Error Analysis of a Modified Discontinuous Galerkin ... · PDF fileError Analysis of a Modi ed Discontinuous Galerkin Recovery Scheme for Di usion ... using Sobolev space energy arguments

Error Analysis of a Modi�ed Discontinuous Galerkin

Recovery Scheme for Di�usion Problems

Donald A. French�, Marshall C. Galbraithy

University of Cincinnati, Cincinnati, OH, 45221

and

Mauricio Osorioz

Universidad Nacional de Colombia, Apartado A�ereo 3840 Medell��n, Colombia

A theoretical error analysis using standard Sobolev space energy arguments is furnishedfor a class of discontinuous Galerkin (DG) schemes that are modi�ed versions of those in-troduced by van Leer and Nomura. These schemes, which use discontinuous piecewisepolynomials of degree q, are applied to a family of one-dimensional elliptic boundary valueproblems. The modi�cations to the original method include a de�nition of the ux functionvia a symmetric L2-projection and the addition of a penalty term. The method is found tohave a convergence rate of O(hq) for the approximation of the �rst derivative and O(hq+1)for the solution. Sample computations with the modi�ed scheme con�rm the analyticalanalysis. The computations also reveal that L2-errors associated with the modi�ed Re-covery scheme using recovery polynomials of degree q + 1 are comparable to the originalRecovery scheme which uses polynomials of degree 2q + 1. However, this is at the cost ofthe additional penalty term.

Nomenclature

Bh Volume integral term of the boundary value problem in bilinear formC Mesh and solution independent constantCqh Continuous piecewise polynomial spaceDqh Discontinuous piecewise polynomial and approximate solution space

Ih Interpolation operator for the recovery function spaceJh Penalty termLj Recovery polynomial de�ned by the modi�ed Recovery methodPr Recovery polynomial spaceU Exact solutionV Generic smooth functionW Analytical solution to the auxiliary problema, b Generic Scalarsc, � Smooth spatial functionsc0, c1 Lower and upper bounds on ccm min(c0,1)fj Recovery polynomial de�ned by the original Recovery methodg Recovery functionh Uniform mesh cell sizeu Approximate solution

�Professor of Mathematical Sciences, Department of Mathematical Sciences, Non-Member AIAA.yPhD Graduate Student, Department of Aerospace Engineering & Engineering Mechanics, AIAA Student Member.zAssistant Professor, Escuela de Matem�aticas, Non-Member AIAA.

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American Institute of Aeronautics and Astronautics

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-1071

Copyright © 2010 by Dr. Donald French, Marshall Galbraith, and Mauricio Osorio . Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

v,w,�,V Generic functions~w Approximate solution to the auxiliary problemx Cartesian coordinatexj Mesh node~x, ~y Generic vectors�h Boundary integral term of the boundary value problem in bilinear form Computational domain�, C� Constants for the arithmetic-geometric mean inequality�h Approximation error equal to u� �hU� Test function�h Interpolation operator for the approximation space�j Mesh cell

Superscripts

0 Derivative with respect to x

I. Introduction

Discontinuous Galerkin (DG) methods have been growing in popularity throughout the ComputationalFluid Dynamics (CFD) literature over the past decade. DG schemes have been known to be suitable for

solving hyperbolic problems, which allow for shocks, because the method uses the discontinuous approxima-tion functions.1 More recently, a number of techniques have been developed to treat the di�usive terms.2

Some of the more commonly used schemes are the local discontinuous Galerkin method3 and the Bassi andRebay4 schemes. These schemes have been analyzed using Sobolev space energy arguments by Arnold etal.2 in a uni�ed framework encapsulating a number of di�usive DG approaches.

The authors use techniques developed by Arnold et al.2 to study a modi�ed version of the recent van LeerRecovery schemes5{7 for di�usion problems. An additional penalty term is added to the modi�ed Recoveryscheme in order to facilitate the error analysis. Furthermore, the authors propose using recovery functionsde�ned by a symmetric projection on the recovery space, rather than the DG space. The authors show thatthe modi�ed scheme has optimal convergence rates of O(hq) and O(hq+1) for the approximation of the �rstderivatives and the solution respectively. These rates are typically expected for DG schemes.2

II. Boundary Value Problem

The method is developed and an analysis provided for the following elliptic problem:

�U 00 + cU = � on = (0; 1), 0 < c0 � c � c1

U 0(0) = 0; U 0(1) = 0:(1)

Here, c0, and c1 are constants and its assumed that �, c, and U are smooth functions. This problemhas been chosen purely for its simplicity. The authors expect that the analysis provided can be extended tomore general elliptic problems.

A. Computational Domain and Approximation Space

The domain 2 [0; 1] is divided into a mesh with cells given by:

0 = x0 < x1 < : : : < xN = 1: (2)

Individual cell intervals and sizes are de�ned denoted by �j = [xj�1; xj ] and hj = xj�xj�1 respectively, withj = 1; : : : N . The mesh is assumed to be quasi-uniformity so that hj �= h := 1=N . We use the approximationspace Dq

h() which consists of discontinuous piecewise polynomials of degree � q over the domain . Thepolynomials are continuous within a cell, and discontinuous across the cell interfaces. The union of two cellsis denoted as

Rj = �j [ �j+1 (3)

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for j = 1; 2; : : : ; N � 1. Space Pr(Rj) will be the set of polynomials of degree r which are non-zero only onthe interval Rj .

Note that the L2-convergence proof requires the use of an interpolation de�ned on the space Cqh(),which consists continuous piecewise polynomials of degree q. This space is continuous within a cell as wellas at cell interfaces. Hence, it is required that q � 1 to achieve continuity across cell interfaces.

B. Weak Form of the Analytical Solution

The boundary value problem in Eq. 1 is put in a weak form by multiplying the equation with � 2 Dqh()

and integrating by parts on each interval �j to form

NXj=1

Z�j

U 0�0 dx�N�2Xj=1

U 0�jxj+1

xj+

NXj=1

Z�j

cU� =

NXj=1

Z�j

�� dx (4)

The summation of the boundary integrals is now rearranged so that it can be expressed as a jump acrossthe cell interface. The jump in an arbitrary variable v at the interface xj is denoted as

[v](xj) = v(xj+)� v(xj�): (5)

The �nal weak form of the analytical boundary value problem in Eq. 1 is

NXj=1

Z�j

U 0�0 dx+

N�1Xj=1

U 0(xj)[�](xj) +

NXj=1

Z�j

cU� =

NXj=1

Z�j

�� dx (6)

Equation 6 is also written in a bilinear form as

Bh(U; �) + �h(U; �) = (�; �): (7)

where

Bh(v; w) =

NXj=1

Z�j

v0w0 dx+

NXj=1

Z�j

cvw dx;

�h(v; w) =

N�1Xj=1

v0(xj)[w](xj); and

(v; w) =

NXj=1

Z�j

vw dx: (8)

C. Numerical Boundary Value Problem

The numerical boundary value problem is formed by, essentially, substituting the numerical approximationu 2 Dq

h() for the analytical solution U in Eq. 6. It is important to note that u is discontinuous on thecell boundaries. Hence, u0(xj) is not de�ned on the boundary, and must be replaced with a numericalapproximation. Several numerical schemes have been devised to treat u0(xj) on cell boundaries, such asthe local discontinuous Galerkin method3 and the Bassi and Rebay4 schemes. An extensive analysis usingSobolev space energy argument for a number of these numerical schemes is provided by Arnold et al.2

The work presented here focuses on providing an analysis using Sobolev energy norms of a modi�edversion of the Recovery scheme presented by van Leer.5{7

D. Original Recovery Scheme (RDG-1x)

The original Recovery scheme by van Leer, shown here using the RDG-1x formulation,7 is de�ned below

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8>>>>>><>>>>>>:

Find u 2 Dqh() and fj(u) 2 Pr(Rj) for j = 1; : : : N � 1 such that

Bh(u; �) + �h(f; �) = (�; �) 8� 2 Dqh()

and, for j = 1; : : : N � 1;RRjfj(u)� dx =

RRju� dx 8� 2 Dq

h():

(9)

Here, the recovery function fj(u) is a polynomial of degree r and, by construction, fj(u) and f 0j(u) arecontinuous on the cell union Rj . While fj is a function of x, it is denoted as fj(u) to indicate that thecoe�cients of fj are functions of the coe�cients in the expansion u. The notation, fj(u)(xj) means that thecoe�cients of fj are functions of u and the polynomial is evaluated at the node xj .

For the analysis, it is important to note that the right hand side of the recovery function projection inZRj

fj(u)� dx =

ZRj

u� dx (10)

is symmetric, i.e. both u and � are in the DG space Dqh(). However, the integral on left hand side of Eq.

10 is not symmetric as � 2 Dqh() and fj 2 Pr(Rj). This lack of symmetry signi�cantly complicates the

error analysis of this scheme, and is the motivation for a modi�ed Recovery scheme. The analysis of themodi�ed Recovery scheme is expected to aid future work to prove an error estimate for the original Recoveryformulation.

E. Modi�ed Recovery Scheme

A modi�ed recovery scheme is presented here that relies on a projection of the recovery function which issymmetric. The recovery polynomial Lj(u) is de�ned through the projectionZ

Rj

Lj(u)g dx =

ZRj

ug dx 8g 2 Pr(Rj): (11)

Unlike the projection in Eq. 10, the left hand side of Eq. 11 is symmetric as both Lj(u) 2 Pr(Rj) andg 2 Pr(Rj). For ease of notation, the vector function operator, L, is de�ned to map L2(R1)�: : :�L2(RN�1)to Pr(R1)� : : :� Pr(RN�1) (i.e. LV = (L1V; : : : ;LN�1V )T ). We will need vector functions ~g as well;

~g = (g1; g2; : : : ; gN�1)T (12)

where gi 2 Pr(Rj). Note, again, that L has a symmetric L2-projection.In addition to the modi�ed projection, the analysis requires the following penalty term, which is written

in bilinear form as

Jh(u; �) := h�pN�1Xj=1

[u](xj)[�](xj) (13)

where p > 0. The necessity of this penalty term will be discussed in section IV.The complete modi�ed Recovery scheme is de�ned by the following problem8>>>>>><>>>>>>:

Find u 2 Dqh() and Lj(u) 2 Pr(Rj) for j = 1; : : : N � 1 such that

Bh(u; �) + �h(L; �) + Jh(u; �) = (�; �) 8� 2 Dqh()

and, for j = 1; : : : N � 1;RRjLj(u)g dx =

RRjug dx 8g 2 Pr(Rj):

(14)

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F. Notation for L2 and Hm Norms

The Sobolev space, Wm;p(S), for an arbitrary domain S, essentially consists of functions with m derivativesin the Lp(S) norm. We will use the following notation for Sobolev space seminorms and norms

jvjWm;p(S) =

�ZS

��@mv@xm

��p dx�1=p

and kvkWm;p(S) =

mX‘=0

ZS

��@‘v@x‘

��p dx!1=p

(15)

where m = 0; 1; 2; : : : and 1 � p � 1. However, we use the simpli�ed notation

jvjm;p;S = jvjWm;p(S) and kvkm;p;S = kvkWm;p(S): (16)

We will primarily be working with Hilbert space, Hm(S), norms which are a special case of the Soblev spacewith p = 2. The norms for these Hm(S) spaces arise naturally from the integral function products that areuesd in the DG formulation. The L2 norm in turn is equivalent the special case of H0(S). See, for instance,Ref. 8 for more details on Sobolev spaces and norms.

G. Theorem

The theorem, and main result of this is paper, is stated as:For the approximations of Eq. 1 by the method of Eq. 14 with r � q, q � 1 and p = 3 + �, there exists a

constant C so that 0@ NXj=1

ju� U j21;2;�j

1A1=2

� Chq (H1-Convergence) (17)

and

ku� Uk0;2; � Chq+1: (L2-Convergence) (18)

The remaining portion of this paper is organized as follows: Section III lists inequalities known in theliterature which will be used to prove the theorem. In section IV we prove the H1-Convergence estimateof Eq. 17 and in section IV we provide the L2-Convergence theorem of Eq. 18. Finally, in section VI weprovide computations on the original and modi�ed recovery schemes to verify the proof. In addition, thecalculations show that the modi�ed Recovery scheme using polynomials of degree q + 1 produces L2-errorscomparable to those of the original Recovery scheme which uses polynomials of degree 2q+ 1. However, themodi�ed Recovery scheme requires the additional penalty term to achieve this accuracy.

III. Auxiliary Estimates

In this section we provide de�nitions for approximation and a series of inequalities that are required forthe analysis of the modi�ed Recovery method. Only a summery will be provided here and complete detailsof these inequalities can be found in Ref. 8.

A. Triangle Inequality

The triangle inequality for the scalars a and b is�jaj2 + jbj2

�1=2 � jaj+ jbj (19)

B. Maximum Function Value

We note that the absolute value of a function, w, evaluated at a point, xj , within the domain S, over whichthe function is de�ned on is always less than the L1 norm of the function, i.e.

jw(xj)j � kwk0;1;S : (20)

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C. Arithmetic-Geometric Mean Inequality

The arithmetic-geometric mean inequality will be used a number of times to separate products of norms.The inequality states that for scalars a and b

jabj = jp

2�ajj bp2�j � �a2 + C�b

2: (21)

where C� = 1=(4�) and � > 0.

D. Squared Sum Inequality

This inequality will be used to separate squared norms of a sum of terms. Using the arithmetic-geometricmean inequality was used on the second step with � = 1=2, it is stated as follows for the scalars a and b

(a+ b)2

= a2 + 2ab+ b2 � a2 + 2

�1

2a2 +

1

2b2�

+ b2 � 2�a2 + b2

�: (22)

E. Cauchy-Schwarz Inequality

Both the vector and inner product forms of the the Cauchy-Schwarz (CS) inequality will be used throughoutthe analysis. The vector form is

j~x � ~yj � k~xkEk~ykE (23)

where k � kE is the euclidean norm. The inner product form, for smooth functions v and w de�ned on thedomain S, is

jZS

vw dxj ��Z

S

jvj2 dx�1=2�Z

S

jwj2 dx�1=2

� kvk0;2;Skwk0;2;S (24)

F. Inverse Estimate

We will use the following inverse estimate

k�0k0;1;S � Ch�3=2k�k0;2;S (25)

where � is a di�erentiable function de�ned over the domain S.

G. Norm Equality

During the analysis, we will take advantage of the following equality between the norm and seminorm of thedi�erentiable function � de�ned on the domain S

k�0k0;p;S = j�j1;p;S (26)

andk�k0;p;S = j�j0;p;S (27)

where 1 � p � 1.

H. Interpolation on the Approximation Space

For smooth functions V 2 Hq+1(), let �hV 2 Cqh() be an interpolant of V where Cqh() is the space ofcontinuous piecewise polynomials of degree � q. Note that Cqh() is a subset of Dq

h(). This means that anoperator de�ned for functions in Dq

h() can also operate on functions in Cqh(). It is also important to notethat the interpolant is exact at the nodes xj 2 , i.e

�hV (xj) = V (xj) for j = 0; : : : N (28)

and therefore that

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[�hV ](xj) = [V ](xj) = 0 for j = 0; : : : N: (29)

We assume that there exists a constant C, independent of V and h, such that

jV � �hV j1;p;�j � ChqkV kq+1;p;�j (30)

and

kV � �hV k0;p;�j � Chq+1kV kq+1;p;�j (31)

where we will use p = 2 or p =1.

I. Interpolation on the Recovery Function Space

For smooth functions V , we also assume there is an interpolation operator Ih on Rj so IhV 2 Pr(Rj) and

jV � IhV j1;p;Rj� ChrkV kr+1;p;Rj

(32)

and

kV � IhV k0;p;Rj� Chr+1kV kr+1;p;Rj

(33)

where we will use p = 2 or p =1.

J. Symmetric Projection Inequality

Since Lj , as de�ned in Eq. 11, is a symmetric projection, it can be shown that, for any V 2 L2(Rj),

kLj(V )k0;2;Rj � kV k0;2;Rj (34)

and

kV � Lj(V )k0;2;Rj � kV � �k0;2;Rj 8� 2 Pr(Rj): (35)

Similar inequalities would need to be developed to prove convergence rates of the original Recovery methodwhich uses the non-symmetric projection in Eq. 10 . This is not a trivial task and is a subject for futurework.

IV. Energy Norm Error Estimate for the Modi�ed Recovery Scheme

In this section we provide an analysis of the modi�ed Recovery scheme in Eq. 13 which proves that thescheme satis�es the H1-convergence of Eq. 17.

A. Bh Inequalities

Before proceeding, we will prove an inequality on Bh which will be needed in the main energy argument.Letting v; w 2 Dq

h(), using the de�nition of Bh in Eq. 8, and the inner product Cauchy-Schwarz inequalityin Eq. 24, we note that

jBh(v; w)j = jNXj=1

Z�j

v0w0 dx+

NXj=1

Z�j

cvw dxj

�NXj=1

jZ�j

v0w0 dxj+NXj=1

jZ�j

cvw dxj

�NXj=1

kv0k0;2;�jkw0k0;2;�j +

NXj=1

kc1=2vk0;2;�jkc1=2wk0;2;�j : (36)

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We can now use the vector form of the Cauchy-Schwarz inequality in Eq. 23 with the following vectors

~xT =�kv0k0;2;�1 ; kv0k0;2;�2 ; : : : ; kv0k0;2;�N ; kc1=2vk0;2;�1 ; kc1=2vk0;2;�2 ; : : : ; kc1=2vk0;2;�N

�~yT =

�kw0k0;2;�1 ; kw0k0;2;�2 ; : : : ; kw0k0;2;�N ; kc1=2wk0;2;�1 ; kc1=2wk0;2;�2 ; : : : ; kc1=2wk0;2;�N

�(37)

to write

jBh(v; w)j �

0@ NXj=1

kv0k20;2;�j +

NXj=1

kc1=2vk20;2;�j

1A1=20@ NXj=1

kw0k20;2;�j +

NXj=1

kc1=2wk20;2;�j

1A1=2

� Bh(v; v)1=2Bh(w;w)1=2 (38)

In addition, using the de�nition cm = min(c0; 1), and the de�nition of the norms in Eq. 15, we note that

Bh(v; v) =

NXj=1

�jvj21;2;�j + kc1=2vk20;2;�j

�

�NXj=1

c1cmjvj21;2;�j + k

�c1cm

�1=2

vk20;2;�j

!

� c1cm

NXj=1

�jvj21;2;�j + kvk20;2;�j

�

� C

NXj=1

kvk21;2;�j (39)

B. Main Energy Argument

To setup the proof, de�ne �h = u� �hU and note that, by using Eq. 22,

NXj=1

ju� U j21;2;�j =

NXj=1

�ju� �hU + �hU � U j21;2;�j

�

� 2

NXj=1

�ju� �hU j21;2;�j + jU � �hU j21;2;�j

�

� 2

NXj=1

�j�hj21;2;�j + jU � �hU j21;2;�j

�(40)

We know from the interpolation inequality in Eq. 31 that

NXj=1

jU � �hU j21;2;�j � Ch2q

NXj=1

kUk2q+1;2;�j � Ch2qkUk2q+1;2; � Ch2q (41)

where we used the fact that kUk2q+1;p; � C.8 However, we do not yet have a bound on the j�hj21;2;�j term.To �nd such a bound, we use the de�nition of Bh in Eq. 8 to write

NXj=1

j�hj21;2;�j � Bh(�h; �h) �NXj=1

j�hj21;2;�j + kc1=2�hk20;2; (42)

Thus, a bound on the interpolation term can be found by �nding a bound on the Bh. To do this, we letL = L(u) for ease of notation, and insert �h and L into Eq. 7, and rearrange it as follows

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Bh(�h; �) + Jh(u; �) = Bh(u; �)�Bh(�hU; �) + Jh(u; �)

+ (Bh(U; �)�Bh(U; �)) + (�h(U; �)� �h(U; �)) + (�h(L; �)� �h(L; �))

= (Bh(u; �) + �h(L; �) + Jh(u; �))� (Bh(U; �) + �h(U; �))

+ Bh(U � �hU; �) + �h(U � L; �) (43)

Recalling now from Eqs. 7 and 14 that

Bh(U; �) + �h(U; �) = (�; �) and Bh(u; �) + �h(L; �) + Jh(u; �) = (�; �); (44)

the right hand side of Eq. 43 reduces to

Bh(�h; �) + Jh(u; �) = (�; �)� (�; �) +Bh(U � �hU; �) + �h(U � L; �)

= Bh(U � �hU; �) + �h(U � L; �): (45)

Next, choose � = �h so that Eq. 45 becomes

Bh(�h; �h) + Jh(u; �h) = Bh(U � �hU; �h) + �h(U � L; �h): (46)

Note that since [�hU ](xj) = 0, we have Jh(u; �hU) = 0 so Jh(u; �h) = Jh(u; u), thus

Bh(�h; �h) + Jh(u; u) = Bh(U � �hU; �h) + �h(U � L; �h): (47)

The goal now is to expand the terms on the right hand so that all terms not involving U can be moved tothe left hand side.

Using the inequality in Eq. 38 and the arithmetic-geometric mean inequality of Eq. 21 with � = 1=2, itcan be shown that

Bh(U � �hU; �h) � (Bh(U � �hU;U � �hU))1=2

(Bh(�h; �h))1=2

� 1

2Bh(U � �hU;U � �hU) +

1

2Bh(�h; �h): (48)

Equation 48 can be used to bound Eq. 47. After some manipulation of Eq. 47, we have

1

2Bh(�h; �h) + Jh(u; u) � 1

2Bh(U � �hU;U � �hU) + �h(U � L; �h): (49)

We now examine the �h-term on the right hand side of 49. Because �hU is continuous, the jump in �hUis zero, i.e. [�hU ](xj) = 0. Therefore, [�h](xj) = [u� �hU ](xj) = [u](xj), and the �h-term reduces to

�h(U � L; �h) = �h(U � L; u): (50)

The absolute value of the right hand of Eq. 50 can be bounded by using Eq. 19 as follows

j�h(U � L; �h)j = j�h(U � L; u)j

=

��N�1Xj=1

(U 0(xj)� Lj(u)0(xj))[u](xj)

��=

��N�1Xj=1

[U 0(xj)� Lj(�hU)0(xj) + Lj(�hU)0(xj)� Lj(u)0(xj)][u](xj)

��=

��N�1Xj=1

[U 0(xj)� Lj(�hU)0(xj) + Lj(��h)0(xj)][u](xj)

��

N�1Xj=1

jU 0(xj)� Lj(�hU)0(xj)j j[u](xj)j+N�1Xj=1

jLj(�h)0(xj)j j[u](xj)j (51)

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Applying the Cauchy-Schwarz and arithmetic-geometric mean inequalities, with � = �h�p, to each of the�nal two sums in Eq. 51 in a manner similar to what was done in Eq. 48 yields

j�h(U � L; �h)j �N�1Xj=1

(C�hpjU 0(xj)� Lj(�hU)0(xj)j2 +�

2h�pj[u](xj)j2)

+

N�1Xj=1

(C�hpjLj(�h)0(xj)j2 +�

2h�pj[u](xj)j2)

� C�hpN�1Xj=1

jU 0(xj)� Lj (�hU)0(xj)j2 + C�h

pN�1Xj=1

jLj (�h)0(xj)j2 + �h�p

N�1Xj=1

j[u](xj)j2

� C�hpT1 + C�h

pT2 + �Jh(u; u): (52)

Rewrite T1 using the inequality in Eq. 20 as

N�1Xj=1

jU 0(xj)� Lj (�hU)0(xj)j2 �

N�1Xj=1

kU 0 � Lj (�hU)0 k20;1;Rj

�N�1Xj=1

kU 0 � Lj(U)0 + Lj(U)0 � Lj (�hU)0 k20;1;Rj

� 2

N�1Xj=1

�kU 0 � Lj(U)0k20;1;Rj

+ kLj(U)0 � Lj(�hU)0k20;1;Rj

�(53)

The last two terms in the summation of the above equation will now be treated individually. The �rstterm is rewritten using the equality of the norms in Eq. 26 to obtain

kU 0 � Lj(U)0k20;1;Rj= jU � Lj(U)j21;1;Rj

: (54)

For the second term, the inverse inequality in Eq. 25 and the projection inequality in Eq. 34 are used toshow that

kLj(U)0 � Lj(�hU)0k20;1;Rj� Ch�3kLj(U)� Lj(�hU)k20;2;Rj

� Ch�3kU � �hUk20;2;Rj: (55)

Hence, the following inequality holds for T1

N�1Xj=1

jU 0(xj)� Lj (�hU)0(xj)j2 �

N�1Xj=1

�CjU � Lj(U)j21;1;Rj

+ Ch�3kU � �hUk20;2;Rj

�: (56)

Using the inequalities in Eqs. 20, 25, and 34 on the T2 term shows that

N�1Xj=1

jLj (�h)0(xj)j2 �

N�1Xj=1

kLj(�h)0k20;1;Rj� Ch�3

N�1Xj=1

kLj (�h) k20;2;Rj� Ch�3

N�1Xj=1

k�hk20;2;Rj: (57)

Substitute Eqs. 56 and 57 into Eq. 52 to obtain

j�h(U � L; �h)j �N�1Xj=1

�ChpjU � Lj(U)j21;1;Rj

+ Chp�3kU � �hUk20;2;Rj

�

+ Chp�3N�1Xj=1

k�hk20;2;Rj+ �Jh(u; u): (58)

Next, substitute Eq. 58 into Eq. 49 with � = 1=2, and hence C� = 1=2,

10 of 21


1

2Bh(�h; �h) + Jh(u; u) � 1

2Bh(U � �hU;U � �hU)

+

N�1Xj=1



�

+ Chp�3N�1Xj=1

k�hk20;2;Rj+

1

2Jh(u; u): (59)

At this point, only the last two terms in Eq. 59 are functions of u. Hence, they must be moved to theleft hand side of the equation. This step is where it becomes evident that the penalty term Jh is requiredin the modi�ed Recovery method. The 1

2Jh term that arose from the �h term needs to be moved to the lefthand side of 59 while keeping all terms on the left hand side positive. One way to achieve this is to introducethe penalty term Jh in the original scheme so that it occurs on the left hand side. The 1

2Jh can then besubtracted from both sides of Eq. 59 to give

1

2Bh(�h; �h) +

1

2Jh(u; u) � 1


+

N�1Xj=1



�

+ Chp�3N�1Xj=1

k�hk20;2;Rj: (60)

Next, the k�hk20;2;Rjterm needs to be moved to the left hand side. This is achieved by �rst choosing

p = 3 + � for some 0 < � << 1 and by recognizing that,

Ch�N�1Xj=1

k�hk20;2;Rj= C2h�

NXj=1

k�hk20;2;�j

� C2h�NXj=1

k�c

c0

�1=2

�hk20;2;�j

� C2h�

c0

NXj=1

�j�hj21;2;�j + kc1=2�hk20;2;�j

�� C

2h�

c0Bh(�h; �h) (61)

Assume that h is su�ciently small such that

C2h�

c0� 1

4: (62)

Thus, we have that

Ch�N�1Xj=1

k�hk20;2;Rj� 1

4Bh(�h; �h) (63)

We can therefore subtract 14Bh(�h; �h) from both sides of Eq. 60 to remove the k�hk20;2;Rj

term from the

right hand side. Furthermore, we half the Jh(u; u) in Eq. 60 such that

1

4Bh(�h; �h) +

1

4Jh(u; u) � 1


+

N�1Xj=1



�(64)

11 of 21


Thus, after multiplying through by 4, and recognizing that the constants on each term are arbitrary, Eq.64 can be written as

Bh(�h; �h) + Jh(u; u) � 2Bh(U � �hU;U � �hU)

+

N�1Xj=1

�Ch3+�jU � Lj(U)j21;1;Rj

+ Ch�kU � �hUk20;2;Rj

�: (65)

This is an optimal order error estimate as norms of the error �h = u � �hU are bounded by errorsinvolving only U � �hU and U � Lj(U).

Consider the Bh term on the right hand side of Eq. 65, we note that by de�nition

Bh(U � �hU;U � �hU) =

NXj=1

jU � �hU j21;2;�j +

N�1Xj=1

kc1=2(U � �hU)k20;2;�j

�NXj=1

jU � �hU j21;2;�j + c1

N�1Xj=1

kU � �hUk20;2;�j : (66)

The two interpolation inequalities of Eqs. 30 and 31 are applied to the right hand side of Eq. 66 to give

Bh(U � �hU;U � �hU) �NXj=1

Ch2qkUk2q+1;p;�j +

N�1Xj=1

Ch2q+2kUk2q+1;p;�j

� Ch2qkUk2q+1;p; + Ch2q+2kUk2q+1;p;

� Ch2q + Ch2q+2: (67)

The norm equality of Eq. 26, the interpolation inequality of Eq. 32, the inverse inequality of Eq. 25,and the interpolation inequality of Eq. 33 are used to modify the last two terms on the right hand side ofEq. 65 as

Ch3+�N�1Xj=1

jU � Lj(U)j21;1;Rj= Ch3+�

N�1Xj=1

jU � IhU + IhU � Lj(U)j21;1;Rj

� Ch3+�N�1Xj=1

�jU � IhU j21;1;Rj

+ jIhU � Lj(U)j21;1;Rj

�

� Ch3+�N�1Xj=1

�jU � IhU j21;1;Rj

+ kIhU 0 � Lj(U)0k20;1;Rj

�

� Ch3+�N�1Xj=1

�Ch2rkUk2r+1;1;Rj

+ Ch�3kIhU � Lj(U)k20;2;Rj

�

� Ch3+�N�1Xj=1

�Ch2r + Ch�3kIhU � U + U � Lj(U)k20;2;Rj

�

� Ch3+�N�1Xj=1

�Ch2r + Ch�3kU � IhUk20;2;Rj

+ Ch�3kU � Lj(U)k20;2;Rj

�

� Ch3+�N�1Xj=1

�Ch2r + Ch2r+2�3kUk2r+1;2;Rj

+ Ch�3kU � Lj(U)k20;2;Rj

�

� Ch3+�N�1Xj=1

�Ch2r + Ch2r�1 + Ch�3kU � Lj(U)k20;2;Rj

�(68)

12 of 21


The inequality related to the symmetric projection of L in Eq. 35 is now used with � = IhU and theinterpolation inequality of Eq. 33 to modify the last term of Eq. 68 as

kU � Lj(U)k20;2;Rj� kU � IhUk20;2;Rj

� Ch2r+2kUk2r+1;2;Rj� Ch2r+2: (69)

Thus Eq. 68 is

Ch3+�N�1Xj=1

jU � Lj(U)j21;1;Rj� Ch3+�

N�1Xj=1

�Ch2r + Ch2r�1 + Ch2r+2�3

�� Ch3+�

�Ch2r + Ch2r�1 + Ch2r�1

�N�1Xj=1

(1): (70)

Recall that h = 1=N , hence N = h�1, therefore

N�1Xj=1

(1) = N � 1 � N = h�1: (71)

Since the smallest exponent of h determines the bound on the error estimate, Eq. 70 reduces to

Ch3+�N�1Xj=1

jU � Lj(U)j21;1;Rj� Ch2+�

�Ch2r + Ch2r�1

�� Ch2+�Ch2r�1 � Ch2r+1+�: (72)

The interpolation inequality of Eq. 31 is now used on the �nal term of Eq. 65 to obtain

Ch�N�1Xj=1

kU � �hUk20;2;Rj� Ch�

N�1Xj=1

Ch2q+2kUk2q+1;p;�j � Ch2q+2+�kUk2q+1;p; � Ch2q+2+�: (73)

Equations 67, 72, and 73 can now substituted into Eq. 65 to give

Bh(�h; �h) + Jh(u; u) � Ch2q + Ch2q+2 + Ch2r+1+� + Ch2q+2+�: (74)

Because we are interested in the worst case, we can drop the two h2q+2 terms to give

Bh(�h; �h) + Jh(u; u) � Ch2q + Ch2r+1+�: (75)

Up till this point, no criteria have been placed on the degree, r, of the Recovery polynomial, Lj(u).However, based on Eq. 75 we choose r � q so that

Bh(�h; �h) + Jh(u; u) � Ch2q: (76)

Thus, by combining Eqs. 76 and 42, we have

NXj=1

j�hj21;2;�j � Bh(�h; �h) + Jh(u; u) � Ch2q (77)

and we have therefore shown that 0@ NXj=1

j�hj21;2;�j

1A1=2

� Chq: (78)

Finally, substitute Eqs. 41 and 78 into Eq. 40 to obtain0@ NXj=1

ju� U j21;2;�j

1A1=2

� Chq (79)

This proves the H1-convergence of Eq. 17. Since the order is the same as would be expected for the directapproximation of U 0 by polynomials of degree � q we say the convergence is optimal.

13 of 21


V. L2-Error Estimate of the Modi�ed Recovery Scheme

In this section we build on the energy norm convergence result using a standard duality argument toprove L2-norm convergence.

A. Slope Inequality

We now prove a simple estimate for the slope of W 2 H2() at the node xj which will be used to prove theL2-norm convergence. Let x 2 �j = [xj�1; xj ]; then by the fundamental theorem of calculus we note that

(W 0(x))2 = (W 0(xj�1))2

+

Z x

xj�1

d

ds

�(W 0(s))2

�ds: (80)

Integrating Eq. 80 over the domain �j

kW 0k20;2;�j = hj (W 0(xj�1))2

+

Z�j

Z x

xj�1

d

ds

�(W 0(s))2

�ds dx: (81)

The di�erentiation within the integral in Eq. 81 is expanded as

kW 0k20;2;�j = hj (W 0(xj�1))2

+ 2

Z�j

Z x

xj�1

W 0(s)W 00(s)ds dx: (82)

The �rst term on the right hand side of Eq. 82 is solved for and the absolute value is taken of the resultingequation

hj jW 0(xj�1)j2 � kW 0k20;2;�j + 2

Z�j

Z x

xj�1

jW 0(s)jjW 00(s)jds dx: (83)

The inner integration bounds are now changed to integrate over �j , and we use the Cauchy-Schwarz inequalityof Eq. 24 on the integrand to obtain

hj jW 0(xj�1)j2 � kW 0k20;2;�j + 2

Z�j

Z�j

jW 0(s)jjW 00(s)jds dx

� kW 0k20;2;�j + 2hjkW 0k0;2;�jkW 00k0;2;�j (84)

The arithmetic-geometric mean of Eq. 21, with � = h�1=2, is now used on the second term on the righthand side of Eq. 84 so that

jW 0(xj�1)j2 � 1

hjkW 0k20;2;�j +

1

hjkW 0k20;2;�j + hjkW 00k20;2;�j

� 2h�1j jW j

21;2;�j + hj jW j22;2;�j (85)

Noting now that h di�ers from any hj by a constant, we have shown that

jW 0(xj�1)j2 � Ch�1jW j21;2;�j + ChjW j22;2;�j ; j = 1; : : : ; N: (86)

B. L2-Convergence

Using standard duality arguments, we consider the auxiliary problem

�W 00 + cW = U � u on ,

W 0(0) = 0; W 0(1) = 0:: (87)

Since Eq. 87 is an elliptic di�erential equation with Neumann boundary conditions, we know from theliterature that the following inequality holds8

kWk2;2; � CkU � uk0;2; (88)

14 of 21


Let ew 2 Cqh be the solution toZ

ew0ev0 dx+

Z

c ewev dx =

Z

(U � u)ev dx; 8ev 2 Cqh: (89)

For Eq. 89, it is also known from the literature that8

kW � ewk1;2; � ChkWk2;2;: (90)

Multiplying Eq. 87 by (U � u), integrating over the domain, and integrating by parts, we have

kU � uk20;2; =

NXj=1

Z�j

(�W 00 + cW )(U � u) dx

=

NXj=1

Z�j

W 0(U � u)0 dx+ �h(W;U � u) +

Z

cW (U � u) dx: (91)

Substitution of ew for � in Eq. 6 yields

NXj=1

Z�j

U 0 ew0 dx+

Z

cU ew dx+ �h(U; ew) =

Z

� ew dx; (92)

and in Eq. 14 gives

NXj=1

Z�j

u0 ew0 dx+

Z

cu ew dx+ �h(u; ew) + Jh(u; ew) =

Z

� ew dx: (93)

However, since [ ew](xj) = 0 for j = 1; 2; : : : ; N � 1, we have �h(u; ew) = 0 and Jh(u; ew) = 0. Subtract Eqs.92 and 93 to obtain

NXj=1

Z�j

ew0(U � u)0 dx+

Z

c ew(U � u) dx = 0: (94)

Because Eq. 94 is zero it can be subtracted from Eq. 91 to give

kU � uk20;2; =

NXj=1

Z�j

(W � ew)0(U � u)0 dx+

NXj=1

Z�j

c(W � ew)(U � u) dx+ �h(W;U � u)

= Bh(W � ew;U � u) + �h(W;U � u) (95)

Using the inequalities in Eqs. 38 and 39, Eq. 95 can be written as

kU � uk20;2; � (Bh(W � ew;W � ew))1=2

Bh(U � u; U � u)1=2 + j�h(W;U � u)j

� C

0@ NXj=1

kW � ewk21;2;�j1A1=20@ NX

j=1

kU � uk21;2;�j

1A1=2

+ j�h(W;U � u)j

� CkW � ewk1;2;0@ NXj=1

kU � uk21;2;�j

1A1=2

+ j�h(W;U � u)j (96)

A bound on the �rst term on the right hand side of Eq. 96 can now be determined by using Eq. 22 andnoting that

15 of 21


NXj=1

kU � uk21;2;�j =

NXj=1

�ku� �hU + �hU � Uk21;2;�j

�

� 2

NXj=1

�ku� �hUk21;2;�j + kU � �hUk21;2;�j

�

� 2

NXj=1

�k�hk21;2;�j + kU � �hUk21;2;�j

�: (97)

Next, using the de�nition of the norm in Eq. 15, the de�nition of Bh from Eq. 8, and Eq. 76, and recallingthat cm = min(c0; 1), we show that

NXj=1

k�hk21;2;�j =

NXj=1

�j�hj21;2;�j + k�hk20;2;�j

�

�NXj=1

1

cmj�hj21;2;�j + k

�c

cm

�1=2

�hk20;2;�j

!

� 1

cmBh(�h; �h)

� Ch2q (98)

By using the de�nition of the norms in Eq. 15 and the interpolation inequalities in Eqs. 30 and 31, we canexpand the second term on the right hand side of 97 as

NXj=1

kU � �hUk21;2;�j = kU � �hUk21;2;

= kU � �hUk20;2; + jU � �hU j21;2;� Ch2q+2 + Ch2q

� Ch2q (99)

Using Eqs. 98 and 99 in Eq. 97, we have0@ NXj=1

kU � uk21;2;�j

1A1=2

� Chq: (100)

Substituting Eq. 100 into Eq. 96

kU � uk20;2; � ChqkW � ewk1;2; + j�h(W;U � u)j: (101)

We can now use Eqs. 90 and 88 to write

kW � ewk1;2; � ChkWk1;2; � ChkU � uk0;2;: (102)

Substituting Eq. 102 into Eq. 101 we thus have

kU � uk20;2; � Chq+1kU � uk0;2; + j�h(W;U � u)j: (103)

Next, consider the �h term on the right hand side of Eq. 103. Using the de�nition of �h in Eq. 8 andthe vector form of the Cauchy-Schwarz inequality in Eq. 23 we have that

j�h(W;U � u)j =

��N�1Xj=1

W 0(xj)[U � u](xj)

�� 0@N�1Xj=1

jW 0(xj)j21A1=20@N�1X

j=1

j[U � u](xj)j21A1=2

: (104)

16 of 21


We can now use the result of Eq. 86, and the triangle inequality of Eq. 19, to rewrite the �rst term on theright hand side of Eq. 1040@ NX

j=1

jW 0(xj)j21A1=2

�

0@ NXj=1

�Ch�1jW j21;2;�i + ChjW j22;�i

�1A1=2

� Ch�1=2jW j1;2; + Ch1=2jW j2;2;� Ch�1=2kWk2;2;: (105)

Here, we dropped the term with h1=2 since the exponent of the h�1=2 is smaller. The last in equality is aquality of the de�nition of the norms. The second term on the right hand side of Eq. 104 can be boundedby �rst considering the de�nition of Jh from Eq. 13, with p = 3 + �, we note that

Jh(U � u; U � u) = h�3��N�1Xj=1

j[U � u](xj)j2: (106)

Multiplying Eq. 106 by h3+� we have

N�1Xj=1

j[U � u](xj)j2 = h3+�Jh(U � u; U � u): (107)

Note that because [U ](xj) = 0, we have that

jJh(U � u; U � u)j = jJh(u; u)j: (108)

Using Eqs. 107, 108 and 76 we therefore have

N�1Xj=1

j[U � u](xj)j2 = h3+�jJh(U � u; U � u)j = h3+�jJh(u; u)j � Ch2q+3+�: (109)

By substituting Eqs. 105 and 109 into Eq. 104, and using Eq. 88, we have

j�h(W;U � u)j � Ch�1=2kWk2;2;�h2q+3+�

�1=2� Chq+3=2�1=2+�kWk2;2;� Chq+1+�kWk2;2;� Chq+1kU � uk0;2; (110)

Substitution of Eq. 110 into Eq. 103 gives

kU � uk20;2; � Chq+1kU � uk0;2;: (111)

Division by kU � uk0;2; yieldskU � uk0;2; � Chq+1; (112)

which proves the L2-convergence of Eq. 18.

VI. Computational Studies

In this section we will provide a computational example to illustrate the theoretical L2-convergence givenby Eq. 18. Solutions were obtained for Eq. 1, with c � 1, on a uniform mesh, and �(x) = 4�cos(2�x).The corresponding analytical solution is U(x) = cos(2�x)� 1. All polynomial products and integrals of thenumerical schemes of Eqs. 9 and 14 were evaluated analytically, and the resulting system of equations wassolved using a block tri-diagonal version of the Thomas algorithm.9

17 of 21


Figures 1 through 4 show the log of the L2-error of the numerical solution vs. the log of mesh spacing h.The L2-error is computed from

L2-error =

0@ NXj=1

Z�j

ju� U j2 dx

1A1=2

(113)

where the integral was also evaluated analytically. To agree with the theoretical L2-convergence rate, thelines in each �gure should have a slope which is � q + 1, i.e. O(hq+1). In addition, recall that Eq. 75indicates that the modi�ed Recovery scheme requires r � q to obtain the optimal convergence rate.

Figure 1 shows the convergence obtained with the modi�ed Recovery method given by Eq. 14 with r = q.The graph in Figure 1a uses a penalty term with p = 3 and con�rms our convergence theorem of Eq. 18.The attened part on the q = 4 curve is a result of rounding errors caused by the integrals in Eq. 113appraoching to machine zero. The squared errors are no smaller than roughly 10�14 in double precision and,after taking a square root are approximately 10�7. As shown in Figure 1b, consistent with the theoreticalanalysis, inclusion of the penalty term is necessary in order to achieve the optimal convergence rate. Inaddition, as expected, the theory does not hold for q = 0 as the interpolant required continuity which canonly be satis�ed with q � 1.

(a) (b)

Figure 1: Convergence plots of L2-errors vs h of the modi�ed Recovery method with r = q.a) Penalty term with p = 3. b) No penalty term.

While the theory indicated that the modi�ed Recovery scheme would archive an optimal convergencerate with of r = q, Figure 2a, with r = q+1, shows a reduction in the L2-error while maintaining the optimalconvergence rate. However, as shown in Figure 2b, the penalty term is still required in order to maintainoptimal convergence. Increasing r beyond q + 1 does not improve the L2-error any further as is shown inFigure 3 where r = 2q + 1.

18 of 21


(a) (b)

Figure 2: Convergence plots of L2 -errors vs h of the modi�ed Recovery method with r = q + 1.a) Penalty term with p = 3. b) No penalty term.

(a) (b)

Figure 3: Convergence plots of L2 -errors vs h of the modi�ed Recovery method with r = 2q + 1.a) Penalty term with p = 3. b) No penalty term.

For comparison, L2-errors of the original RDG formulations of van Leer5{7 are shown in Figure 4. Theconvergence rate of the RDG-1x formulation in Figure 4a appears to be dependent on wether q is odd oreven, while the RDG-2x formulation (Figure 4b) achieves the optimal convergence rate of O(hq+1). Only aslight improvement in L2-error is observed for the RDG-2x formulation when compared with the modi�edRecovery scheme of Figure 2a. However, the modi�ed Recovery scheme requires fewer operations and lessstorage because the degree of the recovery polynomial is only r = q + 1, compared to r = 2q + 1 for theRDG-2x scheme.

19 of 21


(a) (b)

Figure 4: Convergence plots of L2-errors vs h of the original Recovery method with r = 2q + 1.a) RDG-1x. b) RDG-2x.

VII. Conclusions

A modi�ed Recovery scheme was presented that is similar to the original Recovery method developed byvan Leer.5{7 The modi�cations included a symmetric projection operator and a penalty term that is absentfrom the original Recovery method. A theoretical error analyses of the accuracy of the modi�ed Recoverymethods was presented which proved that the scheme achieves a optimal convergence rates in both H1 andL2 norms. The analyses here is similar to, and consistent with, analysis of other numerical schemes studiedin Ref. 2.

Numerical experiments con�rmed that the modi�ed Recovery scheme of Eq. 14 achieved the optimal L2-error convergence of O(hq+1) with r � q. The numerical solution had a minimum L2-error with r = q + 1,and no further improvement was achieved by increasing the degree of r. The L2-errors with r = q + 1 arecomparable to the original RDG-2x formulation of van Leer. However, the computational cost is lower withthe modi�ed Recovery method as it only requires r = q + 1 while the original Recovery method requiresr = 2q + 1.

For future work; the authors speculate that the penalty term of the modi�ed Recovery method could beremoved by using the quasi-standard RDG-2x formulation of the original Recovery method.7 In addition, theimplementation of Dirichlet boundary conditions for the modi�ed Recovery method has not been resolved.Finally, the authors believe that the proofs presented here could be extended to higher dimensions withrelative ease.

Acknowledgments

Dr. French was partially supported by NSF DMS-0515989 as well as the Charles Phelps Taft ResearchCenter. Mr. Osorio was a graduate assistant and supported by a Taft Fellowship during the 2008-9 academicyear when this research was completed, and Mr. Galbraith was supported by the Department of Defense(DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Theauthors also extend great appreciation to Dr. John Benek for volunteering his time and insights.

20 of 21


References

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2Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D., \Uni�ed analysis of discontinuous Galerkin methods for ellipticproblems," SIAM J. Num. Anal., Vol. 39, No. 5, 2001, pp. 1749{1779.

3Cockburn, B. and Shu, C.-W., \The local discontinuous Galerkin method for timedependent convection-di�usion systems,"SIAM J. Numer. Anal., Vol. 35, 1998, pp. 2440{2463.

4Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., and Savini, M., \Ahigh-order accurate discontinuous �nite elementmethod for inviscid and viscous turbomachinery ows," 2nd European Conference on Turbomachinery Fluid Dynamics andThermodynamics, edited by R. Decuypere and G. Dibelius, Technologisch Instituut, Antwerpen, Belgium, 1997, pp. 99-108.

5van Leer, B. and Nomura, S., \Discontinuous Galerkin for Di�usion," AIAA Paper 2005-2108, Jun. 2005.6van Leer, B., Lo, M., and van Raalte, M., \A discontinuous Galerkin method for di�usion based on recovery," AIAA Paper

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