (i.e. v0 = 0). We linearize the momentum

equation by writing

v = v + v0

ρ = ρ+ ρ0

where v, ρ represent small perturbations about the equilibrium values. The

term ρvv can be linearized as

ρvv = ρ(v + v0)(v + v0)

= ρ(vv + vv0 + v0v + v0v0) ≈ 0,

25

since vv is negligible and v0 = 0.

Similarly,

ρv = (ρ+ ρ0)(v + v0)

= ρv + ρv0 + ρ0v + ρ0v0

≈ ρ0v,

since ρv is also assumed negligible. The momentum equation then becomes

∂(ρ0v)

∂t−∇ · τ = 0, (4.1)

where we have dropped the tilde notation for simplicity. Since ρ0 is the constant

equilibrium value, equation (4.1) becomes

ρ0∂v

∂t−∇ · τ = 0. (4.2)

To linearize equation (4.2) and write it in terms of displacement we assume

small oscillations. The partial derivative, ∂∂t

, is related to the total derivative

ddt

by the nonlinear expression [43]

d

dt=

∂

∂t+ v ·∇.

If u(r, t) is the vector field of particle displacement at position r, then in order

to formulate equation (4.2) in terms of displacement, we write

v(r, t) =du

dt(r, t)

=∂u

∂t+ v ·∇u

≈ ∂u

∂t,

26

where ∇u is assumed to be of the same order as v so that v ·∇u is negligible.

So,

∂v

∂t=∂2u

∂t2, (4.3)

and we obtain the linearized form of the momentum equation in terms of

displacement

ρ0∂2u

∂t2−∇ · τ = 0. (4.4)

For wave propagation, it is generally assumed that a time-dependent scalar

field F (x, t) can be separated as

F (x, t) = f(x)e−iωt,

where f is a stationary amplitude function. A similar convention holds for

vector fields. Assuming a time harmonic solution to equation (4.4) and letting

u(x, t) = u(x)e−iωt, we obtain

ω2ρu + ∇ · τ = 0, (4.5)

where τ is the stress tensor. The symmetric stress tensor τ (x) is also known

as the Cauchy stress tensor at the point x ∈ Ωe.

With the assumption of small deformations, the strains are related to

the displacements by the linearized equations, and are defined by

e(u) =1

2(∇u + (∇u)T ). (4.6)

27

By Generalized Hooke’s Law, stress is a linear function of strain,

where the strain assumes small displacements. We then have

τ ij = Cijklekl, (4.7)

where Cijkl is a fourth order elastic stiffness tensor which, for an isotropic

medium, is invariant under rotations and reflections, so that it takes on the

form

Cijkl = λδijδkl + µδjlδjk (4.8)

where λ, µ are the Lame coefficients. Substituting equation (4.8) into (4.7) we

get

τ ij = λδijekk + 2µeij

= λδij∂kuk + µ(∂iuj + ∂jui).

Thus we obtain the equations governing the elastic medium

ω2ρu + ∇ · τ = 0

τ = λ I(∇ · u) + 2µe(u),

where τ is the stress tensor, e(u) is the strain tensor, u is displacement, ρ is

density of the elastic medium, and λ, µ are the Lame coefficients of the elastic

medium.

4.1.2 Acoustic Waves Acoustic waves (sound) are small oscil-

lations of pressure in a compressible ideal fluid (acoustic medium) [35], and are

28

associated with local motions of the particles of the fluid and not with bodily

motion of the fluid itself. The difference between fluids and elastic solids is

that fluids cannot support shear stresses in the absence of internal friction.

In viscous fluids, frictional forces are generated when gradients in velocity are

present. Here we consider inviscid media, so the fluid cannot support any

shear forces. We assume that the fluid is compressible, i.e. the density of the

fluid changes as a consequence of flow processes. The equations governing the

acoustic medium are obtained from fundamental laws for compressible fluids.

In this section, the differential equations governing acoustic wave propagation

in a liquid or gaseous medium are described following the derivation in [65].

As with the derivation of the differential equations governing the elas-

tic medium, we assume that the density, velocity, and pressure, undergo small

fluctuations about their respective mean variables: ρ0,v0, and p0. We also

assume that the fluid as a whole is at rest, i.e. v0 = 0. The simplest con-

stitutive equations encountered in continuum mechanics are those for an ideal

fluid, where the pressure field is isotropic and depends only on density and

temperature; then the stress field τ is represented by

τ = −p(ρ, T )I, (4.9)

where ρ is the density of the fluid and T is the absolute temperature.

Substituting equation (4.9) into the linearized momentum equation (4.2) we

29

obtain

ρ0∂v

∂t−∇ · (−pI) = 0. (4.10)

Taking the curl of equation (4.10), we see [65]

∇× v = 0 ⇒ v = −∇Φ, (4.11)

where Φ is a scalar field called the velocity potential. Equation (4.11) holds

because of the assumption of a nonviscous fluid. Using equation (4.11) in

equation (4.10), we obtain

p = ρ0∂Φ

∂t. (4.12)

From thermodynamics, we can write the pressure as a function of entropy and

density. We assume that the acoustic wave propagation is an adiabatic process

at constant entropy, and that the changes in density are small. Then using the

Taylor series representation, we have

p′ = p0 +

(∂p

∂ρ

)S

dρ ⇒ p =

(∂p

∂ρ

)S

dρ, (4.13)

where p′ = p0 + p. We define the adiabatic compressibility κS via(∂p

∂ρ

)S

= c2 =1

κSρ0

, (4.14)

where c is the speed of the acoustic waves and depends on material properties.

Using equation (4.13), (4.14), and (4.12) in equation (4.10), we obtain the time

dependent wave equation for the velocity potential

∇2Φ− 1

c2∂2Φ

∂t2= 0. (4.15)

30

With the assumption of time-harmonic waves of frequency ω, the wave equation

(4.15) can be transformed to the scalar Helmholtz equation

∇2Ψ− 1

c2i2ω2Ψ = 0, or

∇2Ψ + k2Ψ = 0 (4.16)

where Ψ is amplitude, i =√−1 is the imaginary unit, and

k =ω

c. (4.17)

The physical parameter k is called the wave number. The physical interpre-

tation of the parameter k is the number of waves per 2π units. Hence, k

characterizes the oscillatory behavior of the exact solution. The larger the

value of k, the greater the spacial frequency of the waves.

4.1.3 Boundary Conditions In order to set up a well-posed

problem and to model the physics, the equations need to be complemented by

boundary conditions. The most common boundary conditions imposed on a

model problem are the Dirichlet and Neumann boundary conditions. Holding

the pressure constant on the boundary is an essential, or Dirichlet, boundary

condition of the form p = p0. For a rigid surface, Neumann boundary conditions

are imposed, ∂p∂n = 0, where n is the outer unit normal to ∂Ω. This condition

models a free boundary, i.e., no external forces are acting at the boundary [58].

The physical requirement that all radiated waves are outgoing leads to

the Sommerfeld radiation condition, which can be interpreted as a boundary

condition at infinity. The Sommerfeld condition is usually replaced by its

31

approximation on an artificial boundary of a finite computational domain Ω,

is is given by

∂p

∂n+ ikp = 0.

This condition is also called a radiating boundary condition [58].

4.1.4 Solid-Fluid Interface Conditions When an elastody-

namic wave, which propagates through a homogeneous medium, encounters

a region that is characterized by different physical properties, it will experi-

ence changes in amplitude, wave numbers, and direction of propagation. We

want to impose physically motivated conditions for the description of solid-

fluid interaction. Most often, boundary conditions in elastodynamics require

the displacements and surface traction to be continuous across the boundary

region. The traction vector for an arbitrary region with unit outer normal n

may be written as

t = τ · n, (4.18)

where τ is the stress tensor as in equation (4.5). Let ue,uf , te, tf denote the

displacement and traction in the elastic and fluid medium, respectively. We are

dealing with an inviscid fluid, so we must allow slip in the direction tangential

to the interface. Then the only condition on u is the continuity across the

interface,

n · ue = n · uf . (4.19)

32

The displacement of the fluid is obtained by eliminating uf between the lin-

earized momentum equation (4.10) and equation (4.3). Assuming a time har-

monic solution, we obtain

uf =1

ρfω2∇p. (4.20)

Combining equations (4.19) and (4.20) we obtain the first interface condition

n · ue =1

ρfω2∇p · n. (4.21)

The last two interface conditions can be obtained from the the balance of forces,

namely the pressure is in static equilibrium with the traction normal to the

solid boundary,

n · te = n · tf .

From (4.9) and (4.18), we obtain

n · τ = −pn. (4.22)

Crossing both sides of equation (4.22) with n, gives

n× τ · n = 0. (4.23)

Hence, the fluid cannot support any shearing force. By dotting both sides of

equation (4.22) with n, we obtain

n · τ · n = −p (4.24)

which means that at the wet interface, the fluid pressure is in equilibrium with

the normal traction of the solid.

33

Ωf

Γn

Γn

Γd ΓaΓi Ωe - n

Figure 4.1. Problem setup

4.2 Variational Form of Coupled Problem

We consider the propagation of waves in a composite medium Ω ⊂

Rn, n = 2 or 3, which consists of a fluid part Ωf and a solid part Ωe, so that

Ω = Ωf ∪ Ωe. Denote the interface between the two media by Γ = ∂Ωf ∩ ∂Ωe,

where the boundary Γ is assumed piecewise smooth, and the outward unit

normal is denoted by n.

Our model problem is a channel with rigid walls and an elastic obsta-

cle in the middle, Dirichlet boundary condition for the incoming wave on Γd,

Neumann boundary conditions at rigid surfaces on the sides on Γn, an absorb-

ing boundary at the outgoing side Γa, and interface condition on the boundary

Γi of the scattering obstacle, (see Fig. 4.2).

Mathematically, the model is described by the coupled system of the

elastic wave equations and the acoustic wave equation, derived in Section 4.1.2.

The acoustic field in the fluid is governed by the Helmholtz equation for the

34

pressure p, (4.16),

∆p+ k2p = 0 in Ωf , (4.25)

with the boundary conditions derived in section 4.1.3,

p = p0 on Γd (excitation),

∂p

∂n= 0 on Γn (sound hard surface),

∂p

∂n+ ikp = 0 on Γa (radiation boundary condition). (4.26)

Here, ∆ is the Laplace operator, and k ∈ R is the wave number. The Som-

merfeld radiation condition prevents reflection of outgoing waves from infinity,

and the radiation boundary condition (4.26) is the approximation to the Som-

merfeld radiation condition on an artifical boundary that delimits the finite

computational domain Ω.

Using (4.5) and (4.6) for the isotropic elastic medium we have

∇ · τ + ω2ρeu = 0 in Ωe, (4.27)

where

τ = λI(∇ · u) + 2µe(u) is the stress tensor,

eij(u) = 12

(∂ui

∂xj+ ∂uj

∂xi

)is the strain tensor,

u is the displacement, ρe is the density, ω is the wave frequency, I is the n×n

identity matrix, and λ and µ are the Lame coefficients of the elastic medium.

35

The solid-fluid interface conditions given by (4.21), (4.23), and (4.24),

are [65]

n · u = 1ρf ω2

∂p∂n (continuity)

n · τ · n = −p (balance of normal forces)

n× τ · n = 0 (zero tangential tension)

on Γi. (4.28)

4.3 Weak Formulation

To derive a weak formulation of the acoustic elastodynamic model

problem, we define the spaces

Vf = (p, pΓ) |, p ∈ H1(Ωf ), pΓ ∈ H12 (Γ) | p = 0 on Γd , (4.29)

Ve = (u,n · uΓ) | u ∈ (H1(Ωe))3, uΓ ∈ (H

12 (Γ))3, (4.30)

where the restrictions pΓ = TrΓp and uΓ = TrΓu are the traces of p and u on

Γ.

Multiplying equation (4.25) by a test function (q, qΓ) ∈ Vf , equation

(4.27) by a test function (v,n · vΓ) ∈ Ve, and integrating by parts over Ω we

obtain the following variational form of (4.25), (4.27), and (4.28):

Find (p, pΓ), (p− p0, pΓ) ∈ Vf and (u,n · uΓ) ∈ Ve such that

−∫Ωf

∇p∇q + k2∫Ωf

pq − ik∫Γa

pΓqΓ −∫Γ

ρfω2(n · uΓ)qΓ = 0 (4.31)

−∫Ωe

(λ(∇ · u)(∇ · v) + 2µe(u) : e(v)) + ω2∫Ωe

ρeu · v −

∫Γ

pΓ(n · vΓ) = 0 (4.32)

∀ q, qΓ ∈ Vf and ∀ v,n · vΓ ∈ Ve,

36

where p0 on Γd is understood to be extended to a function in H1(Ωf ).

For the weak formulation of the elastodynamic equation, we use Green’s for-

mula [12] to obtain

∫Ωe

∇ · τ · v = −∫Ωe

τ : ∇v +∫

∂Ωe

n · τ · v

= −∫Ωe

τ :1

2(∇v + ∇vT ) +

∫∂Ωe

n · τ · v.

By multiplying equation (4.32) by ρfω2, the coupled fluid-elastic problem be-

comes

(v,n · vΓ, p− p0, pΓ) ∈ Ve × Vf :

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) = 〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉

∀(u,n · uΓ, p, pΓ) ∈ Ve × Vf , (4.33)

where

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) =∫Ωf

∇p∇q − k2∫Ωf

pq

+ik∫Γa

pΓqΓ + ρfω2∫Γ

(n · uΓ)qΓ

+ρfω2∫Ωe

(λ(∇ · u)(∇ · v) + 2µe(u) : e(v))

−ω4ρfρe

∫Ωe

u · v + ρfω2∫Γ

pΓ(n · vΓ),

(4.34)

and the right-hand side (f ,n · fΓ, r, rΓ) ∈ V0 is a functional defined by

〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉V0 =∫Ωf

rq + k2∫Ωe

f · v

37

+∫Γ

rΓqΓ +∫Γ

|n · fΓ||n · vΓ|,(4.35)

with f ∈ (L2(Ωe))3, r ∈ L2(Ωf ), fΓ ∈ (L2(Γ))3 and rΓ ∈ L2(Γ) given. Since λ

and µ may be very large, we use a scaling of the form u = su′ and v = sv′,

where s is a scalar, so that the leading terms of equation (4.34) differ by a

factor of k2. For the purpose of obtaining error estimates, this scaling is used

in analysis only and not in computations. For computation, we use (4.43)

below, for better covergence of multigrid. Then

a(u′,n · u′Γ, p, pΓ; v′ · v′Γ, q, qΓ) =

∫Ωf


pq

+ik∫Γa

pΓqΓ + ρfω2s∫Γ

(n · u′Γ)qΓ

+ρfω2s2

∫Ωe

(λ (∇ · u′)(∇ · v′) + 2µe(u′) : e(v′))

−ω4ρfρes2∫Ωe

u′ · v′ + ρfω2s∫Γ

pΓ(n · v′Γ). (4.36)

Using (4.17) and by dropping the ’ in equation (4.36), we obtain the variational

form

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) =∫Ωf


pq (4.37)

+ik∫Γa

pΓqΓ + c2ρfk2s∫Γ

(n · uΓ)qΓ (4.38)

+c2ρfk2s2

∫Ωe

(λ(∇ · u)(∇ · v) + 2µe(u) : e(v)) (4.39)

−c4ρeρfk4s2

∫Ωe

u · v + c2ρfk2s∫Γ

pΓ(n · vΓ). (4.40)

38

We choose s so that c2ρfs2 maxλ, 2µ = 1. Hence,

s =1

c√ρf maxλ, 2µ

. (4.41)

Replacing Vf and Ve with conforming finite element spaces, we obtain the

algebraic system −Sf + k2Mf − ikGf −ρfω2sT

−ρfω2sTt −ρfω

2s2Se + ρeρfω4s2Me

p

u

= R (4.42)

In computations we scale to unit diagonal, and s in (4.42) is of the form

s =1

ck√ρf maxλ, 2µ

. (4.43)

In (4.42), p and u are the algebraic representations of p and u, i.e., p and u are

the finite element interpolations of p and u, respectively. The matrix blocks in

(4.42) are defined by

Sf =∫Ωf

∇ph∇qh, Mf =∫Ωf

phqh, Gf =∫Γa

phqh,

Se =∫Ωe

λ(∇ · uh)(∇ · vh) + 2µe(uh) : e(vh),

Me =∫Ωe

uh · vh, T =∫Γ

ph(n · vh).

where we followed the notation of section 3.2. The right-hand side R of the

system (4.42) is defined from the zero right-hand side of the variational form

with the modification for the Dirichlet boundary condition p = p0 on Γd as

follows. Listing the degrees of freedom on Γd first and the remaining variables

39

as second, (4.42) becomes AU = R, where the solution, the right-hand side,

and the matrix are, respectively,

U =

U1

U2

, R =

R1

R2

, A =

A11 A12

A21 A22

.Here, R1 are the degrees of freedom for p0, and R2 = 0. We impose the

constraint U1 = R1 and solve for U2 from the equations in the second block,

which gives the system of equations that is actually solved, I 0

0 A22

U1

U2

=

R1

−A21R1

.This formulation is standard, [55].

4.3.1 Hilbert scale In the analysis, we will need a scale of

spaces analogous to the intermediate spaces between L2(Ω) and H1(Ω) for

a scalar problem. Let

V0 =(L2(Ωe)

)3× L2(Γ)× L2(Ωf )× L2(Γ),

be a Hilbert space equipped with the inner product

〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0 =∫Ωf

pq + k2∫Ωe

u · v

+∫Γ

pΓqΓ +∫Γ

|n · uΓ||n · vΓ|,(4.44)

where the restrictions pΓ and uΓ are the traces of p and u on Γ, respectively.

Let

V1 = Ve × Vf

40

be a Hilbert space equipped with the inner product

〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V1 =∫Ωf

∇p∇q + k2∫Ωe

(∇ · u)(∇ · v)

+k2〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0 ,(4.45)

with the associated norms denoted by ‖(u,n·uΓ, p, pΓ)‖V0 and ‖(u,n·uΓ, p, pΓ)‖V1 .

In general, define for

Vj = (u,n · uΓ, p, pΓ) ∈ (Hj(Ωe))3 ×Hj− 1

2 (Γ)×Hj(Ωf )×Hj− 12 (Γ)

|p = 0 on Γd, uΓ ∈ (H12 (Γ))3, pΓ ∈ H

12 (Γ), (4.46)

where j > 12

so traces exist. Define the scaled norm on Vj, j >12

by

‖(u,n · uΓ, p, pΓ)‖2Vj= k2|u|2(Hj(Ωe))3 + |p|2Hj(Ωf ) + k2j‖(u,n · uΓ, p, pΓ)‖2V0

,

= k2|u|2(Hj(Ωe))3 + |p|2Hj(Ωf ) + k2j+2‖u‖2(L2(Ωe))3

+k2j‖p‖2L2(Ωf ) + k2j∫Γ

|pΓ|2 + k2j∫Γ

|n · uΓ|2.(4.47)

41

5. Analysis

A boundary value problem is well posed if, for a given class of data,

the solution exists, is unique, and depends continuously on the data (i.e. it is

stable). In this chapter we will show that the bilinear form (4.40) is coercive

via the Garding inequality, and that it is continuous. We will also show that

if a solution exists, then the solution is unique. By using intermediate spaces

we will show stability of the solution.

5.1 Garding Inequality for the Coupled Problem

The bilinear form (4.40) associated with the coupled problem is not

coercive, but we will show that adding a sufficiently large constant multiplied

by the V0 inner product makes it coercive over all of V1. From (4.44) and (4.45),

the norms on V0 and V1 are

‖(u,n · uΓ, p, pΓ)‖2V0= k2‖u‖2(L2(Ωe))3 + ‖p‖2L2(Ωf ) +

∫Γ

|p|2 +∫Γ

|n · u|2

‖(u,n · uΓ, p, pΓ)‖2V1= k2‖∇u‖2(L2(Ωe))3 + ‖∇p‖2L2(Ωf )

+k2‖(u,n · uΓ, p, pΓ)‖2V0

= ‖∇p‖2L2(Ωf ) + k2‖p‖2L2(Ωf )

+k2‖∇u‖2(L2(Ωe))3 + k4‖u‖2(L2(Ωe))3

+k2∫Γ

|p|2 + k2∫Γ

|n · u|2.

42

An analogous inequality to the Garding inequality (3.7) is

Lemma 5.1 (Garding Inequality for the coupled problem) Let 0 < k0 <

∞; then there exists β <∞ and α > 0 such that

Re[a(u,n · uΓ, p, pΓ; u,n · uΓ, p, pΓ)] + βk2‖(u,n · uΓ, p, pΓ)‖2V0

≥ α‖(u,n · uΓ, p, pΓ)‖2V1(5.1)

for all (u,n · uΓ, p, pΓ) ∈ V1 and k ≥ k0. We can choose β independently of k,

and

α = min

1, β − 1,

2µCk

maxλ, 2µ, β − c2ρe

maxλ, 2µ

+2µ(Ck − 1)

k2 maxλ, 2µ, β − c

√ρf

maxλ, 2µ

. (5.2)

Proof: We have

Re [a(u,n · uΓ, p, pΓ; u,n · uΓ, p, pΓ)] + βk2‖(u,n · uΓ, p, pΓ)‖2V0

=∫Ωf

|∇p|2 − k2∫Ωf

|p|2 + Re

ck2

√ρf

maxλ, 2µ

∫Γ

(n · u)p

+k2 1

maxλ, 2µ

∫Ωe

(λ|∇ · u|2 + 2µe(u) : e(u))

−c2ρek4 1

maxλ, 2µ

∫Ωe

|u|2 + Re

ck2

√ρf

maxλ, 2µ

∫Γ

p(n · u)

+βk2

∫Ωf

|p|2 + βk4∫Ωe

|u|2 + βk2∫Γ

|p|2 + βk2∫Γ

|n · u|2. (5.3)

Using the inequality 2ab ≥ −a2 − b2, we have

Re

∫Γ

(n · u)p

≥ −1

2

∫Γ

|n · u|2 − 1

2

∫Γ

|p|2. (5.4)

43

Using (5.4) and the generalized Korn’s inequality (3.8) for the integrals over

Ωe, it follows from (5.3) that

Re [a(u,n · uΓ, p, pΓ; u,n · uΓ, p, pΓ)] + βk2‖(u, p)‖2V0

≥ ‖∇p‖2L2(Ωf ) + (β − 1)k2‖p‖2L2(Ωf ) +2µCk

maxλ, 2µk2‖∇u‖2(L2(Ωe))3

− 2µk2

maxλ, 2µ‖u‖2(L2(Ωe))3 +

(β − c2ρe

maxλ, 2µ

)k4‖u‖2(L2(Ωe))3

+

(β − c

√ρf

maxλ, 2µ

)k2

∫Γ

|p|2 +∫Γ

|n · u|2

= ‖∇p‖2L2(Ωf ) + (β − 1)k2‖p‖2L2(Ωf ) +2µCk

maxλ, 2µk2‖∇u‖2(L2(Ωe))3

+

(β − c2ρe

maxλ, 2µ+

2µ(Ck − 1)

k2 maxλ, 2µ

)k4‖u‖2(L2(Ωe))3

+

(β − c

√ρf

maxλ, 2µ

)k2

∫Γ

|p|2 +∫Γ

|n · u|2

≥ α‖(u,n · uΓ, p, pΓ)‖2V1.

The theorem follows by choosing β so that α given by (5.2) with k = k0 satisfies

α > 0.

The Garding inequality provides a lower bound on the form (4.40). We also

need an upper bound, which is the subject of the next lemma.

Lemma 5.2 The bilinear form a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) is continuous

on V1, and

|a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ)|

≤ CB‖(u,n · uΓ, p, pΓ)‖V1‖(v,n · vΓ, q, qΓ)‖V1 ,

∀ (u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ) ∈ V1,

(5.5)

44

where

CB = max

1,c

2

√ρf

maxλ, 2µ,

2µ+ λn

maxλ, 2µ,

c2ρe

maxλ, 2µ

.

Proof: For any n× n square matrix A = [aij], from the Cauchy inequality it

follows that

(trA)2 = (aii)2 = δ2

iia2ii ≤ n a2

ii, (5.6)

where repeating indices imply summation, and δij is the Kronecker symbol.

Since ∇ · u = tr(e(u)), it follows from (5.6) that

|∇ · u|2 = |tr(e(u))|2 = (δiieii)2 ≤ δ2

iie2ii ≤ δ2

iieijeij = n e(u) : e(u). (5.7)

Using (4.40), (5.7), and Schwarz’ inequality, we obtain:

|a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ)| =

∣∣∣∣∣∣∣∫Ωf


pq

+ck2

√ρf

maxλ, 2µ

∫Γ

(n · u)q +∫Γ

p(n · v)

+k2 1

maxλ, 2µ

∫Ωe

(λ(∇ · u)(∇ · v) + 2µe(u) : e(v))

− c2ρek4 1

maxλ, 2µ

∫Ωe

u · v

∣∣∣∣∣∣∣≤ ‖∇p‖L2(Ωf )‖∇q‖L2(Ωf ) + k2‖p‖L2(Ωf )‖q‖L2(Ωf )

+ck2

√ρf

maxλ, 2µ

∫Γ

(n · u)q +∫Γ

p(n · v)

+k2 2µ+ λn

maxλ, 2µ‖e(u)‖(L2(Ωe))3‖e(v)‖(L2(Ωe))3

+k4 c2ρe

maxλ, 2µ‖u‖(L2(Ωe))3‖v‖(L2(Ωe))3 .

(5.8)

45

From (4.6), (5.8), and Cauchy’s inequality, we obtain

|a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ)| ≤ ‖∇p‖L2(Ωf )‖∇q‖L2(Ωf )

+k2‖p‖L2(Ωf )‖q‖L2(Ωf )

+k2 c

2

√ρf

maxλ, 2µ

∫Γ

|n · u|2 +∫Γ

|p|2

+k2 c

2

√ρf

maxλ, 2µ

∫Γ

|q|2 +∫Γ

|n · v|2

+k2 2µ+ λn

maxλ, 2µ‖∇(u)‖(L2(Ωe))3‖∇(v)‖(L2(Ωe))3

+k4 c2ρe

maxλ, 2µ‖u‖(L2(Ωe))3‖v‖(L2(Ωe))3

≤ CB‖(u,n · uΓ, p, pΓ)‖2V1‖(v,n · vΓ, q, qΓ)‖2V1

.

Lemma 5.3 The bilinear form

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) + βk2〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0

is continuous on V1.

Proof: The proof is similar to the proof of Lemma 5.2.

5.2 Existence of Solution

To establish that the variational problem (4.33) has a solution, we

will show that the coupled fluid-elastic variational problem has at most one

solution. Then the existence of a solution will be proven by converting the

46

boundary value problem into an equivalent integral equation and invoking the

Fredholm alternative. Uniqueness of the solution will prove existence.

Consider the variational problem

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ)

+βk2〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0

= 〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉V0 ,

(5.9)

where (f ,n · fΓ, r, rΓ) ∈ V0.

Lemma 5.4 Variational problem (5.9) has a unique solution (u,n · uΓ, p, pΓ) ∈

V1.

Proof: Define F by

F : (v,n · vΓ, q, qΓ) 7→ 〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉V0 . (5.10)

Since V1 is Hilbert space, by Lemma 5.1 and Lemma 5.3 the bilinear form

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) + βk2〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0 is

coercive and continuous. Because F is a continuous linear functional, from

the Lax Milgram theorem, (Section 3.8), it follows that there exists a unique

(u,n · uΓ, p, pΓ) ∈ V1 such that

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) + βk2〈((u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0

= F (v,n · vΓ, q, qΓ) ∀(v,n · vΓ, q, qΓ) ∈ V0.

Let the linear operator G : V0 → V1 be defined by the solution of the variational

47

problem (5.9)

G : (f ,n · fΓ, r, rΓ) 7→ (u,n · uΓ, p, pΓ). (5.11)

Lemma 5.5 The linear operator G defined in (5.11) is bounded from V0 to

V1.

Proof: By Lemma 5.4, the variational form (5.9) has a unique solution

(u,n · uΓ, p, pΓ) ∈ V1. Operator G maps the right-hand side to the solution

(u,n · uΓ, p, pΓ). Thus proving that G is bounded from V0 to V1 is equivalent

to proving ‖(u,n · uΓ, p, pΓ‖V1 ≤ C‖f ,n · fΓ, r, rΓ‖V0 . From the Lax-Milgram

theorem, definition of the dual norm, and the Riesz representation theorem for

Hilbert spaces, we obtain

‖(u,n · uΓ, p, pΓ)‖V1 ≤ C‖F‖V ′1≤ C‖F‖V ′

0≤ C‖(f ,n · fΓ, r, rΓ)‖V0 .

We use the notation A → B as described in section 3.9.

Lemma 5.6 It holds that

V1c→ V0.

Proof: From the definition of the norms on V0 and V1,

‖(u,n · uΓ, p, pΓ)‖2V1= k2‖∇u‖2(L2(Ωe))3 + ‖∇p‖2L2(Ωf )

+k2‖(u,n · uΓ, p, pΓ)‖2V0

≥ C‖(u,n · uΓ, p, pΓ)‖2V0,

48

where C = k2. Hence, V1 → V0. Now we show that the embedding is compact.

We need to show that any sequence in V1 has a convergent subsequence in V0.

Let (un, (n · uΓ)n, pn, (pΓ)n) be a bounded sequence in V1. Then un is a

bounded sequence in (H1(Ωe))3, and has a convergent subsequence unk

in

(L2(Ωe))3. Similarly, since pn is a bounded sequence in H1(Ωf ), there is a

subsequence pmk of pn convergent in L2(Ωf ). From the trace theorem and

the compact embedding of Sobolev spaces [27],

Tr :H1(Ω)→ H1/2(Γ)c→ L2(Γ).

Since pmk is a bounded sequence in H1(Ωf ), there is a subsequence plk

of pmk such that the traces are convergent in L2(Γ), and Tr plk → Tr p in

L2(Γ).

By the same argument, there is a subsequence uik of ulk such that

Tr uik → Tr u in(L2(Γ)

)3.

From Cauchy-Schwartz inequality,

|n · Tr uik − n · Tr u|L2(Γ) ≤ ‖n‖L2(Γ)‖Tr uik − Tr u‖(L2(Γ))3 .

Hence, n ·Tr uik → n ·Tr u in L2(Γ). We can conclude that (uik , pik)→ (u, p)

in V0.

Lemma 5.7 The operator G : V0 → V0 is compact.

49

Proof: By Lemma 5.6, the injection I : V1 → V0 is compact. From Lemma

5.5, G : V0 → V1 is bounded. Since I : V1 → V0 is compact and G : V0 → V1 is

bounded, the operator I G : V0 → V0 is compact.

Lemma 5.8 The operator G : V1 → V1 is compact.

Proof: Consider a bounded sequence (un, (n ·uΓ)n, pn, (pΓ)n) in V1. From

Lemma 5.6, V1c→ V0, so there is a subsequence of (un, (n · uΓ)n, pn, (pΓ)n),

such that (unk, (n · uΓ)nk

, pnk, (pΓ)nk

) → (u0, (n · uΓ)0, p0, (pΓ)0) in V0.

Since G is bounded from V0 to V1, it holds that G(unk, (n·uΓ)nk

, pnk, (pΓ)nk

)→

G(u0, (n · uΓ)0, p0, (pΓ)0) in V1. Hence, G : V1 → V1 is compact.

Lemma 5.9 The variational problem (4.33) is equivalent to

(1

βk2I −G

)U =

1

βk2GF

where U = (u, p), V = (v, q), and F satisfies (5.10).

Proof: The original variational formulation was

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) = 〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉V0

or, a(U, V ) = 〈F, V 〉V0 . By adding to both sides the term βk2〈U, V 〉V0 , we find

that

a(U, V ) + βk2〈U, V 〉V0 = 〈F + βk2U, V 〉V0

50

Then by the definition of G, U = G (F + βk2U) , hence(1

βk2I −G

)U =

1

βk2GF.

Finally, since G : V0 → V1 → V0, we have 1βk2GF ∈ V0.

A relation between the eigenvalues of the bilinear form a(·, ·) and the operator

G follows.

Lemma 5.10 Let λa−βk2 6= 0. Then λa−βk2 is an eigenvalue of the bilinear

form a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ); that is

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) =

(λa − βk2)〈(u,n · uΓ, p, pΓ), (v,n · vΓ, q, qΓ)〉V0 ,

∀(v,n · vΓ, q, qΓ) ∈ V1,

for some (u,n · uΓ, p, pΓ) 6= (0, 0, 0, 0), if and only if 1λa

is an eigenvalue of G.

Proof: From Lemma 5.9 with F = (λa − βk2)U , we obtain

1

βk2U −GU =

λa − βk2

βk2GU,

which is equivalent to 1λaU = GU .

The goal of the following lemma is to show that the variational form a(·, ·) does

not have real eigenvalues.

Lemma 5.11 Let 1λa∈ R be an eigenvalue of G with associated eigenvector

(u,n · uΓ, p, pΓ) ∈ V0. Then p = 0 and ∂p∂n = 0 on Γa.

Proof: Assume 1λa

is a real eigenvalue of G. Then from Lemma 5.10 the form

a(·, ·) has eigenvalue λa − βk2 with eigenvector (u,n · uΓ, p, pΓ) ∈ V0, and

a(u,n·uΓ, p, pΓ; v,n·vΓ, q, qΓ) = (λa−βk2)〈(u,n·uΓ, p, pΓ), (v,n·vΓ, q, qΓ)〉V0 .

51

Choosing the test functions q = p and v = u, we obtain

∫Ωf

|∇p|2 + (βk2 − λa − k2)∫Ωf

|p|2 + ik∫Γa

|p|2

+ k2 1

maxλ, 2µ

∫Ωe

(λ|∇ · u|2 + 2µ|e(u)|2

)

+

(βk2 − λa − k4 c2ρe

maxλ, 2µ

) ∫Ωe

|u|2

+ ck2

√ρf

maxλ, 2µ

∫Γ

(n · u)p+∫Γ

p(n · u)

+ (βk2 − λa)

∫Γ

|p|2 +∫Γ

|n · u|2 = 0, ∀(u,n · uΓ, p, pΓ) ∈ V0.

The terms∫Γ(n ·u)p and

∫Γp(n · u) are complex conjugates of each other; there-

fore, their sum is a real number. All other terms are real numbers with the

exception of ik∫Γa

|p|2, so ik∫Γa

|p|2 = 0. Hence, p = 0 on Γa. From the radiation

boundary condition (4.26), p− ik ∂p∂n = 0, we also have that ∂p

∂n = 0 on Γa.

Lemma 5.12 Let p ∈ H1(Ωf ) and 4p+ k2p = 0 in Ωf . If p = 0 and ∂p∂n = 0

on Γa, then p = 0 in Ωf .

Proof: Since p is a solution of the elliptic equation with constant coefficients

4p+k2p = 0 in Ωf , it holds that p is analytic in Ωf , [39]. That is, at any point

x of Ωf , the Taylor series of p about x converges to p in some neighborhood

N0(x). Consider x ∈ Γa ⊂ ∂Ωf . By the Cauchy-Kowalevski theorem [39, 67],

p = 0 is a unique solution in the class of analytic functions of 4p+ k2p = 0 in

some neighborhood N0(x)∩Γa. This solution satisfies the boundary conditions

52

p = 0 and ∂p∂n = 0 on Γa ⊂ ∂Ωf . Since the neighborhood N0(x), x ∈ Ωf

is compact, there exists a finite subset N1, ..., Nm of the neighborhoods that

form an open covering of Ωf . We will use proof by contradiction to show

that p = 0 on all neighborhoods N1, ..., Nm. Assume that p = 0 on exactly

n < m neighborhoods. Since the domain Ωf is connected, at least one of the

remaining neighborhoods, Nk, has a nonempty intersection with at least one

of the n neighborhoods where p = 0. Since p is analytic and by the Cauchy-

Kowalevski theorem, p = 0 on Nk. Hence p = 0 on n+1 of the neighborhoods,

which is a contradiction.

Consider (u,n · uΓ) ∈ Ve, u the displacement vector and uΓ the trace of u.

Then from the solid-fluid interface conditions (4.28), it follows that on Γi,

n · u = 0 (5.12)

n · τ · n = 0 (5.13)

n× τ · n = 0 (5.14)

Equations (5.13) and (5.14) are equivalent to

τ · n = 0. (5.15)

Assumption 5.13 If u is solution to the elastodynamic equation (4.27) with

the interface conditions (5.12) and (5.15) on ∂Ωe, then u = 0 in Ωe.

The elastodynamic equation (4.27) along with boundary condition (5.15),

∇ · τ + ω2ρeu = 0 in Ωe,

τ · n = 0 on Γ, (5.16)

53

does not necessarily imply that u vanishes in Ωe. It is known [47] that, for

certain geometries and frequencies, there are nontrivial solutions to this prob-

lem. This is equivalent to the assumption that there are no non-radiating

modes [33, 47]. The eigenvalue problem (5.16) has by the Fredholm Alter-

native countably many eigenvalues ρeω2, and its only accumulation point is

infinity. If ω2ρe is not an eigenvalue of (5.16), then we can conclude that u = 0

on ∂Ωe. However, if ω2ρe is an eigenvalue of (5.16), then the question is if

there exist eigenfunctions u 6= 0 such that u ·n = 0. These eigenfunctions are

called non-radiating modes. It has been shown [47] that there are bodies such

that non-radiating modes exist. These bodies must have certain symmetries.

An example of such a body would be a sphere. Another example is a shear

stationary wave in half space for which the amplitude is orthogonal to both

the direction of propagation and to the normal, n. It has been shown [33] that

arbitrary elastic bodies have no non-radiating modes.

Lemma 5.14 The operator G has no real eigenvalues.

Proof: If λa−βk2 ∈ R is an eigenvalue of a(·, ·), by Lemma 5.12 and Lemma

5.13 it follows that the eigenvector associated with λa − βk2 must be the zero

vector, which is not possible. Therefore, by Lemma 5.10 1λa

is not an eigenvalue

of G.

We are now ready to prove the main result of this section.

Theorem 5.15 The variational problem (4.33) has a unique solution for every

given right-hand side F ∈ V ′0 .

54

Proof: From Lemma 5.14, 1λa6= 0 is not an eigenvalue of G. The operator

G is compact. Then, ( 1λaI −G) satisfies the Fredholm alternative. If the

homogeneous problem

4p+ k2p = 0 in Ωf

p = 0 on Γd

∂p

∂n= 0 on Γa,

∇ · τ + ω2ρeu = 0 in Ωe,

τ · n = 0 on Γ,

has only the trivial solution, then from section 3.5, part (I) holds, and ( 1λaI −

G)U = F has a unique solution U for every given right-hand side F ∈ V ′0 .

We have a standard stability estimate.

Theorem 5.16 Let k be fixed, (u,n · uΓ, p, pΓ) ∈ V1, and

a(u,n · uΓ, p, pΓ; v,n · vΓ, q, qΓ) = 〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉V0 (5.17)

∀ (v,n · vΓ, q, qΓ) ∈ V1.

Then

‖(u,n · uΓ, p, pΓ)‖V1 ≤ CR‖(f ,n · fΓ, r, rΓ)‖V0 , (5.18)

holds for all (f ,n · fΓ, r, rΓ) ∈ V0, with

CR = ‖(I − βk2G)−1‖V1→V1‖G‖V0→V1 (5.19)

depending on k and other problem data.

55

Proof: By definition of G, we have

(u,n · uΓ, p, pΓ) = G((f ,n · fΓ, r, rΓ) + βk2(u,n · uΓ, p, pΓ)).

Hence, by linearity of G

(u,n · uΓ, p, pΓ) =(I − βk2G

)−1G(f ,n · fΓ, r, rΓ).

Consequently,

‖(u,n · uΓ, p, pΓ)‖V1

≤ ‖(I − βk2G)−1‖V1→V1‖G‖V0→V1‖(f ,n · fΓ, r, rΓ)‖V0 .

(5.20)

From Lemma 5.8, G is compact from V1 to V1. By Lemma 5.14, operator G

has no real eigenvalues, so 1βk2 is not an eigenvalue of G. From the Fredholm

Alternative, it follows that 1βk2 I−G is bounded from V1 onto V1. From the open

mapping theorem [42],(

1βk2 I −G

)−1is bounded from V1 to V1. Therefore,

‖(I − βk2G)−1‖V1→V1 <∞. (5.21)

From Lemma 5.5 operator G is bounded from V0 to V1, and using equation

(5.21) in (5.20) we get the desired result.

5.3 Intermediate Spaces

In this section we introduce intermediate spaces and norms, which are necessary

for the stability arguments in the next section.

56

5.3.1 Intermediate norms For 0 < j ≤ 1, define the seminorm

|(u,n · uΓ, p, pΓ)|2Vj= k2|u|2(Hj(Ωe))3 + |p|2Hj(Ωf ). (5.22)

Recall the intermediate norms from (4.47),

‖(u,n · uΓ, p, pΓ)‖2Vj= |(u,n · uΓ, p, pΓ)|2Vj

+ k2j‖(u,n · uΓ, p, pΓ)‖2V0,

= k2|u|2(Hj(Ωe))3 + |p|2Hj(Ωf ) + k2j+2‖u‖2(L2(Ωe))3

+k2j‖p‖2L2(Ωf ) + k2j∫Γ

|p|2 + k2j∫Γ

|n · u|2. (5.23)

We show that, at least in a certain range of j, the spaces Vj form a Hilbert

scale. The constants we obtain are of great importance, since we want uniform

equivalence for k →∞. We need the following inequality

Lemma 5.17 For 0 ≤ θ ≤ 1, k ≥ 1 and λ > 0,

2θ−1(k2θ + λ2θ) ≤ (k2 + λ2)θ ≤ (k2θ + λ2θ). (5.24)

Proof: Function f(x) = (1+x)θ

(1+xθ)is a differentiable function. Analyzing its first

derivative, fx = θ(1+x)θ−1(1−xθ−1)(1+xθ)2

, with the constraint 0 ≤ θ ≤ 1 and x ≥ 0, we

find that x = 1 is an absolute minimum, so f(x) > f(1) for all x, and x = 0 is

a maximum, so f(x) < f(0) for all x. This means that 2θ−1 ≤ (1+x)θ

(1+xθ)≤ 1 for

all 0 ≤ θ ≤ 1 and all x ≥ 0, which is equivalent to inequality (5.24) for x = k2

λ2 .

Lemma 5.18 Let Xθ, 0 ≤ θ ≤ 1, be a Hilbert scale. Define the scaled norms

on Xθ by

‖u‖2Xθ,k = ‖u‖2Xθ+ k2θ‖u‖2X0

. (5.25)

57

Then

[(X0, ‖ · ‖X0,k) , (X1, ‖ · ‖X1,k)]θ = (Xθ, ‖ · ‖Xθ,k) , (5.26)

with equivalent norms and the constant of equivalence independent of k ≥ 1:

‖u‖Xθ,k ≥ ‖u‖[(X0,‖·‖X0,k),(X1,‖·‖X1,k)]θ

≥ 21−θ‖u‖Xθ,k

Proof: Let A be the positive definite, self-adjoint, unbounded operator from

X0 to X0, with domain D(A) = X1 such that ‖u‖X1 = ‖Au‖X0 , as presented

in section (3.9), [41]. From the spectral theorem [42], there exists a spectral

decomposition of operator

A =

∞∫0

λdEλ.

Then by definition, we have

‖u‖2X0=

∞∫0

d(Eλu, u),

‖u‖2X1= (A2u, u)X0 =

∞∫0

λ2d(Eλu, u), (5.27)

‖u‖2Xθ= ‖Aθu‖2X0

= (A2θu, u)X0 =

∞∫0

λ2θd(Eλu, u). (5.28)

From 5.25, we obtain

‖u‖2Xθ,k = ‖Aθu‖2X0+ k2θ‖u‖2X0,k

= (A2θu, u)X0 + k2θ(u, u)X0

= ((A2θ + k2θI)u, u)X0 (5.29)

=

∞∫0

(λ2θ + k2θ)d(Eλu, u).

58

The interpolated norm is

‖u‖2[(X0,‖·‖X0,k),(X1,‖·‖X1,k)]θ

=((A2 + k2I)θu, u

)X0

(5.30)

=

∞∫0

(λ2 + k2)θd(Eλu, u).

From (5.30) and (5.29) we only need to prove that the norms are equivalent as

presented in section 3.9, i.e.

((A2 + k2I)θu, u

)≈((A2θ + k2θ)u, u

). (5.31)

Using the inequality (5.24) in (5.29), it follows that

∞∫0

21−θ(k2θ + λ2θ)d(Eλu, u) ≤∞∫0

(λ2 + k2)θd(Eλu, u)

≤∞∫0

(k2θ + λ2θ)d(Eλu, u),

which is equivalent with (5.31).

Lemma 5.19 If Xθ, Yθ, 0 ≤ θ ≤ 1 are two Hilbert scales of spaces, then

[X0 × Y0, X1 × Y1]θ = Xθ × Yθ (5.32)

and

‖(u, v)‖2Xθ×Yθ= ‖u‖2Xθ

+ ‖v‖2Yθ. (5.33)

Proof: ‖u‖2Xθ= ‖Aθu‖2X0

and ‖v‖2Yθ= ‖Bθu‖2Y0

, where Xθ = D(Aθ) and

Yθ = D(Bθ). Now define the operator C such that C : (u, v) 7→ (Au,Bv).

59

Then Cθ : (u, v) 7→ (Aθu,Bθv) and

‖(u, v)‖2Cθ =((Aθu,Bθv), (u, v)

)= (Aθu, u) + (Bθv, v)

= ‖u‖2Xθ+ ‖v‖2Yθ

= ‖(u, v)‖2Xθ×Yθ

We need the following property of interpolation of a subspace, which is known

from the theory of categories [61]. In the theory of categories, an interpolation

method is called a functor, if it maps the pair of spaces (X0, X1) to an inter-

polated space Xθ. We modify and extend category results to get norms with

explicit equivalence constants.

Lemma 5.20 Consider an interpolation functor (X0, X1) 7→ Xθ such that

(3.11) holds. Let (X0, X1) be an interpolation pair, X0 and X1 Banach spaces,

Y ⊂ X0 ∩X1, and suppose there is a linear map T : X0 → X0, T : X1 → X1,

such that Range(T ) = Y and T |Y = I. Denote by Y ∩Xθ the space Y equipped

with the Xθ norm. Then for 0 ≤ θ ≤ 1 and x ∈ Y ,

‖x‖Y ∩Xθ≤ ‖x‖[Y ∩X0,Y ∩X1]θ ≤ ‖T‖

1−θX0→X0

‖T‖θX1→X1‖x‖Y ∩Xθ

(5.34)

In addition, Y ∩Xθ is a closed subspace of Xθ.

Proof: We proceed as in [6] but use norm arguments instead of the theory of

categories. Interpolating the operator I : Y ∩X0 → X0 and I : Y ∩X1 → X1,

the convexity property (3.11) gives the left-hand side inequality in (5.34). The

right-hand side inequality also follows from (3.11) by interpolating the map

T : X0 → Y ∩X0, T : X1 → Y ∩X1. Finally, Y ∩Xθ = x ∈ Xθ|(I − T )x = 0

60

and T : Xθ → Xθ is continuous by (3.11); hence, Y ∩Xθ is a closed subspace

of Xθ.

Lemma 5.21 Let 12< a < b < +∞. Then the spaces Vj, a ≤ j ≤ b, form

a Hilbert scale. Define the scaled norms on Vj by ‖(u,n · uΓ, p, pΓ)‖Vjas in

(4.47). Then [(Va, ‖ · ‖Va), (Vb, ‖ · ‖Vb)] = (Vj, ‖ ·‖Vj

) with equivalent norms and

the constant of equivalence independent of k > 1, and

‖(u,n · uΓ, p, pΓ)‖Vj≤ ‖(u,n · uΓ, p, pΓ)‖[Va,Vb]j

≤ CI‖(u,n · uΓ, p, pΓ)‖Vj. (5.35)

Proof: It is known that Sobolev spaces Hj form a Hilbert scale, [61]. Define

the spaces

Wj = (Hj(Ωe))3 × L2(Γ)×Hj(Ωf )× L2(Γ)

equipped with the norm

‖(u,uΓ, p, pΓ)‖2Wj= k2|u|2(Hj(Ωe))3 + |p|2Hj(Ωf )

+k2j+2‖u‖2(L2(Ωe))3 + k2j‖p‖2L2(Ωf ).

+k2j‖uΓ‖2L2(Γ) + k2j‖pΓ‖2L2(Γ).

The space Vj equals ImT , where T : Wj → Wj, T (u,n · uΓ, p, pΓ) = (u,Tr u ·

n, p,Tr p). Define scales of norms

‖u‖(Hj(Ωe))3 = k2|u|2(Hj(Ωe))3 + k2j+2‖u‖2(L2(Ωe))3 , and

‖p‖Hj(Ωf ) = |p|2Hj(Ωf ) + k2j‖p‖2L2(Ωf ).

61

Let m ≥ 1. From Lemma 5.18 it follows that[((L2(Ωe))

3, ‖ · ‖(L2(Ωe))3,k

),((Hm(Ωe))

3, ‖ · ‖(Hm(Ωe))3,k

)]j/m

=((Hj(Ωe))

3, ‖ · ‖(Hj(Ωe))3,k

), and[(

L2(Ωf ), ‖ · ‖L2(Ωf ),k

),(Hm(Ωf ), ‖ · ‖Hm(Ωf ),k

)]j/m

=(Hj(Ωf ), ‖ · ‖Hj(Ωf ),k

),

with equivalent norms and the constant of equivalence independent of k ≥ 0.

Since the spaces Wj are a cross product of the above spaces, it follows from

Lemma 5.19, that the spaces Wj form a Hilbert scale. Let a and b be fixed,

12< a < b. From the trace and Cauchy inequalities

‖n · uΓ‖2L2(Γ) ≤ ‖n · uΓ‖2Ha− 1

2 (Γ)≤ C1(a,Γ)‖u‖2(Ha(Ωe))3 ,

‖pΓ‖2L2(Γ) ≤ ‖pΓ‖2Ha− 1

2 (Γ)≤ C2(a,Γ)‖p‖Ha(Ωf ),

and from the definition of the norm in Wj, it follows that

‖T‖Wa→Wa ≤ C1(a,Γ), and ‖T‖Wb→Wb≤ C2(a,Γ). (5.36)

Hence, by equality of Vj and ImT we have

‖(u, p)‖2Vj= ‖(u,Tr u · np,Tr p)‖Im T∩Wj

. (5.37)

From Lemma 5.20 for 0 ≤ θ ≤ 1, it follows for j = a+ θ(b− a) that

‖(u,Tr u · n, p,Tr p)‖Im T∩Wj≤ ‖(u,Tr u · n, p,Tr p)‖[Im T∩Wa,Im T∩Wb]j

≤

‖T‖1−θWa→Wa

‖T‖θWb→Wb‖(u,Tr u · n, p,Tr p)‖Im T∩Wj

.

The space Wa is isometric with ImT and the space Wb is isometric with ImT ,

and by (5.36) and (5.37) we obtain

‖(u,n · uΓ, p, pΓ)‖Vj≤ ‖(u,n · uΓ, p, pΓ)‖[Va,Vb]j

≤

62

‖C1(a,Γ)‖1−θ‖C2(a,Γ)‖θ‖(u,n · uΓ, p, pΓ)‖Vj,

where the equivalence constants depend on a and Γ but not on k.

5.3.2 Regularity of Solution for Coupled Problem. Regu-

larity results for the Helmholtz equation and elasticity are well known. We get

regularity results for the coupled problem by using known regularity for the

fluid domain Ωf and the elastic obstacle domain Ωe.

Lemma 5.22 Let domains Ωf and Ωe be Lipschitz. The solution of the cou-

pled problem

∆p = r − k2p in Ωf ,

γp = pd on Γd

∂pΓ

∂n= 0 on Γn

∂pΓ

∂n= −ikp on Γa

∇ · τ = f − ω2ρeu in Ωe,

n · uΓ =1

ρfω2

∂pΓ

∂non Γi (5.38)

n · τ Γ · n = −pΓ on Γi (5.39)

n× τ Γ · n = 0 on Γi

exists, where p on Γd is given in H12+α, 1

2< α < 1. Then (u,n ·uΓ, p, pΓ) is in

V1+α space with α > 12.

Proof: Let u ∈ (H1(Ωe))3 be solution to the elasticity equation with f ∈

(L2(Ωe))3. From the trace theorem, (section 3.6), uΓ ∈ (H

12 (Γ))3, so n · uΓ ∈

63

H12 (Γ). From the wet interface condition (5.38), we observe that pn = ∂pΓ

∂n ∈

H12 (Γ). Since in general, H

12 ⊂ H− 1

2+α1 for 1

2< α1 < 1, then pn ∈ H− 1

2+α1(Γ).

It is known [27, 54, 30] that the solution of ∆p+k2p = r with non-homogeneous

boundary conditions pd ∈ H12+α1(Γ), pn ∈ H− 1

2+α1(Γ), and r ∈ (L2(Ωe))

3, ex-

ists and belongs to H1+α1(Ωf ), for constant 12< α1 < 1. The following bound

on the norm of the solution holds [27],

‖p‖H1+α1 (Ωf ) ≤ C‖r‖H−1+α1 (Ωf ) + C‖pd‖H

12+α1 (Γd)

+ C‖pn‖H− 1

2+α1 (Γn), (5.40)

where pd and pn are the Dirichlet and Neumann boundary conditions. For

the purpose of this presentation constant C is used to denote a general con-

stant that is not the same for all inequalities. Let p ∈ H1(Ωf ) be the so-

lution to the Helmholtz equation with right-hand side r ∈ L2(Ωf ); then by

the trace theorem pΓ ∈ H12 (Γ). From the wet interface condition (5.39),

we then observe that n · τ Γ · n ∈ H12 (Γ). There exists 1

2< α2 < 1, so that

n · τ Γ · n ∈ H− 12+α2(Γ) ⊂ H

12 (Γ). We know [28, 29] that the solution of the

non-homogeneous elasticity operator with constant coefficients with right-hand

side f ∈ (L2(Ωe))3 and normal traction boundary condition n · τ Γ · n ∈

H− 12+α2(Γ), is regular, and belongs to (H1+α2(Ωe))

3 [30, 54]. There exists

a constant C such that

‖u‖(H1+α2 (Ωe))3 ≤ C‖f‖(H−1+α2 (Ωe))3 + C‖uΓ‖(H− 1

2+α2 (Γ))3. (5.41)

Because p ∈ H1+α1(Ωf ), by the trace theorem, pΓ ∈ H12+α1(Γ). Since u ∈

(H1+α2(Ωe))3, then uΓ ∈ (H

12+α2(Γ))3 and uΓ · n ∈ H

12+α2(Γ). Then we

64

conclude that (u,n · uΓ, p, pΓ) ∈ V1+α, where α = minα1, α2.

We need to assume a stronger property, namely the following bound on the

V1+α norm of the solution of the coupled problem.

Assumption 5.23 Assume that there exists a constant CWR such that

|u,n · uΓ, p, pΓ|V1+α ≤ CWR‖f ,n · fΓ, r, rΓ‖V0 . (5.42)

5.4 Discretization and Error Bound

In Theorem 5.15, we proved that the variational problem (4.33) has a

unique solution (u,n ·uΓ, p, pΓ) for every given right-hand side. In this section,

we show that the discrete solution (uh,n · uΓh, ph, pΓh) is also unique. Let Vh

be a finite dimensional subspace of V1 as introduced in section (3.2), and define

the Galerkin approximation (uh,n · uΓh, ph, pΓh) ∈ Vh to (u,n · uΓ, p, pΓ) as

the solution of

a(uh,n · uΓh, ph, pΓh; v,n · vΓ, q, qΓ) =

〈(f ,n · fΓ, r, rΓ), (v,n · vΓ, q, qΓ)〉 (5.43)

∀ (v,n · vΓ, q, qΓ) ∈ Vh ⊂ V1.

Let Ie,h u,n · uΓ ∈ Veh ⊂ Ve and If,h p, pΓ ∈ Vfh ⊂ Vf denote the linear

interpolant of u, and p, respectively [41, 10]. The next lemma gives a bound

on the interpolation error.

65

Lemma 5.24 Let j > 12. If the linear interpolation satisfies

‖u− Ie,hu‖(H1(Ωe))3 ≤ C1hj|u|(Hj+1(Ωe))3 (5.44)

‖p− If,hp‖H1(Ωf ) ≤ C2hj|p|Hj+1(Ωf ), (5.45)

then,

‖(u,n · uΓ, p, pΓ)− Ih(u,n · uΓ, p, pΓ)‖V1 ≤ CAhj|(u,n · uΓ, p, pΓ)|Vj+1

∀(u,n · uΓ, p, pΓ) ∈ Vj,(5.46)

where CA = maxC21(max1, k2+CΓµ(Γ)), C2

2(max1, k2+k2CΓ). Here CΓ

is the constant from the trace inequality (3.9), and µ(Γ) is the measure of Γ.

Proof: Let Ih(u,n ·uΓ, p, pΓ) be the linear interpolant of (u,n ·uΓ, p, pΓ) on

a mesh with mesh size h. From the trace inequality (3.9) and the assumptions

(5.44) and (5.45), we have

‖(u,n · uΓ, p, pΓ) − Ih(u,n · uΓ, p, pΓ)‖2V1=

= k2‖p− If,hp‖2L2(Ωf ) + ‖∇(p− If,hp)‖2L2(Ωf )

+ k4‖u− Ie,hu‖2(L2(Ωe))3 + k2‖∇(u− Ie,hu)‖2(L2(Ωe))3

+ k2∫Γ

|pΓ − If,hpΓ|2 + k2∫Γ

|n · (uΓ − Ie,huΓ)|2

≤ max1, k2 ‖p− If,hp‖2H1(Ωf )

+ k2 max1, k2 ‖(u− Ie,hu)‖2(H1(Ωe))3

+ k2CΓ|pΓ − If,hpΓ|2H1(Ωf ) + k2CΓµ(Γ)|u− Ie,hu|2(H1(Ωe))3

66

≤ k2(max1, k2+ CΓµ(Γ))‖u− Ie,hu‖2(H1(Ωe))3

+ (max1, k2+ k2CΓ)‖p− If,hp‖2H1(Ωf )

≤ C21h

2jk2(max1, k2+ CΓµ(Γ))|u|2(Hj+1(Ωe))3

+ C22h

2j(max1, k2+ k2CΓ)|p|2Hj+1(Ωf )

≤ CAh2j(k2|u|2(Hj+1(Ωe))3 + |p|2Hj+1(Ωf )

)≤ CAh

2j|(u,n · uΓ, p, pΓ)|2Vj+1.

To simplify notation we denote by U = (u,n · uΓ, p, pΓ), V = (v,n · vΓ, q, qΓ),

Uh = (uh,n · uΓh, ph, pΓh), Vh = (vh,n · vΓh, qh, qΓh), W = (w,n ·wΓ, y, yΓ),

and F = (f ,n · fΓ, r, rΓ). We need to assume a bound for the solution of the

adjoint problem, similar to (5.42).

Assumption 5.25 Assume that the adjoint problem

a(V ;W) = 〈(U − Uh),V〉V0 ∀V ∈ V1, (5.47)

has a unique solution W ∈ Vj+1 for every V ∈ V1, and that there exists a

constant CWR and j > 12

such that

|W|Vj+1≤ CWR‖(U − Uh)‖V0 . (5.48)

We have proved that (5.1), (5.5), (5.18), (5.42), and (5.46) hold. We are

now ready to prove the main result of this chapter. In the following, α is

the constant from the Garding inequality (5.1), CB is the constant from the

continuity inequality (5.5), CA is the constant from the linear interpolation

67

lemma 5.24, and CWR is the constant from the regularity estimate (5.42).

They depend on k as follows:

CB = max

1,c

2

√ρf

maxλ, 2µ,

2µ+ λn

maxλ, 2µ,

c2ρe

maxλ, 2µ

= O(1),

α = min

1, β − 1,

2µCk

maxλ, 2µ, β − c2ρe

maxλ, 2µ

+2µ(Ck − 1)

k2 maxλ, 2µ, β − c

√ρf

maxλ, 2µ

= O(1) as k →∞,

CA = maxC21(max1, k2+ CΓµ(Γ)), C2

2(max1, k2+ k2CΓ) ≈ k2,

where β is from Lemma 5.1, independent of k as k becomes unbounded, CΓ is

the constant from trace inequality (3.9), and µ(Γ) is the measure of Γ.

Theorem 5.26 Let (5.1), (5.5), (5.18), (5.42), (5.46), and Assumption 5.25

hold. If

khj ≤ h0 = (α/2β)1/2/CBCACWR, (5.49)

then there is a unique solution to

a(Uh;Vh) = 〈F ,Vh〉V0 , ∀Vh ∈ Vh, (5.50)

and the solution satisfies

‖(U − Uh)‖V1 ≤ C0 infVh∈Vh‖(U − Vh)‖V1 , (5.51)

where we may take C0 = 2CB/α, and

‖(U − Uh)‖V0 ≤ CBCACWRhj‖(U − Uh)‖V1 ,∀ j > 1

2. (5.52)

68

Proof: The proof follows a standard argument [10]. We proved that the

variational problem (4.33) has a unique solution U for every given right-hand

side F . First, assume that the discrete solution Uh exists. Since Vh ⊂ V1, we

have the standard orthogonality condition obtained by subtracting (5.17) from

(5.43),

a(U − Uh;Vh) = 0 ∀Vh ∈ Vh. (5.53)

From the Garding inequality (Lemma 5.1) and from (5.53), it follows that for

any Vh ∈ Vh,

α‖(U − Uh)‖2V1≤ a(U − Uh;U − Uh) + βk2‖(U − Uh)‖2V0

= a(U − Uh;U − Uh) + a(U − Uh;Uh − Vh) + βk2‖(U − Uh)‖2V0

= a(U − Uh;U − Vh) + βk2‖(U − Uh)‖2V0. (5.54)

Using Lemma 5.2 in inequality (5.54), we obtain

α‖(U − Uh)‖2V1≤ CB‖(U − Uh)‖V1‖(U − Vh)‖V1 + βk2‖(U − Uh)‖2V0

. (5.55)

Then, for any Wh ∈ Vh, from (5.53) and Lemma 5.2, applied to the adjoint

problem (5.47), we obtain

‖(U − Uh)‖2V0= 〈(U − Uh), (U − Uh)〉V0

= a(U − Uh;W) = a(U − Uh;W −Wh)

≤ CB‖(U − Uh)‖V1‖(W −Wh)‖V1 . (5.56)

From Lemma 5.24, applied to the adjoint problem

‖(W −Wh)‖V1 ≤ CAhj|W|Vj+1

∀ j > 1

2. (5.57)

69

From Assumption 5.25, substituting (5.48) into (5.57) and using this result in

(5.56), we obtain

‖(U − Uh)‖V0 ≤ CBCACWRhj‖(U − Uh)‖V1 . (5.58)

Hence substituting (5.58) in (5.55) we obtain,

α‖(U − Uh)‖V1 ≤ CB‖(U − Vh)‖V1 + (khj)2β(CBCACWR)2‖(U − Uh)‖V1 .

From (5.49), we find

α‖(U − Uh)‖V1 ≤ 2CB‖(U − Vh)‖V1 ∀Vh ∈ Vh,

and we obtain the desired result. So far, we have assumed the existence of a

solution Uh. By Theorem 5.15, existence and uniqueness are equivalent. If the

solution is not unique, there is a nontrivial solution, Uh, for the right-hand side

F = 0. By (5.18), we have U = 0. However, inequality (5.51) then implies

that Uh = 0, provided that h is sufficiently small. Therefore, we conclude that

(5.50) has a unique solution for h sufficiently small.

Theorem 5.27 From Theorem 5.26, there is a unique solution to (5.50) and

the solution satisfies (5.51). If in addition U ∈ V1+j, j >12, then

‖U − Uh‖V0 ≤ CBCACWRhj‖U − Uh‖V1 ≤ Ch2j‖U‖V1+j

.

Proof: The result follows immediately by substituting (5.46) into (5.52).

70

6. Multigrid Method

Multigrid or multilevel algorithms are very fast solvers used in numer-

ical analysis, physics, dynamics, and computing. In this chapter, we provide

a brief description of multigrid, defining terms, and providing comments on

the structure of multigrid with the goal of setting the stage for multigrid for

coupled fluid-solid scattering. Multigrid has been an active area of research for

almost 30 years, and much literature can be found on the subject.

For the multigrid method, a number of different grids are used on

the domain ranging from coarse to fine. Iterations on different grids reduce

different components of the error: low-frequency errors are damped on coarse

grids, and high-frequency errors are damped on fine grids. An approximate

numerical solution is computed on a coarse grid, which is used as a starting

point for an iterative method on a finer grid. The method alternates between

smoothers and coarse grid corrections, and the recursive combination of the

two results in the multigrid method. A more detailed description of multigrid

methods can be found in [11, 51].

Using multigrid methods for solving the Helmholtz equation modeling

acoustic scattering is classical [9, 32, 44, 25, 26], but it appears that multigrid

methods for the coupled acoustic-elastic problem have not been investigated.

71

In this chapter we formulate such method.

6.1 Multigrid for the Coupled Problem

In this section, we present the multigrid algorithm developed to solve

the coupled problem (4.42). This section provides an overview of the imple-

mented multigrid algorithm. The setting follows mostly Briggs [11] and Mandel

et. al. [51].

Discretization by finite elements of (4.42), leads to a linear system of

equations Au = f , where the coefficient matrix A is complex-symmetric, i.e.,

not Hermitian. For large values of the wave number k, the system of equations

becomes highly indefinite. This indefiniteness has prevented multigrid methods

from being applied to the discrete Helmholtz equation with the same success

as for symmetric-positive definite problems.

Denote by u the exact solution of the algebraic system, and by v the

approximation to the exact solution; then the algebraic error is e = u− v. By

rewriting the original problem, we obtain the residual equation Ae = r, where

r represents residual given by r = f − Av. Let Alvl = fl denote the system of

equations on grid Ωl. Smoothers are simple and inexpensive iterative methods

that reduce the high energy components of the error. The low frequency error

is reduced by projecting the solution onto a smaller space. The grid transfer

operators, the restrictions R and prolongations P , are standard and based on

the finite element space. We will present a variational multigrid method, i.e.

72

the restriction operator is the transpose of the prolongation, and the size of the

prolongation operator P ll+1 from coarse level l+1 to fine level l is computed from

the size of xe, ye, the number of elements in x and y directions, respectively. All

nodes have three degrees of freedom, pressure p, displacement in x direction,

ux, and displacement in y direction, uy. A node is active, if it has nonzero

row or column contributions. For the fluid nodes only p is active, for the

displacement nodes, ux is active in the displacement in the x direction block,

and uy is active in the displacement in the y direction block. Prolongation P

for the coupled problem for level l has size (3 · (1 + xe/2l−1) · (1 + ye/2l−1) by

3 · (1 + xe/2l) · (1 + ye/2l)). A temporary matrix Ptemp is assembled by using

the following bilinear interpolation by averaging neighboring values according

to the stencil,

14

12

14

12

1 12

14

12

14

, (6.1)

which transports the correction obtained on the coarser grid l + 1 to the

fine grid l. We compute the vectors of all odd indices, vx = (1, 3, ..., xn),

vy = (1, 3, ..., yn), and then use the Kronecker tensor product to create a ma-

trix by taking all possible products between the elements of vx and those of vy,

to obtain a large matrix that reproduces all coarse nodes. We use piecewise

bilinear interpolation inside each of the three fields for pressure and displace-

ment separately. The elastic obstacle is rectangular in shape in the middle of

73

fluid, and for a coarse node at a straight part of the wet interface we use the

stencil

14

12

0

12

1 0

14

12

0

, (6.2)

to average neighboring values. For a coarse corner node of the wet interface,

we use

14

12

0

12

1 0

0 0 0

, (6.3)

and for an inside corner of the wet interface we use the stencil

14

12

14

12

1 12

14

12

0

. (6.4)

To obtain the matrix Pfluid, we zero out the entries in Ptemp that correspond

to the obstacle, while keeping contributions from the wet interface. To obtain

matrix Psolid we zero out the entries in Ptemp that correspond to the fluid, and

leave the wet interface contributions unchanged. Final prolongation matix P

has the form

P =

Pfluid 0 0

0 Psolid 0

0 0 Psolid

, (6.5)

74

where 0 represents the zero matrix of size ((1 + xe/2l−1) · (1 + ye/2l−1) by

(1 + xe/2l) · (1 + ye/2l)). The interface is always on the mesh boundary of

the coarse mesh. Given the finest level matrix A0 from the finite element

application, the coarse matrices are created variationally from the recurrence

Al+1 ← (P ll+1)

TAlPll+1. One iteration of the standard V-cycle multigrid algo-

rithm x ← MG(x, b), solving A0x = b, can be described in abstract terms as

follows. Set MG = MGl, where l = 1, ..., L− 1

Algorithm 6.1

Pre-smoothing: Perform m1 smoothing steps on Alul = fl

Coarse grid correction

• compute residual rl = fl − Alul

• restrict fl = (P ll+1)

T rl

• if l+1 = L then solve coarse grid equation by a direct method, else perform

mc iterations of ul+1 ←MGl+1(0, fl+1)

• interpolate vl = Plvl+1

• correct the solution on level l by ul ← ul + vl

Post-smoothing: Perform m2 smoothing steps on Alul = fl

Here, m1 and m2 represent the number of pre-smoothing and post smoothing

steps per multigrid level, and mc is the multigrid cycle parameter.

We observe that the key to satisfactory multigrid performance is the

design and choice of suitable smoothing, which is the central component of

a multigrid algorithm. We test various smoothers on a squared diagonally

75

preconditioned system. A brief description of the iterative methods used as

smoothers follows.

6.1.1 GMRES. Generalized Minimum Residual method is a pop-

ular Krylov method for symmetric and unsymmetric problems [1]. GMRES is

a projection method that approximates the solution of a system of linear equa-

tions A · u = f for u, by minimizing the residual norm ‖f − Au‖ over the

m-th Krylov subspace x0 + spanr, Ar, ..., Am−1r [7, 56, 57]. The approxi-

mation can be obtained as xm = x0 + Vmym, where ym minimizes the function

J(y) = ‖b−A(x0 + Vmy)‖2. The minimizer is inexpensive to compute, since it

requires the solution of an (m+ 1)×m least squares problem, where m is typ-

ically small [56]. GMRES requires more memory than other Krylov methods

because it stores all search directions.

6.1.2 BICG-STAB BICG-STAB was developed to solve non

symmetric linear systems while avoiding the often irregular convergence pat-

terns of the Conjugate Gradient Squared method, CGS [19, 56, 57]. The CGS

method is based on squaring the residual polynomial; however, sometimes this

is slow, and it may lead to substantial buildup of rounding errors or even over-

flow. CGS guarantees that the norm of the residual decreases. BICG-STAB is

a variation of CGS, and was developed as a stabilization of the Arnoldi process.

It uses a 3-term recursion, but a decrease of the norm of the residual is not

guaranteed [56]. BICG-STAB produces iterates whose residual vectors are of

76

the form

rj = ψj(A)φj(A)r0,

where φj(t) is the residual polynomial associated with the Biconjugate Gradient

algorithm, and ψj(t) is a new polynomial that is defined recursively, so that it

stabilizes or smoothes the convergence behavior of the original algorithm.

6.1.3 Gauss-Seidel Gauss-Seidel is a popular multigrid

smoother. We have used Gauss-Seidel in the form xk+1 = xk +Q−1(fk−Axk),

where the matrix Q is the lower triangular part of A that was found by using

the MATLAB function tril(A).

We present results and convergence factors for Gauss-Seidel, GMRES,

and BICG-STAB used as smoothers. Multigrid is often used as a preconditioner

for an iterative method. In our numerical experiments, we use GMRES pre-

conditioned with one full multigrid iteration. Tests are run for different mesh

sizes h; when the mesh is refined the frequency is increased, so that the value

of k3h2 is kept constant. This is needed to avoid pollution by the phase error

and to keep the error decreasing with h for the Helmholtz equation [37]. The

essential ingredients of the multigrid algorithm are as follows. In all the prob-

lems below the standard V-cycle algorithm described in Algorithm 6.1 is used

for the multigrid method. At each multigrid level the number of pre-smoothing

steps always equals the number of post-smoothing steps. The problem on the

coarsest level is solved directly.

77

7. Numerical Results

7.1 Numerical Verification of the Discretization

We now present preliminary numerical experiments from a prototype

implementation in MATLAB. We consider a 2D model problem as in Fig. 4.2.

The domain is the square (0, 1)× (0, 1). The obstacle in the channel is set up

in the center of the fluid domain as a square of size 0.2 m, unless otherwise

specified. In all computational examples, the size of the domain in the x and

y-direction is chosen to be 1 m. The boundary condition on Γd is p0(x, y) = 1.

The origin of the coordinate system is assumed to be in the lower left corner

of Ω. The fluid medium is water with density ρf = 1000 kg m−3 and speed

of sound cf = 1500ms−1. The elastic medium is aluminum with density

ρe = 2700 kg m−3 and Lame elasticity coefficients are λ = 5.5263 · 1010N m−2,

and µ = 2.595 ·1010N m−2. The speed of sound is cp = 6300ms−1 for pressure

waves and cs = 3100ms−1 for shear waves. The fluid domain and the solid

domain are discretized by standard bilinear square finite elements Q1 on a

uniform mesh with mesh size h.

Problem (4.42) is solved directly using sparse LU decomposition in

MATLAB with SYMRCM reordering to reduce the profile bandwidth of the

matrix [52]. Table 7.1 demonstrates the convergence of the solution. The wave

78

h = 110

h = 120

-1.02134046416 - 0.85252801718i -1.04992768430 - 0.83392861101i-1.02134046416 - 0.85252801718i -1.04992768430 - 0.83392861101i0.25394900255 + 1.07327043738i 0.30531713828 + 1.06650378005i0.25394900255 + 1.07327043738i 0.30531713828 + 1.06650378005i-1.51245645214 + 2.40851542272i -1.48018801699 + 2.54931071269i-0.00000000000 + 0.00000000000i -0.00000000000 - 0.00000000000i

h = 140

h = 180

-1.05757604357 - 0.82882031388i -1.05965748637187 - 0.82752600255i-1.05757604357 - 0.82882031388i -1.05965748637186 - 0.82752600255i0.31891339863 + 1.06408783048i 0.32248585398500 + 1.06339927283i0.31891339863 + 1.06408783048i 0.32248585398500 + 1.06339927283i-1.47408877517 + 2.59422086989i -1.47404124589866 + 2.60934086845i0.00000000000 + 0.00000000000i 0.00000000000001 + 0.00000000000i

h = 1160

h = 1320

-1.06024737503 - 0.82720711572i -1.06024377503023 - 0.82720711572i-1.06024737503 - 0.82720711572i -1.06024377503010 - 0.82720711572i0.32354182387 + 1.06320783223i 0.32343982387092 + 1.06320783223i0.32345282387 + 1.06320783223i 0.32343982387133 + 1.06320783223i-1.47467790973 + 2.61466232513i -1.47466790973365 + 2.61466232513i0.00000000000 - 0.00000000000i 0.00000000000005 - 0.00000000000i

Table 7.1. Values of the solution at the 6 sample points as explained in(Fig.7.1), mesh size h is halved at each run and wave number is kept constantk = 5.

79

Ωf

Γn

Γn

Γd ΓaΓiv5,6

Ωe

- n

v v

v v2

1

4

3

Figure 7.1. Configuration of sample points for Table 7.1. Sample points1,2,3,4 are fluid pressure values, and sample points 5 and 6 are ux and uy

displacements, respectively.

80

Error decreases as O(h2)h xh xh − x∗ log|h| log|xh − x∗|1/10 0.25394900255948 -6.9457e-02 -2.30258 -2.667041/20 0.30531713828081 -1.8089e-02 -2.99573 -4.012461/40 0.31891339863201 -4.4924e-03 -3.68887 -5.405361/80 0.32248585398500 -9.1997e-04 -4.38202 -6.991161/160 0.32354182387092 1.6600e-04 -5.07517 -8.703521/320 0.32343982387092 3.4000e-05 -5.76832 -10.28915

Table 7.2. Exact solution is x∗ and solution at mesh size h is xh.

number is chosen to be k = 5. The results are values at sample points for

different mesh sizes, from 10× 10 to 320× 320. Configuration of sample points

for Table 7.1 are displayed in Fig. 7.1. Sample points 1,2,3,4 are fluid pressure

values, and sample points 5,6 are ux and uy displacements, respectively. Denote

the unknown exact solution by x∗ and the solution at mesh size h by xh. Then,

xh − x∗ ≈ Ch2 ⇒ log|xh − x∗| ≈ 2 log|h|+const

To find x∗ by extrapolation, we solved the system of equations

x∗ + Ch2 = xh

x∗ + C(2h)2 = x2h

We use as an example solution points 3 in the lower left corner from Fig. 7.1

to show that the solution converges as 0(h2).

We display the solution for some cases considered. We display the

real part of the pressure amplitude as a function of elevation above the x − y

plane, and the displacement in the x and the y directions as equally spaced

arrows based at the original configuration. The solution obtained by solving

81

1.0e−04

5.0e−04

1.0e−03

5.0e−03

1.0e−02Extrapolated solution converges as O(h2)

1/h

x h−x*

10 20 40 80 160 320

Figure 7.2. Log log plot of difference of solution values xh at sample pointsfor mesh size 10× 10 to 320× 320 and extrapolated exact solution x∗, k=5.

82

fluid pressure

10

20

30

40

10

20

30

40−1.5

−0.5

0

1

1.5

y−axisx−axis

Γd

Γn

Γn

Γ a

solid displacement

10

20

30

4010 20 30 40

x−ax

is

y−axis

Γn

Γn

Γa

Γd

Figure 7.3. Solution for a 40 × 40 mesh, k = 5, right-hand side modified forthe Dirichlet boundary condition p = p0 on Γd

83

fluid pressure

30

60

30

60

−1.5

0

1.5

y−axisx−axis

Γd

Γn

Γn

Γa

solid displacement

30

60

30 60

x−ax

is

y−axis

Γn

Γn

ΓaΓ

d

Figure 7.4. Exact solution for a 64 × 64 mesh, k = 15, right-hand side ismodified for the Dirichlet boundary condition p = p0 on Γd

84

fluid pressure

30

60

30

60

−1.5

0

1.5

y−axisx−axis

Γd

Γn

Γn

Γa

solid displacement

30

60

30 60

x−ax

is

y−axis

Γn

Γn

ΓaΓ

d

Figure 7.5. Three multigrid iterations, 20 smoothing steps, smoother GM-RES, 2 levels to solve a 64 × 64 mesh, k = 15, right-hand side is modified forthe Dirichlet boundary condition p = p0 on Γd.

85

the system of equations directly for a 40 × 40 mesh obstacle (0.2 m), k = 5

and right-hand side modified for the Dirichlet boundary condition p = p0 on Γd

is displayed in Fig. 7.3. For comparison, we display the solution obtained by

applying multigrid with smoother GMRES to the discrete system of equations

for a 64 × 64 mesh obstacle (0.2 m), k = 15 and right-hand side modified

for the Dirichlet boundary condition p = p0 on Γd in Fig. 7.5. We apply 20

smoothing steps, 2 levels, and 3 multigrid iterations to obtain the solution.

For many applications the computational domain is not always sym-

metric. For simplicity, we consider a rectangular scatterer that has a rectan-

gular u-shaped indentation. We test 2 scenarios, one when the gap is on the

x-axis, which we call u-shapeX, and the other with the gap on the y-axis, which

we call u-shapeY. The size of the gap is described as a percentage of the side

on which the gap is present. For instance, Figure 7.6 represents an obstacle

(0.4 m) in the x and (0.2 m) in the y direction. The gap is on the y-axis. The

size of the gap is described as a percentage from the size of the obstacle, in

this case it is 50% in the x direction and 40% in the y direction, respectively.

Figure 7.7 represents an obstacle (0.2 m) in x and (0.4 m) in y direction. The

gap is on the x axis. The size of the gap is 40% in the x direction and 50% in

the y direction. Figure 7.8 represents the solution for a 64 × 64 mesh k = 15

obstacle (0.4 m) in x direction, (0.2 m) in y direction a gap on y axis of 0.4 and

0.5 in the x and y direction, respectively. Figure 7.9 represents the solution for

a u shaped obstacle (0.2 m) in x and (0.4 m) in y direction. Gap on x axis of

86

30

60

30 60

x

y

Γd

Γn

Γn

Γa

Figure 7.6. Contour of an obstacle (0.4 m) and (0.2 m) in the x and the ydirection, respectively. The gap is on the y axis. The size of the gap is 0.5 or50% in the x direction and 0.4 or 40% in the y direction.

30

60

30 60

x

y

Γd

Γn

Γn

Γa

Figure 7.7. Contour of an obstacle (0.2 m) and (0.4 m) in the x and the ydirection, respectively. The gap is on x axis. The size of the gap is 0.4 or 40%in the x and 0.5 or 50% in the y direction.

87

fluid pressure

20

40

60

20

40

60

−2

−1

0

1

2

y−axisx−axis

Γd

Γn

Γn

Γ a

solid displacement

20

40

60

20 40 60

x−ax

is

y−axis

Γn

Γn

Γa

Γd

Figure 7.8. Solution for a 64 × 64 mesh with ushaped obstacle (0.2 m)by (0.4 m). The obstacle has a gap of size 0.4 by 0.5 on the y axis. Thewave number is k = 15, and the right-hand side is modified for the Dirichletboundary condition p = p0 on Γd

88

0.4 and 0.5 in x and y direction respectively. For both figures we display the

real part of the amplitude for the pressure and the real displacement vectors

in x and y direction.

7.2 Computational Results with Multigrid

In the first set of experiments we solve the solid-fluid coupled problem

on a square fluid domain with a square scatterer in the center of the waveguide

of size 0.2 m. In all tests, the multigrid iterations are run until the following

stopping criterion

‖rl‖/‖r0‖ < 10−6, (7.1)

is reached, where ‖rl‖ and ‖r0‖ are the norms of the residual at the lth and 0th

iteration respectively. If norm of the residual increases after two consecutive

iterations, the multigrid iterations are halted and the number of iterations is

saved. We record the relative residual for the solution of the coupled problem

rel res = maxi

|fi −∑

j Aijvj||fi|+

∑j |Aij||vj|

, (7.2)

where v, A, and f are the numerical solution vector, the matrix, and the

right-hand side, respectively, of the coupled system (4.42). In cases when the

iterations converged, the relative residual is of the order 10−6 or smaller. We

record the relative residual errors in the fluid and the elastic parts separately.

The relative residual error is

rel res d = maxi

|fdi −

∑j A

dijv

dj |

|fdi |+

∑j |Ad

ij||vdj |, (7.3)

89

fluid pressure

30

60

30

60

−3

0

3

y−axisx−axis

Γd

Γn

Γn

Γ a

solid displacement

20

40

60

20 40 60

x−ax

is

y−axis

Γn

Γn

Γa

Γd

Figure 7.9. Solution for a 64 × 64 mesh ushaped obstacle (0.2 m) by (0.4 m).The obstacle has a gap of 0.4 by 0.5 on the x axis. The wave number is k = 15,and the right-hand side is modified for the Dirichlet boundary condition p = p0

on Γd

90

where d is pressure or displacement. The right-hand side vector fd, matrix Ad,

and solution vd restricted to either the elastic part or fluid part, are given by

f e, Ae, and ve or f f , Af , and vf , respectively. We define “work” to be the

total number of smoothing steps on the finest level. We observe that increasing

the number of smoothing steps has a stabilizing effect on the iterative process.

To better quantify this, we calculate the residual reduction per unit of work as

resid reduction =

(residual before

residual after

).1/work

(7.4)

In Table 7.3 and Table 7.4, we list the residual reduction per unit of

work for different smoothers, where we decreased h and kept k3h2 constant.

The columns in the tables are organized as follows. The first column lists h the

mesh spacing, the next column is the wave number k, followed by the number

of smoothing steps per multigrid cycle and number of multigrid levels. Then,

we list the iterative method used for smoothing. Column 6 is the number

of iteration required to meet the stopping criterion, and column 7 is residual

reduction computed using (7.4). The last 3 columns in the tables represent the

component wise scaled residual of the coupled problem (7.2), relative residual in

fluid part, and relative residual in elastic part (7.3). We decreased the number

of multigrid levels each time we decreased h. We observe that increasing the

number of smoothing steps improves residual reduction, and norm of residual

decreases from iteration to iteration. When GMRES is used as smoother,

the norm of residual decreases from iteration to iteration, and more multigrid

iterations are performed resulting in better results. The matrix D is the same

91

Decreasing h, k3h2 constant, obstacle size 0.2res red rel rs res fl res el

h k sm lv mth it (7.4) (7.2) (7.3) (7.3)132

10 3 3 BC 2 5.6e-01 2.6e-04 3.2e-04 3.48e-03132

10 10 3 BC 2 7.7e-01 1.5e-04 3.4e-04 2.49e-04132

10 20 3 BC 2 8.5e-01 8.8e-05 3.4e-04 2.60e-04132

10 3 3 GM 4 4.1e-01 1.0e-04 3.0e-04 2.27e-04132

10 10 3 GM 10 4.2e-01 4.2e-07 2.0e-04 4.11e-04132

10 20 3 GM 2 5.0e-01 2.3e-09 2.0e-04 4.11e-04132

10 10 3 GD 4 4.0e-01 2.7e-07 2.0e-04 4.11e-04132

10 20 3 GD 3 5.0e-01 2.3e-09 2.0e-04 4.11e-04132

10 3 3 GA 1 6.1e-02 8.2e-04 1.0e-03 5.21e-03132

10 10 3 GA 1 6.1e-02 8.2e-04 1.0e-03 6.42e-03132

10 20 3 GA 1 6.1e-02 8.3e-04 1.0e-03 5.91e-03164

15 10 4 BC 2 8.3e-01 8.9e-05 1.0e-04 1.57e-03164

15 20 4 BC 2 8.7e-01 8.7e-05 9.9e-05 1.66e-03164

15 3 4 GM 2 8.0e-01 1.2e-04 1.4e-04 2.81e-03164

15 10 4 GM 2 6.7e-01 1.9e-05 6.9e-05 1.29e-04164

15 20 4 GM 6 5.0e-01 9.5e-10 5.3e-05 1.49e-04164

15 3 4 GD 2 4.9e-01 1.2e-04 1.4e-04 1.12e-03164

15 10 4 GD 12 3.0e-01 7.3e-09 5.3e-05 1.49e-04164

15 20 4 GD 2 4.9e-01 6.4e-10 5.3e-05 1.49e-04164

15 3 4 GA 1 5.2e-02 8.3e-04 1.1e-03 4.3e-03164

15 10 4 GA 1 6.5e-02 7.2e-04 1.0e-03 5.3e-031

12825 3 5 BC 2 4.2e-01 5.3e-05 6.3e-05 8.9e-04

1128

25 10 5 BC 2 8.0e-01 2.8e-05 3.1e-05 6.3e-041

12825 20 5 BC 1 8.1e00 3.2e-05 3.9e-05 1.2e-04

1128

25 3 5 GM 2 4.4e-01 2.6e-05 3.1e-05 2.3e-031

12825 10 5 GM 2 7.6e-01 2.1e-05 2.9e-05 5.2e-05

1128

25 20 5 GM 5 7.2e-01 5.8e-07 2.2e-05 6.7e-051

12825 3 5 GD 4 3.1e-01 1.1e-05 1.3e-05 1.6e-04

1128

25 10 5 GD 2 6.1e-01 2.9e-06 2.4e-05 6.0e-051

12825 20 5 GD 2 6.8e-01 1.8e-07 2.2e-05 6.7e-05

Table 7.3. h = step size, k = wave number, sm = number of smoothingsteps, lv = number of levels, mth = iterative method used for the smoothingalgorithm, it = number of multigrid iterations, res red = residual reduction,rel rs = relative residual, res fl = residual in the fluid part, and res el =residual in the elastic part. BC is BICG-STAB, GM is GMRES, GD is GMRESpreconditioned by D−1 (as explained in (7.5)), GA is Gauss-Seidel

92

Decreasing h, k3h2 constant, obstacle (0.2 m)res red rel rs res fl res el

h k sm lv mth it (7.4) (7.2) (7.3) (7.3)1

12825 3 4 BC 1 9.1e-01 1.1e-05 2.1e-03 5.9e-02

1128

25 10 4 BC 1 9.2e-01 6.6e-06 1.1e-03 4.0e-021

12825 20 4 BC 2 7.2e-01 6.7e-06 1.2e-03 6.0e-04

1128

25 3 4 GM 5 7.2e-01 5.8e-07 2.2e-05 6.7e-051

12825 10 4 GM 5 7.2e-01 5.8e-07 2.2e-05 6.7e-05

1128

25 20 4 GM 5 7.2e-01 5.8e-07 2.2e-05 6.7e-051

12825 3 4 GD 2 3.8e-01 3.7e-05 4.4e-05 4.5e-04

1128

25 10 4 GD 2 6.1e-01 2.9e-06 2.4e-05 6.0e-051

12825 20 4 GD 2 6.8e-01 1.8e-07 2.2e-05 6.7e-05

1256

40 10 5 BC 2 8.0e-01 9.4e-06 1.1e-05 6.9e-041

25640 20 5 BC 2 8.6e-01 5.2e-06 6.2e-06 4.6e-05

1256

40 3 5 GM 2 5.4e-01 1.7e-05 2.0e-05 8.4e-051

25640 10 5 GM 2 7.1e-01 3.2e-06 4.0e-06 5.0e-05

1256

40 20 5 GM 2 8.2e-01 2.3e-06 6.3e-06 1.7e-051

25640 10 5 GD 3 6.4e-01 1.3e-06 5.0e-06 1.5e-05

1256

40 20 5 GD 2 7.2e-01 1.8e-07 3.5e-06 2.6e-051

51265 3 6 BC 1 1.1e-01 5.6e-06 6.6e-06 2.5e-04

1512

65 10 6 BC 2 7.5e-01 2.4e-06 2.8e-06 1.7e-051

51265 20 6 BC 2 8.6e-01 2.2e-06 2.6e-06 1.0e-05

1512

65 3 6 GM 3 6.5e-01 1.9e-06 2.1e-04 2.4e-041

51265 10 6 GM 2 2.7e-01 4.3e-06 4.6e-04 8.4e-05

1512

65 20 6 GM 2 7.5e-01 3.3e-07 3.5e-04 5.1e-051

51265 3 6 GD 3 6.3e-01 5.7e-06 6.1e-05 8.6e-04

1512

65 10 6 GD 2 6.5e-01 6.3e-06 6.7e-05 7.5e-041

51265 20 6 GD 2 7.5e-01 3.7e-07 3.9e-05 1.0e-03

Table 7.4. h = step size, k = wave number, sm = number of smoothingsteps, lv = number of levels, mth = iterative method used for the smoothingalgorithm, it = number of multigrid iterations, res red = residual reduction,rel rs = relative residual, res fl = residual in the fluid part, and res el =residual in the elastic part. BC is BICG-STAB, GM is GMRES, GD is GMRESpreconditioned by D−1 (as explained in (7.5)), GA is Gauss-Seidel

93

size as A and is set to be

D([i, i+ nds, i+ 2 · nds], [i, i+ nds, i+ 2 · nds]) =

A([i, i+ nds, i+ 2 · nds], [i, i+ nds, i+ 2 · nds]), (7.5)

where nds represents the number of nodes. GMRES preconditioned by D−1

gives similar results as multigrid with a GMRES smoother. For Gauss-Seidel

as a smoother, we do not get a reduction in residual, but the opposite. This is

due to the fact that the problem is ill-conditioned and the condition number

deteriorates with increasing mesh size, and the speed of convergence deterio-

rates as well. We see that convergence factors for smoothers GMRES, GMRES

preconditioned by D−1, and BICG-STAB improve by as much as a factor of 10

with increased number of smoothing steps per multigrid cycle. These results

are similar to [22], where it was observed that multigrid preconditioned by

GMRES is more efficient than stand-alone multigrid; however, a larger num-

ber of outer GMRES steps require more memory to store more grid vectors.

When BICG-STAB is used as smoother, we observe that increasing the num-

ber of multigrid levels does not affect relative residual significantly. BICG,

however, requires lower storage requirements than GMRES. Using GMRES for

a smoother, however, may work so well because it reduces the component of

error that belongs to negative eigenvalues. We observe that more smoothing

steps have a stabilizing effect on the iterative process.

We followed [22] to find number of multigrid iterations required to

94

h 1/128 1/256 1/512k 25 40 65

level GMS BGS GMT GMS BCS GMT GMS BGS GMT3 5 1 2 2 2 2 2 1 24 5 2 2 2 2 2 2 1 25 5 -1 2 2 5 3 2 2 26 - - - - - - 2 2 2

Table 7.5. Iteration counts for multigrid V cycle to achieve a relativeresidual of order 1e-6, for smoothers GMS=GMRES; BGS=BICG-STAB;GMD=GMRES preconditioned by the inverse of the lower triangular part ofA.

reach the stopping criterion

maxi

|fi −∑

j Aijvj||fi|+

∑j |Aij||vj|

< 10−6. (7.6)

Table 7.5 shows the iteration counts for increasing numbers of levels beginning

with the fine grids containing 128, 256, and 512 elements. Wave number is

increased to keep k3h2 constant. As in [22], when coarser levels are added,

more multigrid iterations are required to reach the stopping criterion.

We tested convergence of numerical solutions for the coupled problem

by recording relative residual and number of multigrid iterations required to

achieve convergence. Results are shown in Table 7.6. Each run, we doubled

h while keeping k3h2 constant and the number of multigrid levels equal to 5.

The test is done for BICG-STAB, GMRES, and GMRES preconditioned by

D−1 as smoothers. We observe that as the mesh spacing decreases the relative

residual decreases. Increasing the number of smoothing steps per multigrid

level refines the solution as expected. As observed in [22], on very coarse

95

Decreasing h, k3h2 constant, obstacle (0.2 m)res red rel rs res fl res el

h k sm lv mth it (7.4) (7.2) (7.3) (7.3)1

12825 3 5 BC 2 4.2e-01 5.3e-05 6.3e-05 8.9e-04

1128

25 10 5 BC 2 8.0e-01 2.8e-05 3.1e-05 6.3e-041

12825 20 5 BC 1 1.0e+00 3.2e-05 3.9e-05 1.2e-04

1128

25 3 5 GM 2 4.4e-01 2.6e-05 3.1e-05 2.3e-031

12825 10 5 GM 2 7.6e-01 2.1e-05 2.9e-05 5.2e-05

1128

25 20 5 GM 5 7.2e-01 5.8e-07 2.2e-05 6.7e-051

12825 3 5 GD 4 3.1e-01 1.1e-05 1.3e-05 1.6e-04

1128

25 10 5 GD 2 6.1e-01 2.9e-06 2.4e-05 6.0e-051

12825 20 5 GD 2 6.8e-01 1.8e-07 2.2e-05 6.7e-05

1256

40 3 5 BC 1 1.5e+00 3.6e-05 4.3e-05 1.0e-031

25640 10 5 BC 2 8.0e-01 9.4e-06 1.1e-05 6.9e-04

1256

40 20 5 BC 2 8.6e-01 5.2e-06 6.2e-06 4.6e-051

25640 3 5 GM 2 5.4e-01 1.7e-05 2.0e-05 8.4e-05

1256

40 10 5 GM 2 7.1e-01 3.2e-06 4.0e-06 5.0e-051

25640 20 5 GM 2 8.2e-01 2.3e-06 6.3e-06 1.7e-05

1256

40 3 5 GD 1 2.1e+00 2.5e-05 2.9e-05 3.4e-041

25640 10 5 GD 3 6.4e-01 1.3e-06 5.0e-06 1.5e-05

1256

40 20 5 GD 2 7.2e-01 1.8e-07 3.5e-06 2.6e-051

51265 3 5 BC 1 1.0e+00 5.6e-06 6.7e-06 2.5e-04

1512

65 10 5 BC 2 7.9e-01 3.0e-06 3.5e-06 6.0e-061

51265 20 5 BC 2 8.4e-01 2.0e-06 2.3e-06 2.1e-05

1512

65 3 5 GM 2 3.0e-01 2.2e-06 2.6e-06 1.2e-051

51265 10 5 GM 2 6.4e-01 5.6e-07 9.7e-07 1.3e-05

1512

65 20 5 GM 2 7.9e-01 4.0e-07 2.6e-06 4.1e-061

51265 3 5 GD 2 3.0e-01 4.3e-06 5.1e-06 3.3e-04

1512

65 10 5 GD 3 6.3e-01 7.6e-07 1.4e-06 8.8e-061

51265 20 5 GD 2 7.5e-01 1.6e-07 1.7e-06 3.5e-06

Table 7.6. h = step size, k = wave number, sm = number of smoothingsteps, lv = number of levels, mth = iterative method used for the smoothingalgorithm, it = number of multigrid iterations, res red = residual reduction,rel rs = relative residual, res fl = residual in the fluid part, and res el =residual in the elastic part. BC is BICG-STAB, GM is GMRES, GD is GMRESpreconditioned by D−1 (as explained in (7.5)).

96

meshes, the Helmholtz operator is poorly approximated. This is also observed

for the coupled problem in Table 7.6 for mesh size 1128

. Finer mesh has a

stabilizing effect on the iterative process because kh is smaller. To observe

the impact of adding coarse meshes we plot the log of the residual reduction

(7.4) as a function of mesh size h and number of coarse levels, while keeping

k3h2 constant in Fig. (7.10) and (7.11). We observe that for Gauss-Seidel

we get divergence and Krylov subspace smoothers are robust and give good

convergence rates. This is consistent with [22] for the Helmholtz problem

where it was observed that finer meshes have a stabilizing effecton the iterative

process. To observe the impact of adding smoothing steps we plot the log of the

residual reduction (7.4) as a function of h and number of smoothing steps, while

keeping k3h2 constant, in Fig. (7.12) and (7.13). We observe that increasing

the number of smoothing steps refines the solution.

The obstacle is made larger, and the fluid domain is smaller to try to

isolate the effect of the elastic domain. The results in Table 7.7 show relative

residual (7.2), and error in the fluid and elastic regions (7.3) for BICG, GMRES

and Gauss-Seidel algorithms when used as smoothers. The obstacle is (0.5 m)

in the middle of the channel, and the number of smoothing steps per multigrid

cycle is 10 for all runs. The mesh is refined and the frequency is increased, while

keeping k3h2 constant. From the relative residual in Table 7.7, we observe that

since Gauss-Seidel amplifies the smooth error components [22], it has a worse

convergence rate compared to GMRES. We note however, that Gauss-Seidel

97

2

3

4

5

2e−82e−7

2e−6 2e−5

10−0.3

10−0.2

10−0.1

multigrid levels

GMRES as smoother

h

resi

dual

red

uctio

n

2

3

4

5

2e−52e−6

2e−72e−8

10−0.4

10−0.3

10−0.2

multigrid levels

GMRES preconditioned by invlower triangular part of A

as smoother

h

resi

dual

red

uctio

n

Figure 7.10. Residual reduction as a function of adding coarse meshes, de-creasing mesh size h while keeping k3h2 constant.

98

23

45

2e−52e−6

2e−72e−8

10−0.17

10−0.13

10−0.09

multigrid levels

BICG−STAB as smoother

h

resi

dual

red

uctio

n

23

452e−8

2e−7

2e−6

2e−5

10−5

100

105

1010

multigrid levels

GAUSS−SEIDEL as smoother

h

resi

dual

red

uctio

n

Figure 7.11. Residual reduction as a function of adding coarse meshes, de-creasing mesh size h while keeping k3h2 constant.

99

3

10

20

2e−6

2e−7

2e−8

2e−9

10e−3

10e−1

smoothing steps

GMRES as smoother

h

resi

dual

red

uctio

n

3

10

20

2e−52e−6

2e−72e−8

10−4

10−3

10−2

10−1

GMRES preconditioned by inv Das smoother

smoothing stepsh

resi

dual

red

uctio

n

Figure 7.12. Residual reduction as a function of adding smoothing steps,decreasing mesh size h while keeping k3h2 constant.

100

3

10

20

2e−62e−7

2e−82e−9

10−3

10−2

10−1

100

smoothing steps

BICG−STAB as smoother

h

resi

dual

red

uctio

n

3

10

20

2e−62e−7

2e−82e−9

10−4

10−3

10−2

10−1

GMRES precondition by invlower triangular part of A

as smoother

smoothing stepsh

resi

dual

red

uctio

n

Figure 7.13. Residual reduction as a function of adding smoothing steps,decreasing mesh size h while keeping k3h2 constant.

101

smoothing is less expensive than GMRES smoothing.

In Table 7.8, we increase wave number k and keep the number of

smoothing steps constant at 10 for each multigrid level. The relative error

(7.2) and the residual reduction for the coupled problem are recorded. Multi-

grid iterations are halted when two consecutive iterations increase the residual.

We observe that for large wave numbers k, residual increases after the first

multigrid iteration for all smoothers tested; however, GMRES and precondi-

tioned GMRES as smoothers give robust performance by returning the lowest

relative residuals. As in [22], we see that for large enough wave numbers, the

increased number of amplified modes causes multigrid with non-Krylov types

of smoothers to fail. Here, more multigrid cycles do not reduce the residual,

and thus we also conclude that the methods fail to converge for high wave

numbers.

Multigrid is often used as a preconditioner for an iterative method.

Table 7.10, lists results for GMRES preconditioned with one standard full

multigrid iteration. We run one multigrid iteration for BICG-STAB, GMRES,

and GMRES preconditioned by the inverse of the lower triangular part of A.

We increase the number of multigrid levels, while decreasing the mesh size h.

The wave number is increased such that k3h2 is kept constant. We observe

that more smoothing steps have a stabilizing effect on the solution. The best

convergence rate is obtained for GMRES preconditioned by the inverse of the

lower triangular part of A and 20 smoothing steps. GMRES preconditioned

102

Decreasing h, k3h2 constant, obstacle (0.5 m)res red rl res rs fl rs el

h k sm lv it (7.4) (7.2) (7.3) (7.3)

Multigrid with 10 BICG smoothing steps132

10.00 10 2 2 8.88e-01 1.76e-04 7.58e-04 2.17e-04164

15.00 10 3 2 8.23e-01 5.22e-05 2.04e-04 8.67e-051

12825.00 10 4 2 1.03e+00 4.73e-05 1.53e-04 9.65e-05

1256

40.00 10 5 2 1.01e+00 2.83e-06 9.35e-05 2.07e-06Multigrid with 10 GMRES smoothing steps

132

10.00 10 2 4 6.80e-01 6.98e-06 4.65e-04 3.56e-04164

15.00 10 3 2 6.65e-01 7.12e-06 2.23e-04 7.38e-051

12825.00 10 4 2 6.71e-01 3.98e-06 6.14e-05 1.90e-05

1256

40.00 10 5 2 7.38e-01 1.69e-06 3.01e-05 3.41e-06Multigrid with 10 Gauss-Seidel smoothing steps

132

10.00 10 2 1 2.00e-01 9.82e-05 1.16e-04 5.63e-04164

15.00 10 3 1 1.90e-01 8.20e-05 1.16e-04 5.63e-041

12825.00 10 4 1 2.00e-01 7.80e-05 1.16e-04 5.63e-04

1256

40.00 10 5 1 1.89e-01 2.32e-05 3.45e-05 2.80e-04

Table 7.7. h = step size, k = wave number, sm = number of smoothing steps,lv = number of levels, it = number of multigrid iterations, res red = residualreduction, rel rs = relative residual, res fl = residual in the fluid part, and resel = residual in the elastic part.

103

Decreasing h, high wave numbers, obstacle (0.2 m)res red rel rs res fl res el

h k sm lv mth it (7.4) (7.2) (7.3) (7.3)1

12850 10 3 BC 1 1.00e-00 1.1e-04 1.3e-04 4.9e-03

1128

50 10 3 GM 3 7.06e-01 1.1e-05 2.9e-05 2.2e-041

12850 10 3 GA 1 1.00e-00 3.4e-04 4.1e-04 2.5e-03

1128

50 10 3 GT 2 7.92e-01 2.8e-05 5.5e-05 1.0e-041

25680 10 4 BC 1 1.00e-00 3.6e-05 4.3e-05 1.0e-02

1256

80 10 4 GM 2 7.19e-01 5.8e-06 8.9e-06 7.0e-051

25680 10 4 GA 1 1.0e-00 1.7e-04 1.6e-04 6.4e-04

1256

80 10 4 GT 3 7.1e-01 4.7e-06 9.2e-06 5.0e-051

512125 10 5 BC 1 7.6e-01 1.1e-05 1.3e-05 1.7e-03

1512

125 10 5 GM 2 6.7e-01 1.8e-06 2.6e-06 1.9e-051

512125 10 5 GA 1 4.6e+21 1.3e-04 1.6e-05 1.2e-04

1512

125 10 5 GT 3 6.3e-01 7.1e-07 1.4e-06 8.6e-061

128100 10 3 BC 1 8.8e-01 4.8e-04 5.9e-04 6.9e-04

1128

100 10 3 GM 1 8.0e-01 1.3e-04 1.7e-04 2.3e-041

128100 10 3 GA 1 6.0e+04 1.4e-03 1.7e-03 4.8e-03

1128

100 10 3 GT 1 7.9e-01 1.3e-04 1.6e-04 1.1e-031

256160 10 4 BC 1 8.1e-01 9.8e-05 1.1e-04 1.0e-02

1256

160 10 4 GM 1 7.7e-01 4.1e-05 5.0e-05 1.1e-041

256160 10 4 GA 1 1.7e+05 1.3e-03 1.7e-03 5.2e-03

1256

160 10 4 GT 1 7.9e-01 1.3e-04 1.6e-04 1.1e-031

512250 10 5 BC 1 8.1e-01 4.3e-05 5.1e-05 1.6e-03

1512

250 10 5 GM 1 7.3e-01 1.1e-05 1.3e-05 4.2e-051

512250 10 5 GA 1 3.3e+23 1.3e-04 1.5e-04 8.9e-04

1512

250 10 5 GT 1 7.3e-01 1.0e-05 1.3e-05 3.6e-04

Table 7.8. h = step size, k = wave number, sm = number of smoothingsteps, lv = number of levels, mth = iterative method used for the smoothingalgorithm, it = number of multigrid iterations, res red = residual reduction,rel rs = relative residual, res fl = residual in the fluid part, and res el =residual in the elastic part. BC is BICG-STAB, GM is GMRES, GT is GMRESpreconditioned by the inverse of the lower triangular part of A, GA=Gauss-Seidel.

104

by one multigrid iteration results are compared to multigrid with a GMRES

smoother; the results are listed in Table 7.9. We observe that less work is re-

quired for GMRES with multigrid preconditioning compared to multigrid with

10 GMRES smoothing steps, to reach comparable results. In [22], it is ob-

served that for the Helmholtz equation, a large number of GMRES smoothing

steps on intermediate levels eliminates lower frequency errors; then multigrid

and GMRES with multigrid preconditioning give comparable results. For the

coupled problem, multigrid with a GMRES smoother gives similar results with

GMRES preconditioned by multigrid, possibly for the same reason.

We changed the shape of the obstacle to be u-shaped in the x direction

as in Figure 7.9. The size of the obstacle gap in the x and y direction is

varied. The gap is centered on the axis on which it resides. We used GMRES

preconditioned by one standard multigrid cycle with 2 smoothing steps (Table

7.11) and multigrid with 2 GMRES smoothing steps per cycle (Table 7.12). We

tested various smoothers on a squared diagonally preconditioned system with 2

and 20 steps. We observed that both iterative methods, multigrid and GMRES

preconditioned by one multigrid cycle, give good results for an obstacle with

a gap in the x direction. However, GMRES preconditioned by one multigrid

cycle gives better results than multigrid; this is probably because GMRES

smoothing eliminates the lower frequency errors.

We tested GMRES preconditioned by standard ILU applied to a 2×2

105

GMRES preconditioned by one multigrid iterationfor square obstacle 0.2 m in the center of the waveguide

res red rl res rs fl rs el

h k sm lv mth it (7.4) (7.2) (7.3) (7.3)1

25640 3 5 BC 1 4.18e-01 1.52e-05 1.80e-05 1.81e-04

1256

40 10 5 BC 1 7.16e-01 7.17e-06 8.56e-06 1.17e-041

25640 20 5 BC 1 8.41e-01 5.75e-06 6.93e-06 8.70e-05

1256

40 3 5 GM 1 3.00e-01 9.02e-06 1.07e-05 1.81e-041

25640 10 5 GM 1 6.75e-01 3.81e-06 4.62e-06 7.60e-05

1256

40 20 5 GM 1 8.06e-01 1.60e-06 4.56e-06 2.65e-051

25640 3 5 GT 1 3.58e-01 2.06e-05 2.46e-05 8.11e-05

1256

40 10 5 GT 1 6.34e-01 1.63e-06 3.65e-06 2.95e-051

25640 20 5 GT 1 7.13e-01 1.33e-07 3.92e-06 2.19e-05

Multigrid V cycle with different smoothers1

25640 3 5 BC 1 3.00e-01 3.62e-05 4.30e-05 1.06e-03

1256

40 10 5 BC 2 8.08e-01 9.43e-06 1.11e-05 6.95e-041

25640 20 5 BC 2 8.66e-01 5.23e-06 6.25e-06 4.67e-05

1256

40 3 5 GM 2 5.45e-01 1.72e-05 2.03e-05 8.46e-051

25640 10 5 GM 2 7.12e-01 3.24e-06 4.08e-06 5.09e-05

1256

40 20 5 GM 2 8.21e-01 2.32e-06 6.34e-06 1.75e-051

25640 3 5 GT 1 8.00e-01 2.53e-05 2.99e-05 3.42e-04

1256

40 10 5 GT 3 6.45e-01 1.34e-06 5.00e-06 1.56e-051

25640 20 5 GT 2 7.22e-01 1.80e-07 3.50e-06 2.61e-05

Table 7.9. h = the step size, k = the wave number, sm = the number ofsmoothing steps, lv = the number of levels, sm =smoother, mth = the itera-tive method used for the smoothing algorithm, it = the number of multigriditerations, res red = the residual reduction, rl res = the relative residual, rs fl= the residual in the fluid part, and rs el = the residual in the elastic part.BC is BICG-STAB, GT is GMRES preconditioned by the inverse of the lowertriangular part of A

106

GMRES preconditioned by one multigrid iterationfor square obstacle 0.2 m in the center of the waveguide

res red rl res rs fl rs el

h k sm lv mth (7.4) (7.2) (7.3) (7.3)164

15 3 3 BC 4.48e-01 6.35e-05 7.84e-05 3.87e-04164

15 10 3 BC 7.63e-01 6.38e-05 7.77e-05 5.01e-04164

15 20 3 BC 8.44e-01 5.75e-05 6.77e-05 7.17e-04164

15 3 3 GM 3.90e-01 6.80e-05 8.19e-05 4.78e-04164

15 10 3 GM 6.63e-01 1.41e-05 5.85e-05 1.46e-04164

15 20 3 GM 6.73e-01 3.39e-07 5.31e-05 1.51e-04164

15 3 3 GT 3.67e-01 5.85e-05 7.13e-05 4.99e-04164

15 10 3 GT 5.45e-01 1.83e-06 5.17e-05 1.52e-04164

15 20 3 GT 5.76e-01 1.35e-08 5.34e-05 1.49e-041

12825 3 4 BC 4.36e-01 2.86e-05 3.43e-05 2.63e-04

1128

25 10 4 BC 7.40e-01 2.02e-05 2.42e-05 3.21e-041

12825 20 4 BC 8.66e-01 1.92e-05 2.71e-05 5.43e-05

1128

25 3 4 GM 3.71e-01 3.53e-05 4.24e-05 3.24e-041

12825 10 4 GM 7.16e-01 1.44e-05 1.99e-05 8.14e-05

1128

25 20 4 GM 7.46e-01 1.07e-06 1.80e-05 7.09e-051

12825 3 4 GT 3.17e-01 3.52e-05 4.21e-05 4.13e-04

1128

25 10 4 GT 6.04e-01 2.13e-06 1.34e-05 6.46e-051

12825 20 4 GT 6.75e-01 1.40e-07 2.19e-05 6.96e-05

1256

40 3 5 BC 4.18e-01 1.52e-05 1.80e-05 1.81e-041

25640 10 5 BC 7.16e-01 7.17e-06 8.56e-06 1.17e-04

1256

40 20 5 BC 8.41e-01 5.75e-06 6.93e-06 8.70e-051

25640 3 5 GM 3.00e-01 9.02e-06 1.07e-05 1.81e-04

1256

40 10 5 GM 6.75e-01 3.81e-06 4.62e-06 7.60e-051

25640 20 5 GM 8.06e-01 1.60e-06 4.56e-06 2.65e-05

1256

40 3 5 GT 3.58e-01 2.06e-05 2.46e-05 8.11e-051

25640 10 5 GT 6.34e-01 1.63e-06 3.65e-06 2.95e-05

1256

40 20 5 GT 7.13e-01 1.33e-07 3.92e-06 2.19e-05

Table 7.10. h = the step size, k = the wave number, sm = the number ofsmoothing steps, lv = the number of levels, mth = the iterative method usedfor the smoothing algorithm, it = the number of multigrid iterations, res red= the residual reduction, rl res = the relative residual, rs fl = the residual inthe fluid part, and rs el = the residual in the elastic part. BC is BICG-STAB,GT is GMRES preconditioned by the inverse of the lower triangular part of A

107

Obstacle with a gap in the x direction, vary gap size,GMRES preconditioned by one multigrid cycle

gap gap rl rs rs fl rs el

h x y x y k conv fctr (7.2) (7.3) (7.3)132

0.4 0.2 0.5 0.4 10 8.57e-01 1.8e-06 2.3e-03 1.2e-03132

0.4 0.2 0.6 0.4 10 8.99e-01 4.3e-06 1.9e-03 1.9e-03132

0.4 0.2 0.8 0.4 10 7.77e-01 4.1e-06 1.4e-03 1.4e-03164

0.4 0.2 0.5 0.4 15 9.38e-01 3.3e-06 3.1e-03 8.4e-04164

0.4 0.2 0.6 0.4 15 9.57e-01 4.2e-06 3.5e-03 1.2e-03164

0.4 0.2 0.8 0.4 15 9.40e-01 4.1e-06 3.2e-03 2.3e-031

1280.4 0.2 0.5 0.4 25 9.66e-01 6.8e-07 2.0e-03 4.3e-04

1128

0.4 0.2 0.6 0.4 25 9.77e-01 6.7e-07 1.9e-03 6.2e-041

1280.4 0.2 0.8 0.4 25 9.75e-01 6.8e-07 1.8e-03 6.2e-04

1256

0.4 0.2 0.5 0.4 35 9.52e-01 2.4e-07 1.3e-03 1.4e-041

2560.4 0.2 0.6 0.4 35 9.53e-01 2.7e-07 1.4e-03 1.4e-04

1256

0.4 0.2 0.8 0.4 35 9.48e-01 2.7e-07 1.4e-03 1.5e-041

5120.4 0.2 0.5 0.4 60 9.30e-01 1.0e-08 1.6e-04 2.9e-05

1512

0.4 0.2 0.6 0.4 60 9.21e-01 1.6e-08 2.3e-04 3.4e-051

5120.4 0.2 0.8 0.4 60 9.31e-01 1.2e-08 2.1e-04 3.1e-05

Table 7.11. h = the step size, x is size of obstacle in x direction, y size ofobstacle in y direction, gap x is size of the gap on the x axis as a percentage,gap y is size of gap on the y axis, k = the wave number, rs red = the residualreduction, rl rs = the relative residual, rs fl = the residual in the fluid part,and rs el = the residual in the elastic part.

108

Obstacle with a gap in the x direction, vary gap sizeOne multigrid cycle with 2 levels and 10 GMRES smoothing steps

x y gap gap rl rs rs fl rs el

h dir dir x y k conv fctr (7.2) (7.3) (7.3)132

0.4 0.2 0.5 0.4 10 8.78e-01 9.9e-05 4.2e-03 1.6e-03132

0.4 0.2 0.6 0.4 10 8.70e-01 6.0e-05 2.4e-03 2.0e-03132

0.4 0.2 0.8 0.4 10 7.60e-01 5.5e-05 1.8e-03 1.3e-03164

0.4 0.2 0.5 0.4 15 9.05e-01 2.4e-05 2.0e-03 9.0e-04164

0.4 0.2 0.6 0.4 15 9.58e-01 8.0e-05 6.8e-03 9.7e-04164

0.4 0.2 0.8 0.4 15 9.14e-01 8.6e-05 6.9e-03 2.1e-041

1280.4 0.2 0.5 0.4 25 9.12e-01 4.3e-06 1.2e-03 1.5e-04

1128

0.4 0.2 0.6 0.4 25 9.92e-01 7.8e-06 2.1e-03 5.1e-041

1280.4 0.2 0.8 0.4 25 9.89e-01 7.4e-06 1.9e-03 5.7e-04

1256

0.4 0.2 0.5 0.4 35 8.32e-01 7.9e-07 3.3e-04 5.2e-051

2560.4 0.2 0.6 0.4 35 8.46e-01 8.3e-07 3.5e-04 5.2e-05

1256

0.4 0.2 0.8 0.4 35 8.42e-01 8.2e-07 3.3e-04 5.3e-051

5120.4 0.2 0.5 0.4 60 7.99e-01 1.9e-07 2.1e-04 2.4e-05

1512

0.4 0.2 0.6 0.4 60 8.15e-01 1.7e-07 2.2e-04 4.2e-051

5120.4 0.2 0.8 0.4 60 7.95e-01 1.8e-07 2.0e-04 2.9e-05

Table 7.12. h = the step size, x is size of obstacle in x direction, y size ofobstacle in y direction, gp x is size of the gap on the x axis as a percentage,gp y is size of gap on the y axis, k = the wave number, rs red = the residualreduction, rl rs = the relative residual, rs fl = the residual in the fluid part,and rs el = the residual in the elastic part.

109

GMRES preconditioned by standard ilu applied to2 × 2 block diagonal for square obstacle 0.2 m

in the center of the waveguideh k res red (7.4) rel res(7.2) rel res fl(7.3) rel res el(7.3)164

15.00 7.71e-01 6.25e-06 2.81e-04 2.37e-041

12825.00 7.88e-01 4.98-06 1.32e-04 6.76e-05

1200

35.00 7.08e-01 2.99e-06 2.28e-04 5.27e-04

Table 7.13. h = the step size, k = the wave number, rel res = the relativeresidual, rel res fl = the relative residual in the fluid part, and rel res el = therelative residual in the elastic part.

block diagonal matrix as follows. Recall the coupled system (4.42), −Sf + k2Mf − ikGf −ρfω2sT

−ρfω2sTt −ρfω

2s2Se + ρfρeω4s2Me

p

u

= R.

Let Lf · Rf and Le · Re be approximate factorizations of −Sf + k2Mf and

−ρfω2s2Se + ρfρeω

4s2Me, respectively. We found that running GMRES pre-

conditioned by the block diagonal matrix (Lf ·Rf )−1 0

0 (Le ·Re)−1

(7.7)

gave good convergence. Comparison convergence rates can be seen in Table

7.13.

For some frequencies, the matrix −Se + ω2Me in the coupled system

(4.42) will be singular. We investigate the 2 by 2 block diagonal matrix when

the scatterer is at resonance, and the matrix block is singular. We compute

the eigenvalues of −Sf + ρeω2Mf from which we find the wave number k. The

eigenvalues are computed by using only the nodes of the matrices Sf and Mf

110

GMRES preconditioned by standard ilu applied to2 by 2 block diagonal for square obstacle 0.2 m

in the center of the waveguide using natural frequencyfor the elastic obstacle

h k res red (7.4) rel res(7.2) rel fl(7.3) rel el(7.3)164

43.6934839795 9.1e-1 1.31e-03 1.32e-03 1.46-141

12837.1100279902 8.9e-1 2.29e-04 1.61e-04 4.22-16

1200

40.5455499405 1e00 4.34-04 3.65e-04 5.71e-16

Table 7.14. GMRES preconditioned by ILU h = the step size, k = the wavenumber, rel res = the relative residual, res fl = the residual in the fluid part,and res el = the residual in the elastic part.

that are active, that is the list of degrees of freedom in the elastic part. The

sparse matrices were treated as full, and the full eigenvalue routines are used to

compute the eigenvalues accurately. Convergence rates for test problems when

the first nonzero eigenvalue is used to make the elastic block singular can be

seen in Table 7.14. We plot the log of the residual reduction for decreasing mesh

size h in the case when k3h2 is constant and in the case in which the elastic

obstacle is at resonance. In Fig. (7.14) we observe that GMRES preconditioned

by the 2 × 2 block diagonal diverges in the case of resonance.

For the obstacle with a gap, we made the size of the gap to be exactly

half of a wavelength. The results can be seen in Figure 7.15. The solution is

for a 64 × 64 mesh with a 0.5 m obstacle in the center of the waveguide with

k = 4π. The gap was made 50 percent of the obstacle size in both directions

and the gap is on the x axis. We tested both GMRES preconditioned by

one multigrid cycle and multigrid with smoother GMRES; results are shown

in Table 7.15. Both methods give good convergence rates. Since numerical

111

1/64

1/1281/200

10−6

10−4

10−2

100

h

GMRES preconditioned bystandard ILU to 2 × 2 block diagonal

rela

tive

resi

dual

natural frequency

k3h2=const

Figure 7.14. Relative residual varies as mesh size h is decreased for the casein which k3h2 is constant and in the case of resonance

Obstacle size 0.5 m in x and 0.5 m in y direction witha gap of half wavelength in the x and in the y directionh k conv fctr rel res(7.2) res fl(7.3) res el(7.3)

GMRES preconditioned by multigrid132

4π 9.38e-01 2.27-05 2.49-03 8.40e-04164

4π 9.05e-01 6.12e-06 7.17-04 9.88-051

1284π 9.75e-01 5.41-07 5.14-05 2.97-05

1256

4π 9.52e-01 9.16-08 8.19e-06 1.31-05One multigrid cycle with 2 levelsand 10 GMRES smoothing steps

132

4π 8.78e-01 2.29-05 2.51-03 5.28-04164

4π 9.05e-01 2.97-06 2.82-04 9.42-051

1284π 9.12e-01 7.11-07 5.05e-05 2.29-05

1256

4π 8.32e-01 1.36-07 9.67-06 6.79-06

Table 7.15. h = the step size, k = the wave number, conv fctr = convergencefactor or the residual reduction, rel res = the relative residual, res fl = theresidual in the fluid part, and res el = the residual in the elastic part.

112

Obstacle size 0.5 m in x and 0.5 m in y direction witha gap one wavelength with gap size 0.5 in x and and 0.5in y 50% is same as 0.25 m in x and in y directionh k conv fctr rel res(7.2) res fl(7.3) res el(7.3)

GMRES preconditioned by multigrid132

8π 9.28e-01 2.64e-05 1.23e-02 9.40e-04164

8π 9.02e-01 6.55e-06 3.13e-03 2.70e-041

1288π 9.65e-01 7.90e-07 3.74e-04 3.70e-05

1256

8π 9.22e-01 1.31e-07 3.47e-05 1.13e-05One multigrid cycles with 2 levelsand 10 GMRES smoothing steps

132

8π 8.28e-01 2.19e-05 9.51e-03 1.22e-03164

8π 8.05e-01 3.74e-06 1.66e-03 1.80e-041

1288π 9.10e-01 2.83e-07 1.11e-04 3.97e-05

1256

8π 9.02e-01 2.13e-07 2.16e-05 1.22e-05

Table 7.16. h = the step size, k = the wave number, conv fctr = convergencefactor or the residual reduction, rel res = the relative residual, res fl = theresidual in the fluid part, and res el = the residual in the elastic part.

113

wavelength is different [37, 35], we record the error as a function of wave number

k when the gap is in the neighborhood of half wavelength. Results are in Table

7.17, and we observe no changes in the relative error.

Since the prototype implementation is done in MATLAB, we do not

report timings. However, we note that more execution time is spent on oper-

ations to generate data and assemble the coupled matrices than the setup of

the iterative method. When the method converges, the iterations take only a

small fraction of the processing time.

7.3 Conclusion and Future Work

An efficient multigrid method for the discrete system was developed

and investigated. The numerical results of this chapter show that Krylov sub-

space smoothers, in particular GMRES, make robust algorithms when used

with a multigrid preconditioner. Also, GMRES preconditioned by multigrid

gives comparable results to multigrid with smoother GMRES, when a large

number of smoothing steps are performed. More smoothing steps have a stabi-

lizing effect on the numerical solution. For the coupled problem, GMRES and

GMRES preconditioned by the inverse of the lower triangular part of A are not

effective on coarse grids where the large eigenvalues of the Helmholtz operator

disappear from the discrete problem [22]. Multigrid methods are found to di-

verge, or give very poor convergence for high wave numbers. Totally different

methods are required for this purpose. For an obstacle with a gap, we find that

114

Varied gap size in the neighborhood of half wavelength sizemultigrid cycle with 2 levels and 10 GMRES smoothing steps

obstacle size 0.5m in x and y directiongap gap rl res rs fl rs el

h x y k conv fctr (7.2) (7.3) (7.3)1

1280.5 0.5 4π 9.12e-01 7.11e-07 5.05e-05 2.29-05

1128

0.5 0.4 4π 9.22e-01 7.01e-07 2.01e-03 5.71e-041

1280.5 0.6 4π 9.19e-01 3.99e-07 1.92e-03 5.17e-04

1128

0.4 0.5 4π 8.92e-01 5.89e-07 3.13e-04 5.72e-051

1280.6 0.5 4π 8.96e-01 7.13e-07 3.25e-04 5.22e-05

1128

0.4 0.6 4π 8.92e-01 1.92e-07 3.93e-04 5.03e-051

1280.6 0.4 4π 8.99e-01 2.99e-07 2.15e-04 2.14e-05

Table 7.17. Residual Reduction per unit of work for one Multigrid cycle. h =the step size, x is size of obstacle in x direction, y size of obstacle in y direction,gp x is size of the gap on the x axis as a percentage, gp y is size of gap onthe y axis, k = the wave number, rs red = the residual reduction, rl rs = therelative residual, rs fl = the residual in the fluid part, and rs el = the residualin the elastic part.

115

fluid pressure

20

40

6020

4060

−10

−5

0

5

10

y−axisx−axis

Γd

Γn

Γn

Γa

solid displacement

20

40

60

20 40 60

x−ax

is

y−axis

Γn

Γn

Γa

Γd

Figure 7.15. Solution for a 64 × 64 mesh, k = 4π, and right-hand sidemodified for the Dirichlet boundary condition p = p0 on Γd. A gap of halfwavelength in x and y direction

116

multigrid with GMRES as a smoother gives slightly better convergence rates

than GMRES preconditioned by one multigrid cycle. Multigrid with smoother

GMRES and multigrid with smoother GMRES preconditioned by the inverse

of the lower triangular part of A and by D−1 were found to be the most robust

on intermediate grids, and to give the best results. However, they require the

most memory to store the fine grid vectors.

Some directions for future work are a 3D finite element implementa-

tion and investigation of the unbounded domain case for the coupled fluid-solid

problem. For high wave numbers we need to research an implementation with

plane wave coarse spaces as in [44]. We also need to investigate the resonance

case with ILU preconditioner and implement a remedy as in [49].

117

REFERENCES

[1] M. F. Adams. A distributed memory unstructured Gauss-Seidel algorithmfor multigrid smoothers. Livermore, CA, November 2001.

[2] R. A. Adams. Sobolev spaces. Academic Press, New York, 1975.

[3] I. Babuska, F. Ihlenburg, E. Paik, and S. Sauter. A generalized finite ele-ment method for solving the Helmholtz equation in two dimensions withminimal polution. Computional Methods in Applied Mechanical Engineer-ing, 128:325–359, 1995.

[4] I. M. Babuska and S. A. Sauter. Is the pollution effect of the FEM avoid-able for the Helmholtz equation considering high wave numbers? SIAMJ. Numer. Anal., 34(6):2392–2423, December 1997.

[5] B. Bagheri, L. R. Scott, and S. Zhang. Implementing and using high-order finite element methods. Finite Elements in Analysis and Design,16:175–189, 1994.

[6] M. S. Baouendi and C. Goulaouic. Commutation de l’intersection et desfoncteurs de l’interpolation. C. R. Acad. Sci. Paris, 265:313–315, 1967.

[7] R. Barrett, M. Berry, and e. a. T. F. Chan. Templates for the Solution ofLinear Systems: Building Blocks for Iterative Methods. SIAM, Philadel-phia, 1994.

[8] J. Bielak, R. C. MacCamy, and X. Zeng. Stable coupling method forinterface scattering problems by combined integral equations and finiteelements. J. of Computational Physics, 119:374–384, 1995.

[9] A. Brandt. Multigrid techniques: 1984 guide with applications to fluid dy-namics. GMD–Studien Nr. 85. Gesellschaft fur Mathematik und Daten-verarbeitung, St. Augustin, 1984.

[10] S. C. Brenner and L. R. Scott. The mathematical theory of finite elementmethods. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

118

[11] W. L. Briggs. A Multigird Tutorial. Society of Industrial and AppliedMathematics, Philadelphia, Pennsylvania, 1987.

[12] P. G. Ciarlet. The finite element method for elliptic equations. NorthHolland, Amsterdam, 1978.

[13] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic ScatteringTheory. Springer-Verlag, Berlin, Heidelberg, New York, 1992.

[14] P. Cummings and X. Feng. Domain decomposition methods for a systemof coupled acoustic and elastic Helmholtz equations. In Eleventh Inter-national Conference on Domain Decomposition Methods, pages 205–213.Domain Decomposition Press, Bergen, Norway, 1999.

[15] L. Demkowicz. Asymptotic convergence in finite and boundary elementmethods: Part 1: theoretical results. Computers Math. Applic, 27(12):69–84, 1994.

[16] L. Demkowicz. Asymptotic convergence in finite and boundary elementmethods: Part 2: The LBB constant for rigid and elastic scattering prob-lems. Computers Math. Applic, 28(6):93–109, 1994.

[17] L. Demkowicz, A. Karafiat, and T. Oden. Solution of elastic scatteringproblems in linear acoustics using h-p boundary element method. Comp.Meth. Appl. Mech. Engrg., 101:251–282, 1992.

[18] L. Demkowicz and T. Oden. Elastic scattering problems in linear acousticsusing an h-p boundary/finite element method. In C. A. Brebbia andM. H. Aliabadi, editors, Adaptive Finite and Boundary Element Methods.Computational Mechanics Publications, 1993.

[19] H. A. der Vorstvan der Vorst. Bi-cgstab: A fast and smoothly convergingvariant of bi-cg for the solution of nonsymmetric linear systems. SIAM J.Sci. Stat. Comput., 13:631–644, 1992.

[20] A. Deraemaeker, I. M. Babuska, and P. Bouillard. Dispersion and pollu-tion of the FEM solution for the Helmholtz equation in one, two and threedimensions. Int. J. for Numer. Meth. in Engng., 46:471–499, 1999.

[21] R. Djellouli, C. Farhat, A. Macedo, and R. Tezaur. Finite element solutionof two-dimensional acoustic scattering problems using arbitrarily shaped

119

convex artificial boundaries. Journal of Computational Acoustics, 8:81–100, 2000.

[22] H. C. Elman, O. G. Ernst, and D. P. O’Leary. A multigrid method en-hanced by Krylov subspace iteration for discrete Helmholtz equations.SIAM J. Sci. Stat. Comput., 23:1291–1315, 2001.

[23] C. Farhat and F. Roux. An unconventional domain decomposition methodfor an efficient parallel solution of large-scale finite element systems. SIAMJ. Sci. Stat. Comput., 13:379–396, 1992.

[24] X. Feng. Analysis of finite element methods and domain decompositionalgorithms for a fluid-solid interaction problem. SIAM, 38(4):1312–1336,2000.

[25] C. I. Goldstein. Multigrid preconditioners applied to the iterative solutionof singularly perturbed elliptic boundary value problems and scatteringproblems. In R. Shaw, J. Periaux, A. Chaudouet, J. Wu, C. Marino, andC. Brebbia, editors, Innovative numerical methods in engineering, Proc.4th Int. Symp., Atlanta/Ga., 1986, pages 97–102, Berlin, Heidelberg, NewYork, 1986. Springer-Verlag.

[26] C. I. Goldstein. Analysis and application of multigrid method precon-ditioners for singularly perturbed boundary value problems. SIAM J.Numer. Anal., 26:1090–1123, 1989.

[27] P. Grisvard. Elliptic problems in nonsmooth domains. Pitman Publishing,Boston, 1985.

[28] P. Grisvard. Problememes aux limits dans les polygones. Mode d’emploi.EDF Bull. de la Direction des Etudes et Recherches Ser. C Mathematiques,Informatique, 1:21–59, 1986.

[29] P. Grisvard. Singularities en elasticite. Arch. Rational Mech. Anal.,107:157–180, 1989.

[30] P. Grisvard. Singularities in boundary value problems. Masson, Paris,1992.

[31] W. Hackbusch. A fast iterative method for solving Helmholtz’s equationin a general region. In U. Schumann, editor, Fast Elliptic Solvers, pages112–124. Advance Publications, London, 1978.

120

[32] W. Hackbusch. On the multigrid method applied to difference equations.Computing, 20:291–306, 1978.

[33] T. Harge. Valeurs propres d’un corps elastique. C.R. Acad. Sci. Paris,1:857–859, 1990.

[34] U. Hetmaniuk and C. Farhat. A fictitious domain decomposition methodfor high frequency acoustic scattering problems. In 12th InternationalConference on Domain Decomposition Methods, pages 365–372. DomainDecomposition Press, Bergen, Norway, 2001.

[35] F. Ihlenburg. Finite element analysis of acoustic scattering. Springer-Verlag, Berlin, Heidelberg, New York, 1998.

[36] F. Ihlenburg and I. M. Babuska. Finite element solution for the Hemholtzequation with high wave number, Part ii: The h-version of the FEM.Technical Note BN-1173, IPCS, University of Maryland, College Park,MD, June 1994.

[37] F. Ihlenburg and I. M. Babuska. Finite element solution for the Helmholtzequation with high wave number, Part i: The h-version of the FEM. Com-put. Math. Appl., 30(9):9–37, 1995.

[38] J. J. Douglas and T. Dupont. Galerkin methods for parabolic equationswith nonlinear boundary conditions. Numer. Math., 20:213–237, 1973.

[39] F. John. Partial Differential Equations. Springer-Verlag, Berlin, Heidel-berg, New York, 1971.

[40] C. Johnson. Numerical solution of partial differential equations by thefinite element method. Cambridge University Press, Sweden, 1994.

[41] S. Krein, J. Petunin, and E. Semenov. Interpolation of linear operators.American Mathematical Society, Providence, Rhode Island, 1982.

[42] E. Kreyszig. Introductory functional analysis with applications. John Wi-ley & Sons, New York, 1989.

[43] L. Landau and E. Lifshitz. Fluid Mechanics. Pergamon Press, Oxford,1987.

[44] B. Lee, T. Manteuffel, S. McCormick, and J. Ruge. Multilevel first-order

121

system least squares (FOSLS) for Hemholtz equation. In Procs. 2nd In-ternational Conf. on Approx. and Num. Meths. for the Solution of theMaxwell Equations, Washington, D.C. John Wiley and Sons, 1993.

[45] B. Lee, T. Manteuffel, S. McCormick, and J. Ruge. First-order systemleast-squares for Hemholtz equation. Society for Industrial and AppliedMathematics J. Sci. Comput, 21:1927–1949, 2000.

[46] R. Leis. Initial Boundary Value Problems in Mathematical Physiscs. J.Wiley & Teubner Verlag, Stuttgart, 1986.

[47] C. Luke and P. Martin. Fluid-solid interaction: acoustic scattering by asmooth elastic obstacle. SIAM J. Appl. Math., 55(4):904–922, 1995.

[48] C. Makridakis, F. Ihlenburg, and I. M. Babuska. Analysis and finite el-ement methods for a fluid-solid interaction problem in one dimension.Mathematical Models and Methods in Applied Sciences, 6(8):1119–1141,1996.

[49] J. Mandel. An iterative substructuring method for coupled fluid-solidacoustic problems. J. of Comp. Phys., 177:95–116, 2002.

[50] J. Mandel. Iterative substructuring with Lagrange multipliers for coupledfluid-solid scattering. Invited Lecture 14th International Conference onDomain Decomposition, March 2002.

[51] J. Mandel, S. McCormick, and R. Bank. Variational multigrid theory. InMultigrid Methods, volume 5, pages 131–177. SIAM, Frontiers of AppliedMathematics, Philadelphia, 1987.

[52] J. Mandel and M. Popa. A multigrid method for elastic scattering.UCD/CCM Report 109, Center for Computational Mathematics, Univer-sity of Colorado at Denver, September 1997.

[53] J. E. Marsden and T. J. Hughes. Mathematical Foundations of Elasticity.Dover, New York, 1994.

[54] W. McLean. Strongly elliptic systems and boundary integral equations.University Press, Cambridge, 2000.

[55] R. Ohayon and C. Soize. Structural acoustics and vibration: Mechanical

122

models, variational formulations and discretization. Academic Press, SanDiego, 1999.

[56] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS PublishingCompany, Boston, 1996.

[57] Y. Saad and M. H. Schultz. Gmres: A generalized minimal residual algo-rithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Com-put., 7:856–869, 1986.

[58] J. Sanchez Hubert and E. Sanchez-Palencia. Vibration and Couplingof Continuous Systems. Springer-Verlag, Berlin, Heidelberg, New York,1989.

[59] R. Tezaur, A. Macedo, and C. Farhat. Iterative solution of large-scaleacoustic scattering problems with multiple right hand sides by a domaindecompositiob method with lagrange multipliers. International Journalfor Numerical Methods in Engineering, 51:1175–1193, 2001.

[60] R. Tezaur, A. Macedo, C. Farhat, and R. Djellouli. Three-dimensionalfinite element calculations in acoustic scattering using arbitrarily shapedconvex artificial boundaries. International Journal for Numerical Methodsin Engineering, 53:1461–1476, 2002.

[61] H. Triebel. Interpolation theory, function spaces, differential operators.Johannes Ambrosius Barth, Heidelberg, second edition, 1995.

[62] P. Vanek, J. Mandel, and M. Brezina. Solving a two-dimensionalHelmholtz problem by algebraic multigrid. UCD/CCM Report 110, Cen-ter for Computational Mathematics, University of Colorado at Denver,October 1997.http://www-math.cudenver.edu/ccmreports/rep110.ps.gz.

[63] P. Vanek, J. Mandel, and M. Brezina. Two-level algebraic multigrid forthe Helmholtz problem. Contemporary Mathematics, 218, 1998.

[64] V. Varadan. Elastic wave scattering. In V. Varadan and V. Varadan,editors, Acoustic, Electromagnetic, and Elastic Wave Scattering - Focuson the T-Matrix Approach; Int. Symp. held at The Ohio State Univ. Perg-amon Press, New York, Oxford, Toronto, Sydney, Frankfurt, Paris, June25-27 1979.

123

[65] V. Varadan and V. Varadan. Acoustic, electromagnetic, and elastic wavescattering. In V. Varadan and V. Varadan, editors, Field Representationsand Introduction to Scattering. North-Holland, Amsterdam, 1991.

[66] M. www.mgnet.org.

[67] E. C. Zachmanoglou and D. W. Thoe. Introduction to partial differentialequations with applications. Dover, New York, 1976.

[68] X. Zeng and J. Bielak. Stable symmetric finite element boundary integralcoupling methods for fluid structure interface problems. Eng. Anal. withBoundary Elements, 15:79–91, 1995.

[69] X. Zeng, J. Bielak, and R. C. MacCamy. Stable variational couplingmethod for fluid-structure interaction in semi-infinite media. J. of Vibra-tion and Acoustics, 114:387–396, 1992.

[70] X. Zeng and F. Zhao. A coupled FE and boundary integral equationmethod based on exterior domain decomposition for fluid-structure inter-face problems. Int. J. Solids Structures, 31:1047–1061, 1994.

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FINITE ELEMENT SOLUTION OF SCATTERING by …math.ucdenver.edu/graduate/thesis/popa_thesis.pdfFINITE ELEMENT SOLUTION OF SCATTERING IN COUPLED FLUID-SOLID SYSTEMS by Mirela O. Popa