A Posteriori Error Estimates and a Local Refinement Strategy for a Finite Element Method to Solve Structural-Acoustic Vibration Problems

Advances in Computational Mathematics 15: 25–59, 2001. 2002 Kluwer Academic Publishers. Printed in the Netherlands.

A posteriori error estimates and a local refinement strategyfor a finite element method to solve structural-acoustic

vibration problems

Ana Alonso a, Anahí Dello Russo a,∗, Claudio Padra b,∗∗ and Rodolfo Rodríguez c,∗∗∗a Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata,

C.C. 172, 1900 La Plata, ArgentinaE-mail: {ana,anahi}@mate.unlp.edu.ar

b Centro Atómico Bariloche, 8400 Bariloche, Río Negro, ArgentinaE-mail: [email protected]

c Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, ChileE-mail: [email protected]

Received 5 February 2001Communicated by Y. Xu

This paper deals with an adaptive technique to compute structural-acoustic vibrationmodes. It is based on an a posteriori error estimator for a finite element method free of spu-rious or circulation nonzero-frequency modes. The estimator is shown to be equivalent, up tohigher order terms, to the approximate eigenfunction error, measured in a useful norm; more-over, the equivalence constants are independent of the corresponding eigenvalue, the physicalparameters, and the mesh size. This a posteriori error estimator yields global upper and locallower bounds for the error and, thus, it may be used to design adaptive algorithms. We proposea local refinement strategy based on this estimator and present a numerical test to assess theefficiency of this technique.

Keywords: fluid–structure interaction, displacement formulation, spurious-modes free FEM,adaptive mesh refinement

AMS subject classification: 65N25, 65N30

1. Introduction

Considerable interest has been recently shown in problems involving fluid–structure interactions. For a survey of current results see the monographs [18,19, andreferences therein].

∗ Member of CIC, Provincia de Buenos Aires, Argentina.∗∗ Partially supported by FONDECYT 7.990.075, Chile.∗∗∗ Corresponding author. Partially supported by FONDECYT 1.990.346 and FONDAP in AppliedMathematics, Chile.

26 A. Alonso et al. / Refinement strategy for structural-acoustic vibration problems

This paper deals with the elastoacoustic vibration problem, which consists of deter-mining the vibration modes of an elastic solid containing a compressible fluid. Differentformulations have been proposed to solve this problem (see [17,20,27], for instance).One of them, the pure displacement formulation, has one attractive feature: it leads toa simple eigenvalue problem involving sparse symmetric matrices. However, nonzero-frequency spurious modes without physical meaning arise when standard finite elementmethods are used to solve it. The presence of these spurious modes has been alreadypointed out long time ago in [16].

An approach to avoid this drawback has been introduced in [6,10]. It consists ofusing lowest-order Raviart–Thomas elements for the fluid displacements and standardCourant elements for those of the solid, both coupled in a weak way across the fluid–solid interface. This method has been successfully extended to deal with, for instance,three-dimensional problems [9], incompressible fluids [7], dissipative acoustics [8], etc.

However, many questions are still open from the computational point of view. Theneed of using adequately refined meshes to take care of the eigenfunctions singularities isvery likely among the most relevant ones. In fact, usually, either the structure or the fluiddomain have reentrant corners and, then, strong singularities arise in their vicinities. Ifthe used meshes were not correctly refined there, much more computational effort wouldbe necessary in order to obtain a numerical solution within a prescribed error tolerance.In general, it is not possible to a priori find an adequate mesh. Instead, a posteriorilocal error estimators are typically used to know where a coarse initial mesh needs tobe refined (see the monograph [25] as a general reference, which includes an extensivebibliography; see also [26] for local upper bounds of the error).

A residual-type error estimator for the lowest-order Raviart–Thomas approxima-tion of the mixed formulation of second-order elliptic spectral problems has been re-cently introduced and analyzed in [13]. An extension to acoustic problems has beengiven in [3], whereas an estimator for structural vibration problems and its applicationto fluid–structure interaction problems have been considered in [2].

In this paper we consider an a posteriori error estimator for the method in [10] tocompute the vibration modes of an elastoacoustic coupled system. We prove that thisestimator is equivalent to an adequate norm of the error, up to higher order terms; moreprecisely, the estimator yields global upper and local lower bounds for the error. Wecombine this estimator with a refinement strategy to design an adaptive scheme, andassess the efficiency of this procedure by means of a numerical test.

The outline of the paper is as follows. In section 2 we introduce the spectral cou-pled problem to be considered and the finite element method. In section 3 we introducethe a posteriori error estimator and prove that it is equivalent to the energy norm of theerror up to higher order terms. We do this in three steps. First, in section 3.1, we proveglobal upper bounds for the error on each media in terms of the estimator and some re-maining terms. Then, in section 3.2, we prove that these terms are of higher order. Wecomplete the equivalence by proving local lower error estimates in section 3.3. Finally,in section 4, we propose an adaptive mesh refinement technique and report the obtainednumerical results.

A. Alonso et al. / Refinement strategy for structural-acoustic vibration problems 27

Figure 1. Fluid and solid domains.

2. The fluid–structure vibration problem

We consider the problem of determining the free vibration modes of a linear elasticstructure containing an ideal acoustic (barotropic, inviscid, and compressible) fluid. Ourmodel problem consists of a 2D polygonal vessel completely filled with fluid as that infigure 1.

Let �F and �S denote the domains occupied by fluid and solid, respectively, �I theinterface between both media, and n its unit normal vector pointing outwards �F. Theexterior boundary of the solid is the union of �D and �N: the structure being fixed along�D and free of stress along �N; we assume |�D| > 0. Finally, nS denotes the unit outwardnormal vector along �N.

Throughout this paper we use standard notation for Sobolev spaces, norms, andseminorms. We also denote H1

�D(�S) the closed subspace of functions in H1(�S) van-

ishing on �D, and H(div,�F) := {u ∈ L2(�F)2: div u ∈ L2(�F)}, endowed with its

corresponding norm defined by ‖u‖2div,�F

:= ‖u‖20,�F

+ ‖div u‖20,�F

.We use the following notation for the physical magnitudes in the fluid:

• u: the displacement field,

• p: the pressure,

• ρF: the density,

• c: the acoustic speed,

and in the solid:

• w: the displacement field,

• ρS: the density,

• λS and µS: the Lamé coefficients, which are defined by

λS := νSES

(1 − 2νS)(1 + νS)and µS := ES

2(1 + νS),

where ES > 0 is the Young modulus and νS ∈ (0, 1/2) the Poisson ratio;


• ε(w): the strain tensor defined by

εij (w) := 1

2

(∂wi

∂xj+ ∂wj

∂xi

), i, j = 1, 2,

• σ (w): the stress tensor which is related to the strain tensor by Hooke’s law:

σij (w) := λS

2∑k=1

εkk(w)δij + 2µSεij (w), i, j = 1, 2.

The classical elastoacoustics approximation for small amplitude motions yieldsthe following eigenvalue problem for the free vibration modes of the coupled systemand their corresponding frequencies ω (see, for instance, [18]):

Problem 2.1. Find ω > 0, u ∈ H(div,�F), w ∈ H1(�S)2 and p ∈ H1(�F), (u,w, p) �=

(0, 0, 0), such that

∇p − ω2ρFu = 0 in �F, (2.1)

p + ρFc2 div u = 0 in �F, (2.2)

div[σ (w)

] + ω2ρSw = 0 in �S, (2.3)

σ (w)n + pn = 0 on �I, (2.4)

w · n − u · n = 0 on �I, (2.5)

σ (w)nS = 0 on �N, (2.6)

w = 0 on �D. (2.7)

The coupling between fluid and structure is taken into account by equations (2.4)and (2.5). The first one relates normal stresses of the solid on the interface with thepressure exerted by the fluid. The second one, called the kinematic constraint, meansthat fluid and solid are in contact at the interface.

In order to obtain a variational formulation for this problem we introduce appro-priate functional spaces. Let

X := H(div,�F) × H1�D(�S)

2,

endowed with the product norm, which we denote ‖ · ‖X . The closed subspace

V := {(u,w) ∈ X : u · n = w · n on �I

}is called the set of admissible coupled fluid–solid displacements.

The following equivalent symmetric variational spectral problem is obtained byintegrating by parts, using (2.2) to eliminate the pressure p in terms of the fluid dis-placement field u, and denoting λ = ω2:


Problem 2.2. Find λ ∈ R and (u,w) ∈ V , (u,w) �= (0, 0), such that∫�F

ρFc2 div u div v +

∫�S

σ (w) : ε(z) = λ

(∫�F

ρFu · v +∫�S

ρSw · z)

∀(v, z) ∈ V.

(2.8)

The following spectral characterization was proved in [6]:

Theorem 2.3. Problem 2.2 has two kinds of solutions:

1. λ0 = 0, with corresponding eigenspace K × {0}, where

K := {u ∈ H(div,�F): div u = 0 in �F and u · n = 0 on �I

}.

2. A sequence of finite-multiplicity eigenvalues λn > 0, n ∈ N, converging to +∞,with corresponding eigenfunctions (un,wn) ∈ V satisfying un = ∇ϕn for someϕn ∈ H1(�F).

The infinite-dimensional eigenspace K × {0} associated to λ0 = 0 consists of purerotational motions which induce neither fluid pressure variations nor structural vibra-tions. They are solution of the present formulation of the coupled problem, because noirrotational constraint was imposed on the fluid displacements allowed in the model.

The second set of eigenfunctions (i.e., those corresponding to λn > 0) is a com-plete orthogonal system of the subspace G of V consisting of those coupled fields beingconservative in the fluid:

G := {(u,w) ∈ V: u = ∇ϕ, ϕ ∈ H1(�F)

}.

In spite of the fact that the rotational eigenfunctions corresponding to λ0 = 0 arenot physically relevant, any suitable numerical approximation of problem 2.2 shouldtake care of them. Otherwise spurious modes might appear. This is the case, for in-stance, when continuous piecewise linear finite elements are used for both, fluid andsolid displacements (see [16]).

A finite element method free of spurious modes was introduced in [6]. It consistsof combining piecewise linear elements for the solid with lowest-order Raviart–Thomaselements for the fluid, coupled in a nonconforming way. This method was further ana-lyzed in [7,24], where optimal a priori error estimates were obtained.

Let {Th} be a family of regular triangulations of �F∪�S (in the sense of a minimumangle condition) such that every triangle of each mesh is completely contained either in�F or in �S, and the end points of �D coincide with vertices of the triangulation. Wedenote by T F

h and T Sh the triangulations induced in �F and �S, respectively.

For each component of the displacements in the solid we use the standard piecewiselinear finite element space

Lh(�S) := {wh ∈ H1(�S): wh|T ∈ P1(T ) ∀T ∈ T S

h

}


and, for the fluid, the Raviart–Thomas space [21]

RTh(�F) := {uh ∈ H(div,�F): uh|T ∈ RT0(T ) ∀T ∈ T F

h

},

where

RT0(T ) := {uh ∈ P1(T )

2: uh(x, y) = (a + bx, c + by), a, b, c ∈ R}.

The degrees of freedom in RTh(�F) are the (constant) values of the normal componentof uh along each edge of the triangulation. The discrete analogue of X is

Xh := RTh(�F) × Lh,�D(�S)2,

with Lh,�D(�S) := {wh ∈ Lh(�S): wh = 0 on �D}.

The conforming finite element spaces V ∩ Xh are too restrictive for this problem,since any pair of displacements in V ∩ Xh has constant normal components along eachwhole edge of the polygonal interface �I. Instead we use as discrete space

Vh :={(uh,wh) ∈ Xh:

∫!

(uh · n − wh · n) = 0 ∀! ⊂ �I

},

where, from now on, ! always denotes an edge of a triangle T ∈ Th. So, we obtain thefollowing discretization of problem 2.2:

Problem 2.4. Find λh ∈ R and (uh,wh) ∈ Vh, (uh,wh) �= (0, 0), such that∫�F

ρFc2 div uh div vh +

∫�S

σ (wh) : ε(zh) = λh

(∫�F

ρFuh · vh +∫�S

ρSwh · zh

)∀(vh, zh) ∈ Vh.

Let us remark that for (uh, vh) ∈ Vh, uh · n and vh · n coincide at the midpointof each edge ! ⊂ �I but, in general, they do not coincide on the whole edge. Hence,Vh �⊂ V and the method turns out to be nonconforming.

The following discrete analogue of theorem 2.3 has been proved in [6]:

Theorem 2.5. Problem 2.4 has two kinds of solutions:

1. λ0 = 0, with corresponding eigenspace Kh × {0}, where

Kh := {uh ∈ RTh(�F): div uh = 0 in �F and uh · n = 0 on �I

}.

2. A set of positive eigenvalues λh, with corresponding eigenfunctions (uh,wh) ∈ Vh,such that uh ∈ K⊥

h (where K⊥h denotes the orthogonal complement of Kh in Vh).

The null space of the discrete problem, Kh×{0}, provides a good approximation ofthat of the continuous one, K×{0}. In fact, K and Kh both consist of fluid displacementsof the form curl ξ , with ξ being constant on each connected component of �I, and withξ ∈ H1(�F) for K and ξ ∈ Lh(�F) for Kh.


Nonexistence of spurious modes and spectral convergence of the solutions of prob-lem 2.4 to those of problem 2.2 have been proved in [6].

Throughout this paper we will restrict our attention to the case of computing an ap-proximation of an eigenpair of problem 2.2 corresponding to a simple eigenvalue λ > 0(i.e., of multiplicity m = 1). Let (u,w) ∈ G be an eigenfunction associated to λ. In thiscase, the results in [6] shows that there is a unique eigenvalue λh > 0 of problem 2.4converging to λ as h goes to zero, and an associated eigenfunction (uh,wh) convergingto (u,w). Optimal order a priori error estimates have been proved in [6,24] including adouble order for the eigenvalues. Finally,

ph := −ρFc2 div uh

provides an approximation to the fluid pressure given by (2.2): p = −ρFc2 div u.

From the point of view of applications, the kinematic constraint∫!(uh ·n−wh ·n) =

0 in the definition of the discrete spaces Vh can be conveniently imposed by means of apiecewise constant Lagrange multiplier (see [7,9]). In fact, let

Ch := {δh ∈ L2(�I): δh|! ∈ P0(!) ∀! ⊂ �I

}(2.9)

and consider the following discrete mixed spectral problem:

Problem 2.6. Find λh ∈ R, (uh,wh) ∈ Xh, and γh ∈ Ch, (uh,wh, γh) �= (0, 0, 0), suchthat ∫

�F

ρFc2 div uh div vh +

∫�S

σ (wh) : ε(zh) +∫�I

γh(vh · n − zh · n)

= λh

(∫�F

ρFuh · vh +∫�S

ρSwh · zh

)∀(vh, zh) ∈ Xh, (2.10)∫

�I

δh(uh · n − wh · n) = 0 ∀δh ∈ Ch. (2.11)

The following lemma shows that this is equivalent to the original discrete eigen-value problem:

Lemma 2.7. (λh, (uh,wh)) ∈ R × Vh is a solution of problem 2.4 if and only if thereexists (a unique) γh ∈ Ch such that

(λh, (uh,wh), γh

) ∈ R × Xh × Ch is a solution ofproblem 2.6.

Proof. Clearly, any solution of (2.10)–(2.11) provides a solution of problem 2.4.The converse is also true. In fact, let λh and (uh,wh) be an eigenpair of prob-

lem 2.4. Equation (2.11) is satisfied since (uh,wh) ∈ Vh, whereas equation (2.10)is also true for (vh, zh) ∈ Vh independently of the particular value of γh ∈ Ch. Letϕ! denote the nodal basis functions of RTh(�F) associated with the edge !. ThenXh = Vh ⊕ 〈{(ϕ!, 0): ! ⊂ �I}〉. Thus, it is enough to prove that there exists a uniqueγh ∈ Ch such that (2.10) holds for (vh, zh) = (ϕ!, 0) ∀! ⊂ �I. To prove this, let


{χ!: ! ⊂ �I} denote the canonical basis of Ch (i.e., χ!|! ≡ 1 and χ!|�I\! ≡ 0). Then, bywriting the unknown multiplier in this basis, γh = ∑

!⊂�Ic!χ!, it is clear that it is enough

to verify that there are unique coefficients c! such that (2.10) holds for (vh, zh) = (ϕ!, 0)∀! ⊂ �I; that is

|!|c! =∑!′⊂�I

(∫�I

χ!′ϕ! · n)c!′ = λh

∫�F

ρFuh · ϕ! −∫�F

ρFc2 div uh divϕ! ∀! ⊂ �I.

Since these equations are uniquely solvable, we conclude the proof. �

Notice that problem 2.6 can also be considered as a conforming discretizationof the following variational mixed eigenvalue problem, posed on the spaces Y :={(u,w) ∈ X : u · n|�I ∈ L2(�I)}, endowed with the norm defined by ‖(u,w)‖2

Y :=‖(u,w)‖2

X + ‖u · n‖2L2(�I)

, and C := L2(�I):

Problem 2.8. Find λ ∈ R, (u,w) ∈ Y , and γ ∈ C, (u,w, γ ) �= (0, 0, 0), such that∫�F

ρFc2 div u div v +

∫�S

σ (w) : ε(z) +∫�I

γ (v · n − z · n)

= λ

(∫�F

ρFu · v +∫�S

ρSw · z)

∀(v, z) ∈ Y, (2.12)∫�I

δ(u · n − w · n) = 0 ∀δ ∈ C. (2.13)

Proceeding as in [7, section 8], it can be shown that this is equivalent to prob-lem 2.2, with the Lagrange multiplier γ being the pressure at the fluid–solid interfacep|�I . Thus, γh can be seen as an approximation of this interface pressure.

3. A posteriori error estimator

From the point of view of applications, it is highly important to be able to designmeshes correctly refined to reduce the approximation errors as much as possible withthe lowest computational effort.

The standard approach to attain this goal is to compute an approximation of theeigenpair of interest on an initial coarse mesh Th and to use the obtained approximateeigenpair to compute indicators of some local measure of the error for each elementT ∈ Th, to know which of them should be further refined.

We choose a weighted H(div,�F) × H1(�S)2 norm on X to measure the error of

the computed eigenfunction (uh,wh):

ρFc2‖u − uh‖div,�F + 2µS‖w − wh‖1,�S .

The weight in the H(div,�F) norm is chosen to control the error of the approximatepressure in L2(�F) norm: ‖p − ph‖0,�F = ‖ρFc

2 div (u − uh)‖0,�F . The other weight ischosen to scale accordingly the H1(�S) seminorm of the solid displacements.


In what follows we will define error indicators ηT for each element T ∈ Th. Forthose elements T ∈ T F

h , ηT := ηFT will approximate ρFc

2‖u − uh‖div,T , whereas forelements T ∈ T S

h , ηT := ηST will approximate 2µS‖w − wh‖1,T . These indicators are

expected to satisfy the following properties:

1. Reliability. They should provide an upper estimate of the global error:

ρFc2‖u − uh‖div,�F + 2µS‖w − wh‖1,�S � C

(∑T ∈Th

η2T + h.o.t.

)1/2

, (3.1)

where h.o.t. denotes higher order terms (i.e., terms which becomes negligible incomparison with the other ones in the estimate, when the mesh-size becomes small).

2. Efficiency. They should provide lower error estimates, as local as possible, in orderto indicate which elements should be effectively refined:

ηT � C(ρFc

2‖u − uh‖div,T∩�F+ 2µS‖w − wh‖1,T∩�S

+ h.o.t.), (3.2)

where T is the union of T and a few neighboring elements.

3. Low computational cost. The effective computation of ηT should be inexpensive incomparison with the overall computation of (uh,wh) and λh.

Here and thereafter, C denotes a generic constant, not necessarily the same at each oc-currence, but always independent of the mesh-size. This constant will also be indepen-dent of the physical parameters and the particular approximate eigenvalue, if we do notmention it.

3.1. Definition of the error indicators. Reliability of the error estimates

To define these error indicators, we begin by proving an upper error estimate like(3.1). The following lemma provides a Helmholtz decomposition of the error on thefluid domain and upper estimates for each term of this decomposition:

Lemma 3.1. There exist ϕ ∈ H1(�F) and χ ∈ K such that

u − uh = ∇ϕ + χ,

with

‖∇ϕ‖0,�F �C

[∥∥div (u − uh)∥∥

0,�F+ ∥∥(w − wh) · n

∥∥0,�I

+(∑!⊂�I

|!| ‖uh · n − wh · n‖20,!

)1/2]

and

‖χ‖0,�F � C

( ∑T⊂�F

∑!⊂∂T \�I

|!| ∥∥[uh · t!]!∥∥2

0,!

)1/2

,


where [·]! denotes the jump across the edge !, and t! a vector tangent to !.

Proof. Since u − uh ∈ H(div,�F), the following L2(�F) orthogonal decompositionholds:

u − uh = ∇ϕ + χ,

where ϕ ∈ H1(�F)/R is the solution of the Neumman problem

(ϕ = div (u − uh) in �F,

∂ϕ

∂n= (u − uh) · n on �I,

and χ ∈ K. By applying Lax–Milgram lemma, we have

‖∇ϕ‖0,�F �C[∥∥div (u − uh)

∥∥0,�F

+ ∥∥(u − uh) · n∥∥−1/2,�I

]�C

[∥∥div (u − uh)∥∥

0,�F+ ∥∥(w − wh) · n

∥∥0,�I

+‖wh · n − uh · n‖−1/2,�I

],

the latter because u · n and w · n coincide on �I.To estimate the third term above, we use that

∫!

uh · n = ∫!

wh · n for all ! ⊂ �I.So, given ψ ∈ H1/2(�I), let ψh be its L2 projection onto Ch, then∫

�I

(wh · n − uh · n)ψ =∫�I

(wh · n − uh · n)(ψ − ψh)

�C

(∑!⊂�I

|!| ‖wh · n − uh · n‖20,!

)1/2

‖ψ‖1/2,�I,

where we have used a standard error estimate for the L2 projection and the fact that∑!⊂�I

‖ψ‖21/2,! � ‖ψ‖2

1/2,�I. Hence,

‖wh · n − uh · n‖−1/2,�I� C

(∑!⊂�I

|!| ‖wh · n − uh · n‖20,!

)1/2

.

It remains to estimate ‖χ‖0,�F . Since χ ∈ K, then χ = curl ξ , with ξ ∈ H1(�F)

being constant on each connected component of �I (see, for instance, [14]). Letξ I ∈ H1(�F) be a Clément-type interpolant of ξ taking the same constant values asthis function on each connected component of �I and satisfying∥∥ξ − ξ I

∥∥0,! � C|ξ |1,T |!|1/2 ∀! ⊂ ∂T , T ∈ T F

h ,

where T is the union of the triangles in �F sharing a vertex with T (see [11], for in-stance). Then curl ξ I ∈ Kh ⊂ K, and hence, it is L2(�F) orthogonal to u, uh and ∇ϕ.Therefore, since curl (uh|T ) = 0 ∀T ∈ T F

h , then


‖χ‖20,�F

= −∫�F

uh · curl(ξ − ξ I)

= −∑T⊂�F

∫∂T

uh · t(ξ − ξ I

)

= −1

2

∑T⊂�F

∑!⊂∂T \�I

∫!

[uh · t!]!(ξ − ξ I

)

�C

( ∑T⊂�F

∑!⊂∂T \�I

|!| ∥∥[uh · t!]!∥∥2

0,!

)1/2

|ξ |1,�F .

Hence, since ‖χ‖0,�F = ‖ curl ξ‖0,�F = |ξ |1,�F, we conclude the lemma. �

It is well known that the mixed finite element method of Raviart–Thomas isstrongly related with the nonconforming approximation by Crouzeix–Raviart elements(see [12]). We will use this relation in our arguments to estimate ‖div (u − uh)‖0,�F byfollowing the ideas introduced in [13] for a posteriori error estimates for mixed approx-imation of eigenvalue problems.

Consider the following weak formulation of problem 2.1, which is easily obtainedby using (2.2) and (2.5) to eliminate the fluid displacements (see, for instance, [18,27]):

Problem 3.2. Find λ ∈ R and (p,w) ∈ H1(�F) × H1�D(�S)

2, (p,w) �= (0, 0), such that

1

ρF

∫�F

∇p · ∇q +∫�S

σ (w) : ε(z) −∫�I

p z · n

= λ

(1

ρFc2

∫�F

pq +∫�S

ρSw · z +∫�I

q w · n)

∀(q, z) ∈ H1(�F) × H1�D(�S)

2.

It is simple to show that, for λ > 0, problems 2.2 and 3.2, are equivalent to prob-lem 2.1. Thus, the strictly positive eigenvalues of both problems coincide and the asso-ciated eigenfunctions can be chosen with the same solid displacements w. In such case,if (u,w) and (p,w) are the eigenfunctions associated to λ > 0 of problems 2.2 and 3.2,respectively, then the fluid variables are related by (2.1) and (2.2): p = −ρFc

2 div u andu = 1/(ρFλ)∇p.

We will use in our proofs an approximate solution of problem 3.2 obtained bydiscretizing the fluid pressure by piecewise linear Crouzeix–Raviart elements enrichedwith local cubic bubbles. Let us denote

CRh(�F) := {qh ∈ L2(�F) : qh|T ∈ P1(T ) ⊕ B(T ) ∀T ∈ T F

h ,

qh is continuous at the midpoints of each ! ⊂ ∂T},

where B(T ) is the space of cubic bubble functions defined on T (i.e., cubic polynomialfunctions vanishing on ∂T ). Let


P : L2(�F)−→Qh := {qh ∈ L2(�F): qh|T ∈ P0(T ) ∀T ∈ T F

h

},

P�I : L2(�I)−→ Ch,

with Ch as defined in (2.9), and

,T : L2(T )2 −→ RT0(T ) ∀T ∈ T Fh ,

be the corresponding L2 orthogonal projections. Consider the following discretizationof problem 3.2:

Problem 3.3. Find λh ∈ R and (ph, wh) ∈ CRh(�F) × Lh,�D(�S)2, (ph, wh) �= (0, 0),

such that

1

ρF

∑T⊂�F

∫T

,T ∇(ph|T ) · ,T ∇(qh|T ) +∫�S

σ (wh) : ε(zh) −∫�I

P�I ph zh · n

= λh

(1

ρFc2

∫�F

P ph Pqh +∫�S

ρSwh · zh +∫�I

P�Iqh wh · n)

∀(qh, zh) ∈ CRh(�F) × Lh,�D(�S)2.

The above problem is equivalent to problem 2.4 in a sense made precise in thefollowing lemma, which has been proved in [1] by adapting to this case the argumentsin [4]:

Lemma 3.4. Problems 3.3 and 2.4 attain the same strictly positive eigenvalues λh. Thecorresponding associated eigenfunctions (ph, wh) and (uh,wh) can be chosen such that

wh = wh. (3.3)

Moreover, in such case, it holds that

P ph = −ρFc2 div uh = ph, (3.4)

,T∇(ph|T ) = λhρFuh ∀T ∈ T Fh , (3.5)

and

P�I ph = γh, (3.6)

with γh being the approximate interface pressure obtained from problem 2.6.

Let us remark that there is no need of actually computing ph to calculate the errorindicators defined below. In fact, ph only plays a role in proving an upper estimate like(3.1), since, in the next subsection, several terms like, for instance, ‖p − ph‖0,T areshown to be of a higher order than the error of the approximate eigenfunctions.

Now we are able to give an estimate of the error for ρFc2‖div (u − uh)‖0,�F:


Lemma 3.5. For all T ∈ T Fh ,

ρFc2‖div (u − uh)‖0,T � C

(λhρF|T |1/2‖uh‖0,T + ‖p − ph‖0,T

).

Proof. The proof is essentially contained in that of theorem 3.1 in [3] (see in particular(3.22) in that reference), which in its turn depend only on properties (3.4)–(3.6) in theprevious lemma. �

We are in order to define the error indicators for the elements on the fluid domain.For each T ∈ T F

h , let

ηFT :=

(λ2hρ

2F|T | ‖uh‖2

0,T +∑!⊂∂T

ρ2Fc

4|!| ∥∥JF!

∥∥20,!

)1/2

,

where

JF! :=

{[uh · t!]! if ! �⊂ �I,

uh · n − wh · n if ! ⊂ �I.

Let

ηF :=[ ∑T⊂�F

(ηFT

)2]1/2

.

The following theorem states an upper estimate for the error on the fluid domain interms of ηF plus two more terms which will be proved to be of higher order in the nextsubsection:

Theorem 3.6. The following estimate holds:

ρFc2‖u − uh‖div,�F � C

[ηF + ρFc

2‖(w − wh) · n‖0,�I + ‖p − ph‖0,�F

],

where C only depends on the regularity of the mesh.

Proof. It is an immediate consequence of lemmas 3.1 and 3.5, and the definition of theerror estimator ηF. �

Regarding the structure, the error indicator we are going to use is similar to one forthe standard linear elasticity equations (see [2]). For each T ∈ T S

h , let

ηST :=

(λ2hρ

2S|T | ‖wh‖2

0,T + 1

2

∑!⊂∂T

|!| ∥∥JS!

∥∥20,!

)1/2

,

with

JS! :=

[σ (wh)n!]! if ! �⊂ ∂�S,

0 if ! ⊂ �D,

2σ (wh)nS if ! ⊂ �N,

2[γhn + σ (wh)n

]if ! ⊂ �I,


with n! being a unit vector normal to the edge !. Let

ηS :=[ ∑T⊂�S

(ηST

)2]1/2

.

The following theorem shows that the error on the solid domain can be bounded aboveby this error estimator plus two other terms which will be also proved to be of a higherorder than the error in the next subsection:


2µS‖w − wh‖1,�S

� C

[ηS + ρS‖λw − λhwh‖0,�S + √

2µS

∣∣∣∣∫�I

(γ − γh)(w − wh) · n

∣∣∣∣1/2],

where C only depends on the regularity of the mesh.

Proof. Because of Korn’s inequality, there exists a constant C > 0 such that

2µSC‖w − wh‖21,�S

� 2µS

∫�S

ε(w − wh) : ε(w − wh)

�∫�S

σ (w − wh) : ε(w − wh).

Let ew := w − wh and eLw be its Lagrange interpolant. Then, by using (2.10) with

(vh, zh) = (0, eLw) and (2.12) with (v, z) = (0, ew − eL

w), straightforward computationsyield ∫

�S

σ (w − wh) : ε(w − wh)

=∫�S

σ (w − wh) : ε(ew − eL

w

) +∫�S

σ (w − wh) : ε(eL

w

)=

∫�S

ρS(λw − λhwh) · ew +∫�I

(γ − γh) ew · n +∫�S

ρSλhwh · (ew − eL

w

)−

∫�S

σ (wh) : ε(ew − eL

w

) +∫�I

γh(ew − eL

w

) · n.

Regarding the last two terms, integrating by parts on each triangle T ⊂ �S and usingthe definition of JS

! we obtain

−∫�S

σ (wh) : ε(ew − eL

w

) +∫�I

γh(ew − eL

w

) · n =∑T⊂�S

∑!⊂∂T

∫!

JS! · (

ew − eLw

).


Therefore, Cauchy–Schwarz inequality, standard estimates for the interpolation errorew − eL

w, and the definition of the error estimator ηS lead to

2µS‖w − wh‖21,�S

� C

[ηS|ew|1,�S + ρS‖λw − λhwh‖0,�S‖ew‖1,�S +

∣∣∣∣∫�I

(γ − γh) ew · n

∣∣∣∣].

Hence, a straightforward computation yields the estimate. �

3.2. Analysis of the higher order terms in the reliability estimates

Our next goal is to prove that the terms

ρFc2∥∥(w − wh) · n

∥∥0,�I

and ‖p − ph‖0,�F

in theorem 3.6, as well as

ρS‖λw − λhwh‖0,�S and

∣∣∣∣∫�I

(γ − γh)(w − wh) · n∣∣∣∣1/2

in theorem 3.7, are negligible. Since the results of these two theorems are valid inde-pendently of the normalization used for the eigenfunctions, we will choose a particularconvenient one, and then we will prove that the terms above are of a higher order thanthe errors in the corresponding estimates.

Consider the following seminorm on X :

|(u,w)|X := ∥∥ρFc2 div u

∥∥0,�F

,

which corresponds to the L2(�F) norm of the fluid pressure. We choose this seminormto normalize the eigenfunctions. Namely:

|(u,w)|X = 1 and |(uh,wh)|X = 1. (3.7)

Clearly, | · |X is not a norm on X , but this is not a drawback unless the eigenfunction(u,w) of the continuous problem were divergence free. However, for compressible flu-ids, this never happens in practice. Moreover, this normalization is the standard one inacoustics problems where the fluid pressure is the most relevant field to be computed.

Optimal order error estimates are known for the eigenfunctions normalized withthe natural norm of X . These estimates depend on the following regularity result provedin [6,24]:

Theorem 3.8. Let (u,w) be an eigenfunction of problem 2.2 associated with an eigen-value λ > 0, and p be the corresponding fluid pressure given by (2.2). Then there existconstants s ∈ (1/2, 1], t ∈ (0, 1], and C > 0, such that u ∈ Hs(�F)

2, w ∈ H1+t (�S)2,

p ∈ H1+s(�F), and

‖u‖s,�F + ‖w‖1+t,�S + ‖p‖1+s,�F � C(‖u‖0,�F + ‖w‖0,�S

).


In this theorem, s is either 1, if �F is convex, or any s < π/θ , with θ beingthe largest reentrant corner, otherwise. On the other hand, t depends on the reentrantangles of �S, the angles between �D and �N, and the Lamé coefficients (see [15]).Finally, C depends on the eigenvalue λ. (Throughout this subsection, C will denotedifferent constants always independent of the mesh-size h, but in general depending onthe eigenvalue λ and the physical parameters.)

Let

(u, w) := (u,w)

‖(u,w)‖X and (uh, wh) := (uh,wh)

‖(uh,wh)‖X .

The following error estimates hold (see [6,24]):

Theorem 3.9. There exist strictly positive constants C and h0 such that, if h � h0, then∥∥(u, w) − (uh, wh)∥∥X � Chr

and

|λ − λh| � Ch2r ,

where r := min{s, t}, with s and t being the regularity constants in theorem 3.8.

The following lemma shows that similar estimates are valid for the eigenfunctionsnormalized by (3.7):

Lemma 3.10. There exist strictly positive constants C and h0 such that, if h � h0, then∥∥(u,w) − (uh,wh)∥∥X � Chr.

Proof. Let us denote

α := (u,w), αh := (uh,wh), α := (u, w), and αh := (uh, wh).

Straightforward computations yield

‖α − αh‖X � ‖α‖X ‖α − αh‖X + ‖αh‖X∣∣ ‖α‖X − ‖αh‖X

∣∣= ‖α‖X

(‖α − αh‖X +

∣∣∣∣1 − ‖αh‖X‖α‖X

∣∣∣∣).

Now, because of (3.7), |α|X = 1/‖α‖X and |αh|X = 1/‖αh‖X . Then∣∣∣∣1 − ‖αh‖X‖α‖X

∣∣∣∣ =∣∣∣∣ |αh|X − |α|X

|αh|X

∣∣∣∣ � ‖αh‖X |α − αh|X � ρFc2‖αh‖X ‖α − αh‖X ,

where we have used that | · |X � ρFc2‖ · ‖X . Therefore,

‖α − αh‖X � ‖α‖X(1 + ρFc

2‖αh‖X)‖α − αh‖X .


Now, since ‖αh‖X � ‖α − αh‖X + ‖α‖X , then we have

‖α − αh‖X � ‖α‖X(1 + ρFc

2‖α‖X)‖α − αh‖X + ρFc

2‖α‖X ‖α − αh‖X ‖α − αh‖X .By virtue of theorem 3.9, there exists a constant C such that ‖α − αh‖X � Chr . Hence,we can choose h0 such that CρFc

2‖α‖Xhr0 < 1/2, and we finally obtain

‖α − αh‖X � 2‖α‖X(1 + ρFc

2‖α‖X)‖α − αh‖X � C ′hr ∀h < h0.

Thus, we conclude the proof. �

The following theorem shows that the approximations γh of the interface pressure,obtained by solving problem 2.6, converge with the same order as that of the displace-ments:

Theorem 3.11. Let (λ, (u,w), γ ) and (λh, (uh,wh), γh) be eigenpairs of problems 2.8and 2.6, respectively, such that λh → λ as h → 0 and the eigenfunctions being chosensatisfy ‖(u,w) − (uh,wh)‖χ � C hr . Then,

‖γ − γh‖0,�I � Chr,

with r as in theorem 3.9.

Proof. The proof is an immediate extension of that of theorem 8.1 in [7]. �

Let (ph,wh) be the eigenfunction of problem 3.3 chosen as in lemma 3.4. Sincewe are considering the normalization given by (3.7) in the continuous problem (whichin the case of problem 3.3 reads ‖p‖0,�F = 1) we use the same norm for (ph,wh). Thus,let

(ph,wh) := (ph,wh)

‖ph‖0,�F

, (3.8)

then ‖ph‖0,�F = 1.It was shown in [1] that ph provides an accurate approximation of the fluid pres-

sure p. In fact, a higher order convergence result in a weaker norm was proved in thatreference. To state this result, which we will use below, we introduce for any ε ∈ (0, 1/2)the space

Hε := L2(�F) × L2(�S)2 × Hε(�I),

endowed with the corresponding product norm which we denote ‖ · ‖Hε. We also denote

β := (p,w,w · n|�I

) ∈ Hε and βh := (ph,wh,wh · n|�I

) ∈ Hε,

as well as

β := (p, w, w · n|�I) := β

‖β‖Hε

and βh := (ph, wh, wh · n|�I) := βh

‖βh‖Hε

,


the corresponding normalized triples.The following higher order convergence result has been proved in [1]:

Theorem 3.12. For each ε ∈ (0, 1/2), there exist strictly positive constants C and h0

such that, if h � h0, then ∥∥β − βh

∥∥Hε

� Chr+rε ,

where r is as in theorem 3.9 and rε := (1 − 2ε)min{1/2, t}, with t as in theorem 3.8.

The following lemma shows that a similar estimate holds for β − βh:

Lemma 3.13. For each ε ∈ (0, 1/2), there exist strictly positive constants C ′ and h0

such that, if h � h0, then ∥∥β − βh

∥∥Hε

� C ′hr+rε ,

with r and rε as in the previous theorem.

Proof. By repeating the arguments in the proof of lemma 3.10, using ‖p‖0,�F insteadof |α|X , and the inequality ‖p‖0,�F � ‖β‖Hε

instead of |α|X � ρFc2‖α‖X , we obtain∥∥β − βh

∥∥Hε

� 2‖β‖Hε

(1 + ‖β‖Hε

)∥∥β − βh

∥∥Hε

� C ′hr+rε ∀h < h0,

for h0 such that C‖β‖Hεhr+rε0 < 1/2, with C being the constant in theorem 3.12. Thus

we conclude the proof. �

The following estimate will be also used:

Lemma 3.14. There exist strictly positive constants C and h0 such that, if h � h0, then∣∣1 − ‖ph‖0,�F

∣∣ � Ch2r ,

with r as above.

Proof. Since ‖ph‖0,�F = ‖ph‖0,�F = 1, we may write

‖ph − ph‖20,�F

= ‖ph‖20,�F

+ ‖ph‖20,�F

− 2∫�F

phph = 2

(1 −

∫�F

phph

).

Then, from (3.4), the fact that ph ∈ Qh, (3.8), and the previous equation, we have

1 =∫�F

|ph|2 =∫�F

P ph ph =∫�F

ph ph = ‖ph‖0,�F

∫�F

ph ph

= ‖ph‖0,�F

(1 − 1

2‖ph − ph‖2

0,�F

).


Now, because of lemmas 3.10 and 3.13 we know that, for h small enough,

‖ph − ph‖0,�F � ‖ph − p‖0,�F + ‖p − ph‖0,�F

� ρFc2∥∥(u,w) − (uh,wh)

∥∥X + ∥∥β − βh

∥∥Hε

� ρFc2Chr + Chr+rε

�C ′hr.

Therefore, if we choose h0 such that C ′hr0 < 1, the previous equation leads to

1 � ‖ph‖0,�F = 1

1 − (1/2)‖ph − ph‖20,�F

� 1 + ‖ph − ph‖20,�F

∀h < h0,

which together with the previous inequality yields the lemma. �

Now we are able to improve theorems 3.6 and 3.7:


ρFc2‖u − uh‖div,�F � C

(ηF + h.o.t.

),

where C only depends on the regularity of the mesh, and

h.o.t. = ρFc2∥∥(w − wh) · n

∥∥0,�I

+ ‖p − ph‖0,�F .

Furthermore, for each ε ∈ (0, 1/2), there exist strictly positive constants C ′ and h0 suchthat

h.o.t. � C ′hr+rε ∀h < h0,

with r and rε as in theorem 3.12.

Proof. From (3.3) and (3.8) we have that

h.o.t. � ρFc2∥∥(w − wh) · n

∥∥0,�I

+ ‖p − ph‖0,�F

+ ρFc2∥∥(wh − wh) · n

∥∥0,�I

+ ‖p − ph‖0,�F

� ρFc2∥∥β − βh

∥∥Hε

+ ρFc2∣∣1 − ‖ph‖0,�F

∣∣∥∥βh

∥∥Hε

.

Therefore, since ∥∥βh

∥∥Hε

� ‖β‖Hε+ ∥∥β − βh

∥∥Hε

,

we conclude the theorem from lemmas 3.13 and 3.14, and the fact that r � rε . �


2µS‖w − wh‖1,�S � C(ηS + h.o.t.

),


where C only depends on the regularity of the mesh, and

h.o.t. = ρS‖λw − λhwh‖0,�S + √2µS

∣∣∣∣∫�I

(γ − γh)(w − wh

) · n

∣∣∣∣1/2

.

Furthermore, for each ε ∈ (0, 1/2), there exist strictly positive constants C ′ and h0 suchthat

h.o.t. � C ′hr+rε/2 ∀h < h0,

with r and rε as in theorem 3.12.

Proof. From (3.3) and (3.8) we have now

ρS‖λw − λhwh‖0,�S � ρS|λ − λh|‖wh‖0,�S

+ λρS‖β − βh‖Hε+ λρS

∣∣1 − ‖ph‖0,�F

∣∣‖βh‖Hε.

Therefore, since

‖wh‖0,�S �∥∥(u,w)

∥∥X + ∥∥(u,w) − (uh,wh)

∥∥X ,

from lemma 3.10 and theorem 3.9, by using lemmas 3.13 and 3.14 as in the previoustheorem, we obtain

ρS‖λw − λhwh‖0,�S � Chr+rε .

On the other hand, from Cauchy–Schwartz inequality and theorem 3.11, by using againlemmas 3.13 and 3.14 as before, we have∣∣∣∣

∫�I

(γ − γh)(w − wh) · n

∣∣∣∣ � Ch2r+rε .

Thus, the theorem follows from the two previous inequalities. �

Remark 3.17. Since the previous theorems are valid for arbitrary small ε > 0, thenrε ≈ r, if t � 1/2 (i.e., if the singularities of the structure displacements are such thatw /∈ H

3/2(�S)

2). Otherwise, rε ≈ 1/2. In any case, the h.o.t. are actually of higher orderthan the error estimates given by theorem 3.9.

3.3. Efficiency of the error indicators

In order to prove that the error indicators defined above provide local lower errorestimates satisfying (3.2), we will use the following lemmas:

Lemma 3.18. For each T ∈ T Fh ,

λhρF|T |1/2‖uh‖0,T � C[ρFc

2∥∥div (u − uh)

∥∥0,T + ρF‖λu − λhuh‖0,T |T |1/2].


Proof. It is essentially contained in that of theorem 3.2 in [3] (see in particular (3.31)in that reference). �

Lemma 3.19. For each edge ! such that ! = T1 ∩ T2, with T1, T2 ∈ T Fh ,

|!|1/2∥∥[uh · t!]!

∥∥0,! � C‖u − uh‖0,T1∪T2 .

Proof. It is essentially part of theorem 3.2 in [13]. �

Lemma 3.20. For each edge ! ⊂ �I, let TF ∈ T Fh and TS ∈ T S

h such that TF ∩ TS = !.Then

|!|1/2‖uh · n − wh · n‖0,! � C(‖u − uh‖div,TF + ‖w − wh‖1,TS

).

Proof. It is simple to show that there is a unique ψ ∈ P3(TF) vanishing on the twoedges !′ �= ! of TF and satisfying∫

!

ψφ =∫!

(uh · n − wh · n) φ ∀φ ∈ P1(!),∫TF

ψ = 0.

Furthermore, standard homogeneity arguments yield

‖ψ‖0,TF �C|!|1/2‖uh · n − wh · n‖0,!,

|ψ |1,TF �C|!|−1/2‖uh · n − wh · n‖0,!.

Then, by taking φ = (uh · n − wh · n) ∈ P1(!) and using that u · n = w · n on !, we have

‖uh · n − wh · n‖20,!

= −∫!

(u − uh) · nψ +∫!

(w − wh) · nψ

= −∫TF

div (u − uh) ψ −∫TF

(u − uh) · ∇ψ +∫!

(w − wh) · nψ

�∥∥div (u − uh)

∥∥0,TF

‖ψ‖0,TF + ‖u − uh‖0,TF|ψ |1,TF + ‖w − wh‖0,!‖ψ‖0,!

� C ‖uh · n − wh · n‖0,![|!|1/2

∥∥div (u − uh)∥∥

0,TF+ |!|−1/2‖u − uh‖0,TF

+ |!|1/2|w − wh|1,TS + |!|−1/2‖w − wh‖0,TS

],

where we have used the estimates above for ‖ψ‖0,TF and |ψ |1,TF , and a standard localtrace inequality to estimate ‖w − wh‖0,! and ‖ψ‖0,!. Therefore,

|!|1/2‖uh · n − wh · n‖0,! �C[‖u − uh‖0,TF + |!| ∥∥div (u − uh)

∥∥0,TF

+ ‖w − wh‖0,TS + |!| |w − wh|1,TS

],

and we conclude the lemma. �


Lemma 3.21. For each T ∈ T Sh ,

λhρS|T |1/2‖wh‖0,T � C

[2µS

1 − 2νS|w − wh|1,T + ρS‖λw − λhwh‖0,T |T |1/2

].

Proof. Let ψ := whbT , with bT ∈ B(T ) a cubic bubble scaled as to satisfy∫T

wh · ψ =∫T

|wh|2bT = λh‖wh‖20,T |T |.

Then, standard homogeneity arguments yield:

‖ψ‖0,T �Cλh‖wh‖0,T |T |,|ψ|1,T �Cλh‖wh‖0,T |T |1/2.

Since ψ vanishes on ∂T , we have∫T

σ (wh) : ε(ψ) =∫∂T

σ (wh)n! · ψ = 0,

whereas, extending ψ by zero outside of T and using (2.8) with (0,ψ) ∈ V , we obtain∫T

σ (w) : ε(ψ) = λ

∫T

ρSw · ψ .

So, as a consequence of all this, we have

λ2hρS|T |‖wh‖2

0,T

= −∫T

ρS(λw − λhwh) · ψ +∫T

σ (w − wh) : ε(ψ)

� Cλh‖wh‖0,T |T |1/2

[ρS‖λw − λhwh‖0,T |T |1/2 + 2µS

1 − 2νS|w − wh|1,T

],

which allows us to conclude the lemma. �

Lemma 3.22. For each edge ! such that ! = T1 ∩ T2, with T1, T2 ∈ T Sh ,

|!|1/2∥∥[σ (wh)n!]!

∥∥0,!

� C

[2µS

1 − 2νS|w − wh|1,T1∪T2

+2∑

i=1

ρS‖λw − λhwh‖0,Ti |Ti|1/2

].

Proof. For JS! = [σ (wh)n!]!, let ψ! ∈ H1

0(T1 ∪ T2)2 be such that∫

!

JS! · ψ! = |!|∥∥JS

!

∥∥20,!,∫

Ti

φ · ψ! = 0 ∀φ ∈ P1(Ti)2, i = 1, 2.


The function ψ! can be taken as a continuous piecewise quadratic polynomial augmentedwith local bubbles of degree four. Standard homogeneity arguments yield:

‖ψ!‖0,Ti � C|!|1/2∥∥JS

!

∥∥0,!|Ti|1/2, i = 1, 2,

|ψ!|1,Ti � C|!|1/2∥∥JS

!

∥∥0,!, i = 1, 2.

Extending ψ! by zero outside of T1 ∪ T2 and using (2.8) with (0,ψ!) ∈ V , we obtain∫T1∪T2

σ (w) : ε(ψ!) = λ

∫T1∪T2

ρS w · ψ!,

whereas, by integrating by parts on each triangle, we have∫T1∪T2

σ (wh) : ε(ψ!) =∫!

JS! · ψ!.

Hence,

|!|∥∥JS!

∥∥20,!

= −∫T1∪T2

σ (w − wh) : ε(ψ!) +∫T1∪T2

ρS (λw − λhwh) · ψ!

� C|!|1/2∥∥JS

!

∥∥0,!

[2µS

1 − 2νS|w − wh|1,T1∪T2

+2∑

i=1

ρS‖λw − λhwh‖0,Ti |Ti|1/2

].


Lemma 3.23. For each edge ! of a triangle T ∈ T Sh such that ! ⊂ �N,

|!|1/2∥∥σ (wh)nS

∥∥0,! � C

[2µS

1 − 2νS|w − wh|1,T + ρS‖λw − λhwh‖0,T |T |1/2

].

Proof. The lemma can be proved with an argument like that used in lemma 3.22, withobvious modifications for ! ⊂ �N. �

Lemma 3.24. For each edge ! ⊂ �I, let TF ∈ T Fh and TS ∈ T S

h be such that TF ∩TS = !.Then

|!|1/2∥∥γhn + σ (wh)n

∥∥0,! �C

[ρFc


∥∥0,TF

+ 2µS

1 − 2νS|w − wh|1,TS

+ ρS‖λw − λhwh‖0,TS|TS|1/2 + ρF‖λu − λhuh‖0,TF

|TF|1/2

].


Proof. Let ψ! ∈ H10(TF ∪ TS)

2 be chosen as in the proof of lemma 3.22, with T1 =TF and T2 = TS. The arguments of that proof apply with obvious modifications. Inparticular, integrating by parts on TF and TS, and using that now JS

! = 2[γhn + σ (wh)n],we obtain∫

TF

ρFc2div uhdivψ! +

∫TS

σ (wh) : ε(ψ!) = −1

2

∫!

JS! ·ψ! +

∫!

(γh − ph)ψ! · n. (3.9)

The second term in the right-hand side of this equation can be bounded as follows:∫!

(ph − γh)ψ! · n = (ph|! − γh|!

) ∫TF

divψ!

�∣∣ph|! − γh|!

∣∣ |TF|1/2|ψ!|1,TF,

since(ph|! − γh|!

) ∈ P0(!). So, since ph = P ph and γh = P�I ph, we have

ph|! − γh|! = 1

|!|∫!

(P ph − ph)

�C|ph|1,TF � CλhρF‖uh‖0,TF,

where the last inequality can be proved with arguments identical to those in lemma 3.18.Hence, ∣∣∣∣

∫!

(ph − γh)ψ! · n

∣∣∣∣ � CλhρF‖uh‖0,TF |TF|1/2|ψ!|1,TF .

Therefore, proceeding as in the proof of lemma 3.22 we obtain from (3.9)

1

2|!|∥∥JS

!

∥∥20,! =

∫!

(γh − ph)ψ! · n +∫TF

ρFc2div (u − uh)divψ!

−∫TF

ρF (λu − λhuh) · ψ! +∫TS

σ (w − wh) : ε(ψ!)

−∫TS

ρS (λw − λhwh) · ψ!

�C|!|1/2∥∥JS

!

∥∥0,!

[ρFc

2‖div (u − uh)‖0,TF+ 2µS

1 − 2νS|w − wh|1,TS

+ ρS‖λw − λhwh‖0,TS|TS|1/2 + ρF‖λu − λhuh‖0,TF

|TF|1/2

].


As a direct consequence of all the previous lemmas we have the following theorem:

Theorem 3.25. For all T ∈ Th, let T := ⋃ {T ′ ∈ Th: T ′ shares an edge with T }. Thefollowing holds:


1. For T ⊂ �F, if ρFc2 � 2µS, then

ηFT �C

[ρFc

2‖u − uh‖0,T∩�F+ ρFc


∥∥0,T

+ 2µS‖w − wh‖1,T∩�S+ ρF‖λu − λhuh‖0,T |T |1/2

],

where the constant C only depends on the regularity of the mesh.

2. For T ⊂ �S, if 0 < νS � ν0, with ν0 < 1/2, then

ηST �C

[2µS|w − wh|1,T∩�S

+ ρFc2∥∥div (u − uh)

∥∥0,T∩�F

+∑

T ′⊂T∩�S

ρS‖λw − λhwh‖0,T ′ |T ′|1/2

+ ρF‖λu − λhuh‖0,T∩�F

∣∣T ∩ �F

∣∣1/2],

where the constant C only depends on the regularity of the mesh.

Remark 3.26. The previous theorem shows that the local estimators satisfy (3.2), wherethe higher order terms are

h.o.t. = ρF‖λu − λhuh‖0,T |T |1/2 if T ⊂ �F

and

h.o.t. =∑

T ′⊂T∩�S

ρS‖λw − λhwh‖0,T ′∣∣T ′∣∣1/2

+ ρF‖λu − λhuh‖0,T∩�F

∣∣T ∩ �F

∣∣1/2if T ⊂ �S.

In the first case, the h.o.t. can be bounded as follows:

ρF‖λu − λhuh‖0,T |T |1/2 � ρF|λ| ‖u − uh‖0,T |T |1/2 + ρF|λ − λh| ‖uh‖0,T |T |1/2

� |T |1/2 λ

c2

(ρFc

2‖u − uh‖0,T∩�F

) + |λ − λh|λh

ηFT .

The first term in the right-hand side is of higher order than the estimated local error,since it is equal to part of this error times |T |1/2λ/c2. The second term is of higher orderthan the local estimator ηF

T , according to theorem 3.9.In the second case the higher order terms are not so local, since they involve norms

on T instead of T . Anyway, both terms can be dealt with as above: they split into twoparts, one of higher order than some local error and the other of higher order than thesum of the estimators corresponding to elements T ′ sharing an edge with T .

Remark 3.27. Two assumptions on the physical parameters have been made in theo-rem 3.25 to obtain lower estimates with constants only depending on the regularity ofthe mesh: ρFc

2 � 2µS and νS � ν0 < 1/2. None of them is restrictive in practice.Regarding the first one, in most cases of interest ρFc

2 is much smaller than 2µS (for


instance, for air–steel interaction, ρFc2/(2µS) ≈ 10−6, while for water–steel interaction,

ρFc2/(2µS) ≈ 10−2). The second assumption just means that the elastic solid material

is not nearly-incompressible.

4. Numerical results

We report in this section the results of some numerical computations which showhow the performance of the previously described finite element method can be remark-ably improved by a mesh refinement process based on the error indicator ηT .

The process starts with a quasi-uniform triangulation T0. At step k, a new meshTk+1 is obtained from Tk by computing the error indicators ηT ∀T ∈ Tk, and refiningthose elements with

ηT � β maxT ′∈Tk

ηT ′,

where β ∈ (0, 1) is a given refinement parameter. This strategy is based on the heuristicargument that a mesh is optimal when the error is equally distributed throughout its ele-ments (see [5,22]). We have followed the “longest edge” refinement procedure describedin [23] which guarantees that, for every k, the minimum angle of Tk is not smaller thanone half the minimum angle of T0. In our experiments we have taken β = 0.5 and wehave started the process from a very coarse uniform mesh (see figure 7).

As a test problem we have computed the natural vibration modes of a compressiblefluid contained into an elastic L-shaped cavity. The geometrical data can be seen infigure 2.

Figure 2. Geometrical data of the domains.


Table 1Vibration frequencies computed on uniformly refined meshes.

Mode N = 20224 N = 44928 N = 79360 order ωex

S1 446.5056 443.1521 441.6671 0.67 438.5072S2 988.4996 981.9213 978.9936 0.66 972.7157S3 1351.6052 1342.1216 1337.9079 0.67 1328.8590F1 1617.2572 1615.2832 1614.3258 0.54 1611.7293S4 2080.0428 2068.2220 2063.0377 0.69 2052.3072S5 2347.2062 2343.6824 2342.0873 0.64 2338.5034F2 2740.9260 2737.9373 2736.6192 0.68 2733.8383

We have used the following physical parameters for the solid:

• density: ρS = 7700 kg/m3,

• Young modulus: ES = 1.44 · 1011 Pa,

• Poisson coefficient: νS = 0.35,

which corresponds to steel, and for the fluid:

• density: ρF = 1000 kg/m3,

• sound speed: c = 1430 m/s,

which corresponds to water in normal conditions of temperature and pressure.In table 1 we present the vibration frequencies ω corresponding to the lowest-

frequency vibration modes computed on several embedded uniform meshes. The pa-rameter N denotes the number of degrees of freedom (d.o.f.) of each mesh. Since noanalytical expression is known, the computed vibration frequencies have been extrapo-lated to obtain highly accurate approximations ωex of the exact ones. This extrapolationtechnique has also been used to estimate the order of convergence in powers of 1/N .

The orders of convergence are much smaller than 1, which is the order predictedby the theory for regular eigenfunctions when uniform meshes are used. This agreeswith the fact that �S and �F have reentrant corners and, thus, eigenfunctions with sin-gularities should be expected. In fact, according to [15], the constants in theorem 3.8 ares ≈ 0.67 and t ≈ 0.54. Therefore, since for uniform meshes h ≈ C/

√N , theorem 3.9

yields for this test

|λ − λh| � C

N0.54.

It can be seen from table 1 that the computed orders of convergence are all in the rangepredicted by the theory, although only that of the vibration mode F1 is very close to thetheoretical bound.

Figures 3–6 show the deformed structure and the fluid displacement field corre-sponding to the first four vibration modes in table 1.

We present now the results obtained with the adaptive mesh-refinement techniquedescribed above. Figure 7 shows the initial mesh we have used.


Figure 3. Deformed structure and fluid displacement field for the vibration mode S1.



Table 2 shows the computed vibration frequencies of the first three vibration modesS1, S2, and S3, obtained after eight steps of the refinement process. This table alsoincludes the orders of convergence estimated by means of a least squares error fitting ofthe last six computed vibration frequencies for each mode. These orders are now close


Figure 6. Deformed structure and fluid displacement field for the vibration mode F1.

Figure 7. Initial mesh for the adaptive scheme.

Table 2Vibration frequencies computed on adaptively refined meshes for S1, S2, and S3.

S1 S2 S3

k N ωh |ωex − ωh| N ωh |ωex − ωh| N ωh |ωex − ωh|0 400 569.285 130.778 400 1202.026 229.311 400 1678.228 349.3691 592 494.916 56.409 563 1091.803 119.087 453 1544.089 215.2302 703 478.543 40.036 746 1055.906 83.191 603 1456.997 128.1383 1116 464.710 26.203 1170 1021.096 48.380 945 1404.947 76.0884 1911 453.130 14.622 2005 1001.151 28.436 1185 1383.194 54.3355 2500 448.643 10.136 2858 990.592 17.876 1543 1367.670 38.8116 3513 445.827 7.320 3941 985.315 12.599 2258 1353.802 24.9437 5119 443.347 4.840 5424 981.641 8.925 2887 1346.937 18.0788 7463 441.463 2.956 7696 978.455 5.739 4235 1340.316 11.457

order = 1.13 order = 1.14 order = 1.25


Table 3Vibration frequencies computed on adap-

tively refined meshes for F1

F1

k N ωh |ωex − ωh|0 400 1630.1086 18.37931 582 1626.5905 14.86122 614 1616.4748 4.74553 819 1608.6195 3.10984 1345 1612.6325 0.90325 1830 1608.8417 2.88766 2588 1609.5408 2.18857 3758 1611.7988 0.06948 7403 1611.5525 0.1768

order = 1.52

Figure 8. Error curves of the adaptive process for modes S1, S2, S3, and F1.

to 1 (in fact, they are even larger than 1). Thus, the optimal order of convergence wasattained for these vibration modes.

It can be observed by comparing tables 1 and 2, how significant is the reductionof the computational effort needed to obtain a solution within a prescribed accuracy.


Figure 9. Some refined triangulations for the vibration mode S1.

In fact, in the three cases, approximate vibration frequencies with similar errors wereobtained on adaptively refined meshes with about only 10% the number of d.o.f. neededto attain the same errors on uniform meshes.

The obtained results for the remaining vibration modes are essentially the same,except for F1. Table 3 shows the results for this mode, which exhibits a somewhat erraticbehavior. The errors of the computed vibration frequencies reduced very significantlyduring the first steps. At step k = 4, the relative error had already decreased to lessthan 0.06%, in spite of the coarseness of the used mesh (1345 d.o.f.). However, in thefollowing steps, the error tended to decrease in an unpredictable nonmonotonic way.

In spite of this fact, the least squares error fitting of the last six computed vibrationfrequencies for this mode shows an impressive order of convergence: 1.52 (although thenonmonotonic behavior of the estimated error makes the whole process a bit unreliable).Moreover, after 8 steps, the error on the adaptively created mesh (7403 d.o.f.) is lessthan one tenth the error on a uniform mesh with ten times more degrees of freedom(79360 d.o.f.). Thus, in spite of its erratic behavior, the mesh refinement process isremarkably efficient to reduce the computational effort needed to calculate this mode.



Figure 8 summarizes tables 2 and 3. This figure shows that the slope of the errorcurves corresponding to the vibration modes S1, S2, and S3 tend to align with the lineof slope −1 (i.e., the orders of convergence approach to 1). Figure 8 also shows theerratic behavior of the mode F1, as well as the significant error reduction obtained forthis mode.

Finally, figures 9–12 show some of the refined meshes generated in the adaptiveprocess for the computation of each of these four vibration modes.

All these results allow us to conclude that the error indicator detects efficientlythose parts of the domain where the mesh needs to be refined.

References

[1] A. Alonso, A. Dello Russo, C. Padra and R. Rodríguez, Accurate pressure post-process of a finiteelement method for elastoacoustics (to appear).

[2] A. Alonso, A. Dello Russo and V. Vampa, A posteriori error estimates in finite element solution of



structure vibration problems with applications to acoustical fluid–structure analysis, Comput. Mech.23 (1999) 231–239.

[3] A. Alonso, A. Dello Russo and V. Vampa, A posteriori error estimates in finite element acousticanalysis, J. Comput. Appl. Math. 117 (2000) 105–119.

[4] D.N. Arnold and F. Brezzi, Mixed and non-conforming finite element methods: implementation,postprocessing and error estimates, M2AN 19 (1985) 7–32.

[5] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J.Numer. Anal. 15 (1976) 736–754.

[6] A. Bermúdez, R. Durán, M.A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibrationanalysis of fluid–solid systems without spurious modes, SIAM J. Numer. Anal. 32 (1995) 1280–1295.

[7] A. Bermúdez, R. Durán and R. Rodríguez, Finite element analysis of compressible and incompressiblefluid–solid systems, Math. Comp. 67 (1998) 111–136.

[8] A. Bermúdez, R. Durán, R. Rodríguez and J. Solomin, Finite element analysis of a quadratic eigen-value problem arising in dissipative acoustics, SIAM J. Numer. Anal. 38 (2000) 267–291.

[9] A. Bermúdez, L. Hervella-Nieto and R. Rodríguez, Finite element computation of the vibration modesof a fluid–solid system, J. Sound Vibrat. 219 (1999) 277–304.

[10] A. Bermúdez and R. Rodríguez, Finite element computation of the vibration modes of a fluid–solidsystem, Comp. Methods Appl. Mech. Eng. 119 (1994) 355–370.


Figure 12. Some refined triangulations for the vibration mode F1.

[11] P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal.Numér. 9 (1975) 77–84.

[12] M. Crouzeix and P.A. Raviart, Conforming and non conforming finite element methods for solvingthe stationary Stokes equations, RAIRO Anal. Numér. R3 (1973) 33–76.

[13] R. Durán, L. Gastaldi and C. Padra, A posteriori error estimators for mixed approximations of eigen-value problems, Math. Models Methods Appl. Sci. 9 (1999) 1165–1178.

[14] V. Girault and P.A. Raviart, Finite Element Methods for Navier–Stokes Equations (Springer, Berlin,1986).

[15] P. Grisvard, Elliptic Problems for Non-smooth Domains (Pitman, Boston, 1985).[16] L. Kiefling and G.C. Feng, Fluid–structure finite element vibration analysis, AIAA J. 14 (1976) 199–

203.[17] H. Morand and R. Ohayon, Substructure variational analysis of the vibrations of coupled fluid–

structure systems. Finite element results, Internat. J. Numer. Methods Eng. 14 (1979) 741–755.[18] H.J.-P. Morand and R. Ohayon, Fluid–Structure Interaction (Wiley, Chichester, 1995).[19] R. Ohayon and C. Soize, Structural Acoustics and Vibration (Academic Press, New York, 1998).[20] L. Olson and K. Bathe, Analysis of fluid–structure interaction. A direct symmetric coupled formula-

tion based on the fluid velocity potential, Comp. Struct. 21 (1985) 21–32.


[21] P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, in:Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, Vol. 606 (Springer,Berlin, 1972) pp. 292–315.

[22] W.C. Rheinboldt, On a theory of mesh-refinement processes, SIAM J. Numer. Anal. 17 (1980) 766–778.

[23] M.C. Rivara, Mesh refinement processes based on the generalized bisection of simplices, SIAM J.Numer. Anal. 21 (1985) 493–496.

[24] R. Rodríguez and J. Solomin, The order of convergence of eigenfrequencies in finite element approx-imations of fluid–structure interaction problems, Math. Comp. 65 (1996) 1463–1475.

[25] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques(Wiley-Teubner, Chichester/Stuttgart, 1996).

[26] J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations,Math. Comp. 69 (2000) 881–909.

[27] O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 2 (McGraw-Hill, London, 1991).

Documents

A Posteriori Error Estimates and a Local Refinement Strategy for a Finite Element Method to Solve Structural-Acoustic Vibration Problems