Finite-volume schemes for Friedrichs systems in multiple space dimensions: A priori and a posteriori error estimates

Finite-Volume Schemes for Friedrichs Systems inMultiple Space Dimensions: A Priori andA Posteriori Error EstimatesVladimir Jovanovic and Christian RohdeMathematisches Institut, Albert-Ludwigs-Universitat Freiburg, Hermann-Herder-Straße 10, D-79104 Freiburg im Breisgau, Germany

Received 18 December 2002; accepted 5 January 2004Published online 18 May 2004 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20026

We consider a class of finite-volume schemes on unstructured meshes for symmetric hyperbolic linearsystems of balance laws in two and three space dimensions. This class of schemes has been introduced andanalyzed by Vila and Villedieu (1998). They have proven an a priori error estimate for approximations ofsmooth solutions. We extend the results to weak solutions. This is the base to derive an a posteriori errorestimate for finite-volume approximations of weak solutions. © 2004 Wiley Periodicals, Inc. Numer Methods

Partial Differential Eq 21: 104–131, 2005

Keywords: finite-volume method; linear systems of balance laws; a priori and a posteriori error estimate

I. INTRODUCTION

Finite-volume schemes on unstructured grids are most frequently used to approximate solutionsof systems of hyperbolic balance laws in multiple space dimensions. The convergence analysisfor the scalar case is well developed, even though not complete up to now (cf. the text books[1, 2] and references therein). In contrast, very few is known for the case of systems. In thisarticle we draw attention to linear systems of balance laws, namely (m � m)-systems ofFriedrichs type with m � � ([3]). We consider the spatially d-dimensional case with d � 2, 3,and coordinates x � (x1, . . . , xd)T. For T � 0, let A1, . . . , Ad, B : �d � [0, T] 3 �m�m andf : �d � [0, T] 3 �m be given (matrix-valued) functions. We suppose that the matrices A1(x,t), . . . , Ad(x, t) are symmetric for all (x, t) � �d � [0, T]. Then a Friedrichs system for theunknown vector-valued function u : �d � [0, T] 3 �m takes the form

Correspondence to: Christian Rohde, Mathematisches Institut, Albert-Ludwigs-Universitat Freiburg, Hermann HerderStraße 10, D-79104, Freiburg im Breisgau, Germany (e-mail: [email protected])

© 2004 Wiley Periodicals, Inc.

�u

�t�x, t� � �A1�x, t�u�x, t��x1 � · · · � �Ad�x, t�u�x, t��xd

� B�x, t�u�x, t� � f �x, t�

� ��x, t� � �d � �0, T ��.(1.1)

We consider the Cauchy problem for (1.1), that is, for u0 : �d 3 �m, we assume

u�x, 0� � u0�x� �x � �d�. (1.2)

Specific regularity assumptions on the data and a short review on analytical results for (1.1),(1.2) will be given in Section II below. Here we note in passing that physically relevantexamples of systems of type (1.1) are the wave equation written as a first-order system,Maxwell’s equations of linear electrodynamics, or relaxation systems for nonlinear hyperbolicconservation laws.

In [4, 5] Vila and Villedieu analyze a class of finite-volume schemes on unstructured gridsfor the system (1.1), (1.2). Suppose that there is a solution u of (1.1), (1.2) and denote thecorresponding finite-volume approximation by uh : �d � [0, T] 3 �m, where h is the gridparameter. Then the following a priori error estimate result is proven. For each compact set Q� �d there is some constant C � 0 such that

�u � uh�L2�Q��0,T�� Ch1/2. (1.3)

The constant C depends on Q, T, u and parameters of the numerical method but not on h. Theestimate (1.3) is proven in [5] for u being a smooth classical solution of (1.1), (1.2) and A1, . . . ,Ad, B, f, u0 sufficiently smooth.

In this article we will extend the results of Vila and Villedieu in two directions. First we willextend the a priori error estimate (1.3) to the case of problems with less regular functions f, u0

and (consequently) a less regular weak solution in [C([0, T]; Hs(�d))]m, s � (0, 1]. Second wepresent an a posteriori error estimate that could be the foundation of a locally adaptive algorithmto solve (1.1), (1.2).

Let us give a more detailed outline of the article’s content. In Section II we introducenecessary notations and collect all analytical results for the Cauchy problem (1.1), (1.2), whichwe need in the sequel. The numerical scheme and the basic results from [5] are introduced anddiscussed in Section III. This includes the precise statement for (1.3), which will be improvedin the subsequent sections. Using Bramble-Hilbert like techniques we present and prove inSection IV the estimate (1.3) for weak solutions in [C([0, T]; H1(�d))]m (Theorem 4.1). Froma technical point of view we represent the error by means of measure terms that has also beenused in [5] and has been introduced in [6] in the context of nonlinear scalar equations.

In the next step we derive an error estimate for weak solutions in [C([0, T]; Hs(�d))]m,s � (0, 1] (Theorem 4.5). This is done by interpolation techniques. In this way we can also treatdiscontinuous solutions of the problem. This concludes the first part of the article on a priorierror estimates. In the second part of the article, that is Section V, we present the detailed newa posteriori error analysis. Our a posteriori error estimate itself is summarized in Theorem 5.1.

The proofs rely heavily on a discrete version of an L2-comparison result for smooth solutionsof (1.1). The L2-norm might not look natural in the framework of hyperbolic balance lawsassociated usually with the L1-norm. However, L2 is the correct function space for hyperbolicsystems with m � 1 in space dimension d � 1. The results of Brenner and Rauch [7, 8] show

ERROR ESTIMATES FOR FRIEDRICHS SYSTEMS 105

(for appropriate initial data) that if a (weak) solution of (1.1) is in C([0, T], Lp(�d)), p � [1, 2)� (2, �], then the matrices Al and Ak commute for k, l � {1, . . . , d}, that is,

AkAl � AlAk.

Note that the successful convergence analysis for finite-volume schemes for scalar conservationlaws where the matrices commute trivially relies on the L1-theory of Kruzkov [9].

To end the introduction we mention some related results on Friedrichs systems obtained forfinite-element approaches. In [10] the authors consider a boundary value problem for theFriedrichs system in the stationary case. Using the Galerkin method the authors derive anefficient and reliable, residual-based a posteriori error estimate in the L2-norm for a locallygenerated component of the global error (the so-called cell error). The finite-volume method forinitial boundary value problem for (1.1) is analyzed in [11], and an error estimate like (1.3) isproven. Another approach to a posteriori error estimates for finite-element methods is describedin [12].

Note: During the review process for this article a new version of [5] has been published [13].In this version the authors extend the convergence result from [5] to the case of weak solutionsin [C([0, T]; H1(�d))]m � [C1([0, T]; L2(�d))]m. However, their results do not cover discon-tinuous weak solutions.

II. NOTATIONS, ASSUMPTIONS, AND BASIC ANALYTICAL RESULTS

In this section we introduce some notations valid for the whole article and specify allassumptions on the coefficients in the initial value problem (1.1), (1.2). All analytical results thatturn out to be necessary in the rest of the section are reviewed.

A. Notations and Assumptions

For s � 0 and m � � we denote by [L2(�d)]m, [Hs(�d)]m � [H2s(�d)]m the usual Lebesgue and

Sobolev spaces equipped with the norms � � �2, � � �H s, respectively. Cb0,1(�d � [0, T]) is the set

of bounded, Lipschitz continuous functions on �d � [0, T]. Furthermore, for l � �, we needthe Bochner spaces [Cl([0, T]; X )]m and [L2((0, T ); X )]m, where X is an arbitrary function spaceon �d. The corresponding norms are denoted by � � �C l([0,T];X ) and � � �L2((0,T );X ). If M : �d � [0,T] 3 �m�m, then

�M�� :� sup�x,t��d��0,T�

�M�x, t��,

where � � � denotes the matrix norm induced by the Eucledean norm on �m.Provided A1, . . . , Ad in (1.1) are bounded and sufficiently smooth, we define for (x,

t) � �d � [0, T]

div A�x, t� � �i�1

d�Ai

�xi�x, t�, � � sup��

i�1

d

niAi�

�

�n � �n1, . . . , nd�T � �d, �n� � 1�,

�12�B � BT � div A��, � 1 � �div A�� B��. (2.1)

106 JOVANOVIC AND ROHDE

Define also for a sufficiently smooth function z : �d � (0, T ) 3 �m and t � 0,

�Dz� � , t��22 � �

�d

�i�1

d � �z

�xi�x, t�� 2

dx. (2.2)

Finally, for x0 � �d and �, R � 0, the cone �(x0, R) is given by

��x0, R� � �x, t��x � x0� � R � ��T � t�, t � �0, T�. (2.3)

To be precise we introduce the following assumptions on the coefficients in (1.1), (1.2) that willhold throughout the article.

Assumption 2.1. Consider the initial value problem (1.1), (1.2).

(i) The mappings B, A1, . . . , Ad : �d � [0, T] 3 �m�m satisfy

B, A1, . . . , Ad � C��d � �0, T�, �m�m�,

Ai(x, t)T � Ai(x, t)(i � 1, . . . , d) � (x, t) � �d � [0, T],

�i�1

d

��tj�x

Ai�� tj�x

B�� , � � �0d, j � �0.

(ii ) The functions u0 and f satisfy u0 � [L2(�d)]m and f � [L2((0, T ); L2(�d))]m.

A function u � [C1(�d � [0, T])]m is called a classical solution of (1.1), (1.2) if (1.1) and(1.2) hold pointwise. We say that u � [C([0, T]; L2(�d))]m is a weak solution of (1.1), (1.2) if(1.1) holds in the distributional sense in �d � (0, T ) and if u(x, 0) � u0(x) holds for almost allx � �d. We cite the following theorem on existence, uniqueness, and regularity of weaksolutions that can be deduced from the results in the book [14].

Theorem 2.2. Let Assumption 2.1 be satisfied. For s � 0, suppose that we haveu0 � [Hs(�d)]m and f � [L2((0, T); Hs(�d))]m.

Then there exists a unique weak solution u � [C([0, T]; Hs(�d))]m of (1.1), (1.2). Further-more, there is a positive constant C � C(T, B, Ai) such that

supt��0,T�

�u� � , t��Hs � C��u0�Hs � � f �L2��0,T �;Hs��. (2.4)

If u0 � [C0�(�d)]m and f � [C0

�(�d � [0, T])]m, then u is a classical solution and lies in the space[C0

�(�d � [0, T])]m.The a priori estimate (2.4) is obtained using the theory of pseudodifferential operators.

However, one can derive by the energy method more precise estimates in 2-D foru0 � [H1(�2)]m, f � [L2((0, T ); H1(�2))]m, and consequently u � [C([0, T]; H1(�2))]m:


supt��0,T�

e��0t�u� � , t��2 � �u0�2 � 2 �0

T

e��0s�f � � , s��2 ds, (2.5)

where �0 � 12

�div A�� B��, and

supt��0,T�

e��1t�Du� � , t��2 � �Du0�2 � 2 �0

T

e��0s�Df � � , s��2 ds

� 2T�j�1

2 ��i�1

2

�ij2Ai � �jB�

�

21/2

supt��0,T�

e��0t�u� � , t��2, (2.6)

where �1 � ��* � 12

�div A�� B�� with � � �2�2, �ij � 12

(��jAi�� iA

j��)(i, j � {1, 2})and � � �* is the operator norm induced by the Euclidean norm on �2.

These inequalities will be later used for the a posteriori estimates. Following [2] weintroduce two useful measures.

Definition 2.3. Let Assumption 2.1 be satisfied and let v � [L2(�d � (0, T))]m be given. Theweak consistency measure v : [C1([0, T]; L2(�d))]m � [C0([0, T]; H1(�d))]m 3 � and thedissipation measure �v : Cb

0,1(�d � [0, T]) 3 � are defined for � � [C1([0, T]; L2(�d))]m

� [C0([0, T]; H1(�d))]m and � � Cb0,1(�d � [0, T]) by

v, �� 0

T ��d

v � �t� � �i�1

d

Aiv � �i� dxdt � �0

T ��d

�Bv � f � � � dxdt

� ��d

u0 � �� , 0� dx, �v, �� 1

2 �0

T ��d

�v�2�t� � �i�1

d

vTAiv�i� dxdt

� �0

T ��d

1

2vT�div A � B � BT�v � f � v� dxdt �

1

2 ��d

�u0�2�� , 0� dx.

Using these measures we can estimate the L2-error between a solution of (1.1), (1.2) and anarbitrary approximation in a compact form. We demonstrate this for a smooth classical solution.Recall the notations (2.1) and (2.3).

Proposition 2.4. Let u0 � [C0�(�d)]m, f � [C0

�(�d � [0, T])]m, v � [L2(�d � (0, T))]m and �v, v the corresponding measures from Definition 2.3. Suppose that u � [C0

�(�d � [0, T])]m is theclassical solution of (1.1), (1.2). Then we have

1

2 �C�x0,R�

e�t�u � v�2 dxdt � �v, �� v, �u�, (2.7)


1

2 �0

T ��d

e�t�u � v�2 dxdt � �v, �� v, �u�. (2.8)

The functions � : �d � [0, T] 3 � and � : [0, T] 3 � are defined by

��x, t� � e�t�T � t��,R�x, t�, � �t� � e�t�T � t� ��x, t� � �d � �0, T��, (2.9)

where

��,R�x, t� � ��x � x0� � R � ��T � t��, ��s� � �1: s � 01 � s/�: s � �0, � �0: s � �

,

for the constants , � given by (2.1) and R, � � 0 arbitrary.Note that ��,R is a Lipschitz regularization of the indicator function for the cone �(x0, R).Proof of Proposition 2.4. Define h(u, v) � 1

2�u � v�2 and fi(x, t, u, v) � 1

2(u � v)TAi(x,

t)(u � v) for i � 1, . . . , d. Let � � �u and � � Cb0,1(�d � [0, T]) with �( � , T ) � 0. A

straightforward calculation shows that

��0

T ��d

h�u, v��t� � �i�1

d

fi�x, t, u, v��i� dxdt

� �1

2 �0

T ��d

�u � v�T�div A � B � BT��u � v�� dxdt � �v, �� v, ��.

Furthermore, we deduce from the inequality

�1

2 �0

T ��d

�u � v�T�div A � B � BT��u � v�� dxdt � �0

T ��d

h� dxdt,

that

��0

T ��d

h�u, v��t� � �i�1

d

fi�x, t, u, v��i� dxdt � �0

T ��d

h� dxdt � �v, �� v, ��. (2.10)

From the definition of � we compute the derivatives

�t��x, t� � ��x, t� � e�t��,R�x, t� � �e�t�T � t��x � x0� � R � ��T � t��,�i��x, t�

� ��x � x0� � R � ��T � t��xi � x0i

�x � x0�e�t�T � t� �i � 1, . . . , d �


Since, because of the definition (2.1) of �,

�i�1

d �xi � x0i��x � x0�

�fi�x, t, u, v�� h�u, v�

for (x, t) � �d � [0, T], x � x0, and �� 0 a.e. on �, we have

��0

T ��d

h�u, v��t� � �i�1

d

fi�x, t, u, v��i� dxdt � �0

T ��d

h�u, v�� dxdt

� �0

T ��d

e�t��,Rh�u, v� dxdt � �0

T ��d

e�t�T � t��x� � R � ��T � t��

� ��i�1

d

fi�x, t, u, v�xi � x0i

�x � x0�� h�u, v�� dxdt � �

0

T ��d

h�u, v�� dxdt

� �0

T ��d

e�t��,Rh�u, v� dxdt.

Now, replacing the test function � in (2.10) by � one obtains (2.7).Similarly, the choices � � � and � � �u in (2.10) lead to (2.8). y

III. FINITE-VOLUME DISCRETIZATION AND THE THEOREM OF VILA ANDVILLEDIEU

We approximate the weak solution of the Friedrichs system by a finite-volume scheme onunstructured meshes. This construction of the scheme follows [5].

For some index set � � � let a family {Ki}i�� of open nonempty sets be given. This familyis called a triangulation if each element is a convex polyhedron, �i�� Ki � �d, and

Ki � Kj � A @i, j � �, i � j, h :� supi��

diam�Ki� � �.

We denote the family {Ki}i�� by �h and introduce the following notations for K � �h.

�K�: area of K,e � �K: an edge of K with length �e�,

nK,e � (nK,e1 , . . . , nK,e

d )T: unit outward normal to the edge e of K,Ke: neighboring cell of K with K� � K� e � e,

�(K ): Set of all edges of K.

For N � �, let 0 � t0 � t1 � . . . � tN � T be a partition of the interval [0, T]. We denote�t n � t n�1 � t n for n � � � {N}, � � {0, . . . , N � 1}.


Let l � �. For each n � {0, . . . , N}, K � �h, and e � �(K ) we define

SKn �

1

�tn�K� �tn

tn�1 �K

S�x, t� dxdt, Sen �

1

�tn�K� �tn

tn�1 �e

S��, t� d�dt. (3.1)

Definition 3.1. Suppose that Assumption 2.1 holds. Let a triangulation �h and a partition of[0, T] as defined above be given.

The piecewise constant finite-volume approximation uh : �d � [0, T ) 3 �m is given by

uh�x, t� � vKn for �x, t� � K � �tn, tn�1�. (3.2)

Thereby, the vectors vKn � �m are given for n � 0, K � �h by

vK0 �

1

�K� �K

u0�x� dx, (3.3)

and iteratively for n � �, K � �h by

vKn�1 � vK

n ��tn

�K� �e��K �

�e�gK,en �vK

n , vKe

n � � �tBKn vK

n � �tnf Kn . (3.4)

The numerical flux gK,en : �m � �m 3 �m is defined for K � �h and e � �(K ) by

gK,en �u, v� � �CK,e

n v � DK,en u �u, v � �m�,

where

CK,en � ��AK,e

n �� I, DK,en � �AK,e

n �� I, AK,en � �

i�1

d

nK,ei �Ai�e

n. (3.5)

The operators �� and �� stand for positive and negative part of a symmetric matrix. � is anarbitrary positive number.

The definitions (3.5) lead to a consistent upwind numerical flux. Note that we have

CK,em � DKe,e

n , (3.6)

for K � �h and e � �(K ). This leads to

gK,en �v, w� � �gKe,e

n �w, v� �v, w � �m�,

and ensures that the scheme (3.4) is conservative. Moreover the finite-volume scheme is linear.


The formula (3.4) can be rewritten in a nonconservative form that turns out to be moreconvenient: using Green’s formula yields ¥e��(K ) �e�AK,e

n � �K�(div A)Kn . Together with (3.6) we

are led to

vKn�1 � vK

n ��tn

�K� �e��K �

�e�CK,en �vK

n � vKe

n � � �tn��BKn � �div A�K

n �vKn � f K

n �. (3.7)

In the rest of this section we explain some results from [5] in greater detail to provide thenecessary setting for our considerations. To obtain these results the following conditions on thetime step and the triangulation are supposed.

Since the scheme is explicit, we need the time step to be restricted by a CFL-like condition.In our case we suppose for some � � (0, 1)

supK��h,e��K �

n��

�tn��K��K� �CK,e

n � � 1 � �. (3.8)

Here it is ��K� � ¥e��(K ) �e�. For the a priori error analysis we assume additionally that the timestep is constant, that is, for �t � 0,

�tn � �t, tn � n�t, (3.9)

and for from (2.1) the timestep condition

�t �12

. (3.10)

Finally, the triangulation �h satisfies also some regularity conditions: there exists a real numbera � 0 such that

�K� � ahd �K � �h� (3.11)

and

�e� � ahd�1 �K � �h, e � ��K ��. (3.12)

Note that in the case when �h consists of simpleces (triangles, tetrahedra), (3.12) is aconsequence of (3.11).

Now, we recall the following discrete analogue to (2.4) with respect to the L2-norm.

Proposition 3.2. Let Assumption 2.1 hold. Consider the finite-volume approximation fromDefinition 3.1 and suppose that the conditions (3.8), (3.9), (3.10), and (3.11) hold.

Then there exists a constant C � C(T, Ai, B, �) such that we have

supt��0,T �

�uh� � , t��2 � C��u0�2 � � f �L2��0,T �;L2��. (3.13)


Furthermore, the discrete time derivatives of uh satisfy

�n��

�K��h

�K��vKn�1 � vK

n �2 � C��u0�22 � �f�L2��0,T �;L2�

2 �, (3.14)

�n��

�K��h


n �2�n � CQh�� C�t��u0�22 � �f�L2��0,T �;L2�

2 �. (3.15)

Here � is chosen as in (2.9) and �n � �t nt n�1

� (s) ds/�t. The meaning of the symbol Qh is statedin Proposition 4.2 below, where (3.15) is actually needed the first time.

Proof. For (3.13) and (3.14) see [5, Proposition 3.2]. For the first inequality (3.14) see theproof of Theorem 4.1 (also in [5]). y

The starting point of our a priori and a posteriori error analysis is the following propositiontogether with Theorem 2.4 in which the comparison function v is chosen to be the finite-volumeapproximation uh.

Proposition 3.3. Let Assumption 2.1 be satisfied and consider the finite-volume approximationuh from Definition 3.1. Suppose that the CFL condition (3.8) and (3.9) hold. If we choose v �uh in Definition 2.3, then we get

uh, ��

l�1

7

Rhl ��, 2 �uh

, �� l�1

7

Ehl �� Qh��, (3.16)

where � : �d � [0, T]3 [0, �) and � : �d � [0, T]3 �m are smooth functions with compactsupport in x. Here we used

Rh1��

n��

�K��h


n � � ��K�tn�1� � �Kn �, Rh

2�� n��

�K��h

�e��K �

�t�e��vKn

� vKe

n � � CK,en ��e

n � �Kn �, Rh

3�� d

�uh�x, 0� � u0�x�� x, 0� dx,

Rh4��

n��

�K��h

�t�K�vKn � � �

e��K �

�e��K� AK,e

n �en �

1

�t�K� �tn

tn�1 �K

�i�1

d

�Ai��xidxdt�,

Rh5��

n��

�K��h

�t�K�vKn � � 1

�t�K� �tn

tn�1 �K

�div A�� dxdt � �div A�Kn �K

n�,

Rh6��

n��

�K��h

�t�K�vKn � � 1

�t�K� �tn

tn�1 �K

B� dxdt � BKn �K

n�,

Rh7��

n��

�K��h

�t�K�� 1

�t�K� �tn

tn�1 �K

f � � dxdt � fKn � �K

n�,

and


Eh1��

n��

�K��h

�K��vKn�1�2 � �vK

n �2��K�tn�1� � �Kn �,

Eh2��

n��

�K��h

�e��K �

�t�e��en � �K

n ��vKn � vKe

n � � CK,en �vK

n � vKe

n �,

Eh3��

�d

��uh�x, 0��2 � �u0�x��2��x, 0� dx,

Eh4��

n��

�K��h

�t�K�vKn � � �

e��K �

�e��K� �e

nAK,en �

1

�t�K� �tn

tn�1 �K

�i�1

d

��Ai�xidxdt�vK

n ,Eh5��

� �n��

�K��h

�t�K�vKn � � 1

�t�K� �tn

tn�1 �K

��2 div A � B � BT� dxdt

� �Kn �2�div A�K

n � BKn � �BK

n �T��vKn ,

Eh6�� 2 �

n��

�K��h

�t�K�� 1

�t�K� �tn

tn�1 �K

�f dxdt � �Kn fK

n�vKn ,

Eh7�� 2 �

n��

�K��h

�t�K��Kn �vK

n�1 � vKn � � ��BK

n � �div A�Kn �vK

n � fKn �.

The nonnegative term Qh is defined by

Qh�� n��

�K��h

�e��K �

�t�e��vKn � vKe

n �CK,en

2�K

n .

For the definition of �Kn , �e

n, �Kn , �e

n we refer to (3.1). Denote �K(t) � 1/�K� �K �(x, t) dx. By� � �CK,e

n we mean the norm on �m generated by the positive definite matrix CK,em , that is, �v�CK,e

n :�(CK,e

m v, v)1/2 for v � �m.Proof. See [5, Proposition 4.1]. y

Remark 3.1. In fact the relations (3.16) hold for the test functions �, � from Definition 2.3.Note that Proposition 3.3 does apply in the case of nonuniform time steps �t n, that is, we do nothave to take �t � const.

Estimating the terms in (3.16) and applying the stability result (2.8) Vila and Villedieuobtained the following theorem.

Theorem 3.4. Let Assumption 2.1 hold and let u0 � [Hs(�d)]m and f � [C1([0, T]; Hs(�d))]m

for s � 1 � d/2. Consider the finite-volume approximation uh from Definition 3.1 and supposethat (3.8), (3.9), and (3.11) hold.

For an arbitrary compact subset Q of �d there exists a constant C � C(Q, T, �, Ai, B,�Du�L�

) � 0, such that the error estimate

�u � uh�L2�Q� �0,T �� Ch1/2


holds. u � [C([0, T]; Hs(�d))]m � [C1([0, T]; Hs�1(�d))]m is the weak solution of the initialvalue problem (1.1), (1.2).

Proof. See [5, Theorem 4.1]. yNote that the Sobolev embedding theorem and the fact that u is a weak solution ensure that

u � [C([0, T]; Hs(�d))]m � [C1([0, T]; Hs�1(�d))]m is a classical solution, in particular, that u �[C1(�d � [0, T])]m holds.

IV. THE IMPROVED A PRIORI ERROR ESTIMATE

In this section we show that the Theorem 3.4 can be generalized to weak solutions in the space[C([0, T], Hs(�d))]m for s � (0, 1]. Part A treats the case s � 1 and leads to exactly the rate h1/2

as in Theorem 3.4. Part B uses the interpolation theory to obtain the rate hs/2 for s � [0, 1]. Atthe same time we reduce the required regularity for u0 and f.

Throughout the subsection we consider the finite-volume scheme from Definition 3.1 for afixed triangulation �h and a uniform time step.

A. The Error Estimate for Weak Solutions in [C([0, T], H1(�d))]m

Theorem 4.1. Let Assumption 2.1 hold and let u0 � [H1(�d)]m and f � [L2((0, T); H1(�d))]m.Consider the finite-volume approximation uh from Definition 3.1 and suppose that (3.8), (3.9),(3.10), (3.11), and (3.12) hold.

Then there exists a constant C � C(T, �, A1, A2, B) � 0 such that the error estimate

�u � uh�L2��d� �0,T �� Ch1/2��u0�H1 � �f�L2��0,T �;H1�� (4.1)

holds. u � [C([0, T]; H1(�d))]m is the weak solution of the initial value problem (1.1), (1.2) andQh is defined by (4.3).

The proof of Theorem 4.1 will be delivered at the end of the subsection. It relies on a numberof observations and lemmas that we present in advance.

Proposition 4.2. Suppose that we have u0 � [C0�(�d)]m, f � [C0

�(�d � [0, T])]m such that theweak solution u of (1.1), (1.2) satisfies u � [C0

�(�d � [0, T])]m. Suppose further that (3.8) and(3.9) hold.

Then, for � given by (2.9) and � � �u, we have

�0

T ��d

e�t�u � uh�2 dxdt � �Qh�� l�1

7

��hl �� 2Rh

l ��. (4.2)

The terms Rhl are defined in Proposition 3.3. Further we have

�h1��

n��

�K��h

�K��vKn�1�2 � �vK

n �2�� tn�1� � �n�,

�h2�� 0, �h

3�� d

��uh�x, 0��2 � �u0�x��2�� 0� dx,


�h4��

n��

�K��h

�t�K�vKn � � �

e��K �

�e��K� �nAK,e

n �1

�t�K� �tn

tn�1 �K

�i�1

d

��Ai�xidxdt�vK

n ,

�h5��

n��

�K��h

�t�K�vKn � � 1

�t�K� �tn

tn�1 �K

� �2 div A � B � BT� dxdt

� �n�2�div A�Kn � BK

n � �BKn �T��vK

n ,

�h6�� 2 �

n��

�K��h

�t�K�� 1

�t�K� �tn

tn�1 �K

�f dxdt � �nfKn�vK

n ,

�h7�� 2 �

n��

�K��h

�t�K��n�vKn�1 � vK

n � � ��BKn � �div A�K

n �vKn � fK

n �.

The nonnegative term Qh is now defined by

Qh�� n��

�K��h

�e��K �


n �CK,en

2�n. (4.3)

Finally, the important term �n is given by

�n �1

�t �tn

tn�1

� �t� dt. (4.4)

Proof. It follows from relations (3.16) for � � � and � � �u, Remark 3.1 and (2.8). yIn the sequel we need the inequality

maxt��tn,tn�1�

� �t� � 2e�t�n for n � �. (4.5)

The subsequent lemma presents an adaption of the Bramble-Hilbert Lemma ([15]) to our needs.

Lemma 4.3. Assume that the unstructured mesh �h satisfies (3.11) and (3.12). If K � �h,then, for each function z � [H1(K )]m we have

�K

�z � zK�2 dx � ch2 �K

�Dz�2 dx, (4.6)

�e

�z � zK�2 d� � ch �K

�Dz�2 dx, �e � ��K ��, (4.7)


where zK � 1/�K� �K z dx, �Dz(x)�2 � ¥i�1d ��iz(x)�2 and the constant c depends only on d and

m. In the special case, if d � 2 and if �h consists of triangles, then, under the mesh condition(3.11), it is

�K

�z � zK�2 dx �64�2

9�K

2 �K

�Dz�2 dx, (4.8)

�e

�z � zK�2 d� � 32 2 � 20 6

3�2 �

2

3 �K3

�K� �K

�Dz�2 dx �e � ��K ��, (4.9)

for �K given by

�K2 � �

e��K �

�e�2. (4.10)

We have computed the constants on (4.8) and (4.9) explicitly since they are needed in the aposteriori error estimate in this form.

Proof of Lemma 4.3. Due to the approximation argument, we can assume thatz � [C1(K� )]m, z � z(x). Denote by c a generic constant that depends only on d and m. Thanksto convexity of K � �h, following the arguments from [15], one can deduce that

�z � zK�L2�K � � c�diam K �d�1

�K� �Dz�L2�K �.

This proves (4.6). For K � �h, e � �(K ), let Ke � K be a simplex with the following property:its edge is e and it has a vertex which is the vertex of K with a maximal distance from e. Denoteby hK,e the height of Ke which corresponds to e. Then, we have

hK,ehd�1 � hK,e�diam K �d�1 � �K� � ahd. (4.11)

Hence, hK,e � ah. On the other hand, applying (3.12), we get

�Ke� � chK,e�e� � chd. (4.12)

Let further

K0 :� �x � �d : �i�1

d

xi � 1, xi � 0, i � 1, . . . , d�and � (y), : K0 3 Ke is a bijective affine mapping. It is well known that �D� � ch.

Therefore,

�Dy�z � �� ch� �Dxz� � �


and

�e

�v�2 d�x � chd�1 �e0

�v � �2 d�y,

for arbitrary v � [C1(K� )]m, where e0 :� �1(e) is an edge of K0. Using the last two inequalities,the trace theorem, (4.12) and (4.6), one obtains

�e

�z � zK�2 d�x � chd�1 �e0

�z � � zK�2d�y � chd�1 �K0

�z � � zK�2 dy

� chd�1 �K0

�Dy�z � ��2 dy � chd�1 �K0

�z � � zK�2 dy � chd�1 �K0

��Dxz� � �2 dy

�chd�1

�Ke� �Ke

�z � zK�2 dx �chd�1

�Ke� �Ke

�Dz�2 dx � ch�1 �K

�z � zK�2 dx � ch �K

�Dz�2 dx

� ch �K

�Dz�2 dx.

Thus, (4.7). One can get (4.8), (4.9) in a similar way by evaluating the constants in (4.11) andin the trace inequality. y

The Lemma 4.3 will now be applied to the function � � �u from Proposition 4.2.

Lemma 4.4. Let the assumptions of Proposition 4.2 hold together with (3.11), (3.12), andassume that the weak solution u of (1.1), (1.2) lies in the space [C0

�(�d � [0, T])]m.Then there exists a constant C � C(a, �, T, Ai, B) � 0 such that for each K � �h and

e � �(K) we have the following estimates for � � �u with � given by (2.9):

�tn

tn�1 �K

�� x, t� � �Kn �2 dxdt � Ch2 �

tn

tn�1 �K

��u

�t� 2

� �Du�2 � �u�2 dxdt,

�tn

tn�1 �e

�� , t� � �Kn �2 d�dt � Ch �

tn

tn�1 �K

��u

�t� 2

� �Du�2 � �u�2 dxdt. (4.13)

Proof. According to statement (4.6) in Lemma 4.3, we get for t � [0, T]

�K

�� x, t� � �K�t��2 dx � Ch2 �K

�D� �x, t��2 dx.

So we have


�tn

tn�1 �K

�� x, t� � �K�t��2 dxdt � Ch2 �tn

tn�1 �K

�D��2 dxdt. (4.14)

On the other hand, for t � [t n, t n�1],

��K�t� � �Kn �2 � � 1

�t�K� �tn

tn�1 �K

�s

t ��

�t�x, �� d�dxds� 2

��t

�K� �tn

tn�1 �K

��

�t� 2

dxdt.

From the last inequality it follows

�tn

tn�1 �K

��K�t� � �Kn �2 dxdt � �t2 �

tn

tn�1 �K

��

�t� 2

dxdt. (4.15)

The inequalities (4.14), (4.15), and �� (x, t) � �Kn �2 � 2�� (x, t) � �K(t)�2 � 2��K(t) � �K

n �2 imply

�tn

tn�1 �K

�� x, t� � �Kn �2 dxdt � C �

tn

tn�1 �K

�t2��

�t� 2

� h2�D��2 dxdt.

The definition of � and (3.8) yield the assertion of the lemma. The second inequality in thisLemma one creates similarly. y

Remark 4.2. Note that the relations (4.13) also hold if

u � �C1��0, T�; L2��d��m � �C��0, T�; H1��d��m.

Using the last lemma we can present the proof of Theorem 4.1. In particular it enables us toexpress the error in terms of the norms supt�[0,T]�u( � , t)�H 1 instead of supt�[0,T]�u( � ,t)�� supt�[0,T]�Du( � , t)��. These estimates permit to extend the convergence proof to weaksolutions.

Proof of Theorem 4.1. In the first step of the proof let us assume that we have in additionto the conditions of the theorem u0 � [C0

�(�d)]m and f � [C0�(�d � [0, T])]m. Then, according

to Theorem 2.2, there is a classical solution u � [C0�(�d � [0, T])]m of (1.1), (1.2). Furthermore

the estimate (2.4) for s � 0, 1 and (1.1) imply that there is a constant C � C(T, Ai, B) such that

�0

T ��d

�u�2 dxdt � C��u0�22 � �f�L2��0,T �;L2�

2 �, (4.16)

�0

T ��d

��u

�t� 2

� �Du�2 dxdt � C��u0�H12 � �f�L2��0,T �;H1�

2 �. (4.17)


To prove the assertion of the theorem in the case of smooth solutions, we will estimate the termsin the sum of the right-hand side term of (4.2) in Proposition 4.2. Recall that � � �u.

Estimate for �h1 � 2Rh

1. Consider the decomposition Rh1(�) � Rh

1,a � Rh1,b with

Rh1,a � �

n��

�K��h

�K�� tn�1� � �n��vKn�1 � vK

n � � uK�tn�1�,

Rh1,b � �

n��

�K��h


n � � ��nuK�tn�1� � ��u�Kn �].

To estimate Rh1,b observe that we have thanks to (4.5)

��nuK�tn�1� � ��u�Kn )� �

1

�K��t �tn

tn�1 � �K

u�x, tn�1� � u�x, t� dx�� t� dt

� C�n1

�K� �tn

tn�1 �K

��u

�t� dxdt � C�n �t

�K� 1/2�tn

tn�1 �K

��u

�t� 2

dxdt 1/2

.

With the last inequality we get for Rh1,b

�Rh1,b� � C �

n��

�K��h

��t�K��1/2�vKn�1 � vK

n ��n�tn

tn�1 �K

��u

�t�2

dxdt1/2

. (4.18)

Applying the Schwarz-Cauchy inequality to the sum in (4.18), then using (3.15) and inequality(4.17) lead to

�Rh1,b� � C�t1/2 �

n��

�K��h

�n�K��vKn�1 � vK

n �21/2�0

T ��d

��u

�t�2

dxdt1/2

� C�h1/2Qh1/2��u0�H1 � � f �L2��0,T �;H1�� h��u0�H1 � � f �L2��0,T �;H1��

2�.

Using the splitting vKn�1 � vK

n � 2uK(t n�1) � vKn�1 � vK

n � 2(vKn � uK

n ) � 2(uKn � uK(t n�1)),

we obtain from

�vKn � uK

n � �1

��K��t�1/2 �tn

tn�1 �K

�u � uh�2 dxdt 1/2

,

�uKn � uK�tn�1�� t

�K� 1/2�tn

tn�1 �K

��u

�t� 2

dxdt 1/2

,

and the estimates (3.14), (4.17) the inequality


�n��

�K��h

�t�K��vKn�1 � vK

n � 2uK�tn�1��2 � C��u � uh�L2��d� �0,T ��2 � �t��u0�H1

2 � �f�L2��0,T �;H1�2 ��.

The last inequality together with (3.14) reads

��h1�� 2Rh

1,a�

� �n��

�K��h

��n � � �tn�1��K��vKn�1 � vK

n ��vKn�1 � vK

n � 2uK�tn�1��

� �t1/2 �n��

�K��h


n �2 1/2

��u � uh�L2��d� �0,T ��

� �t1/2��u0�H1 � �f�L2��0,T �;H1��)

� C�h1/2�u � uh�L2��d� �0,T ��u0�H1 � �f�L2��0,T �;H1�� h��u0�H1 � �f�L2��0,T �;H1��2�.

From the last and the estimate for Rh1,b we get

��h1�� 2Rh

1�� C�h1/2Qh1/2��u0�H1 � �f�L2��0,T �;H1��

� h1/2�u � uh�L2��d� �0,T ��u0�H1 � �f�L2��0,T �;H1�� h��u0�H1 � �f�L2��0,T �;H1��2�.

Estimate for Rh2. In order to obtain estimates in this case we have to consider the term

�en � �K

n :

��en � �K

n � � � 1

�t �tn

tn�1 1

�e� �e

u��, t� d� �1

�K� �K

u�x, t� dx� �t� dt�� max

t��tn,tn�1�

� �t�1

�t�e�1/2 �tn

tn�1 �e

�u��, t� �1

�K� �K

u�x, t� dx�2

d�1/2

.

The estimates (4.7), (4.5), and the regularity assumption (3.11) imply

��en � �K

n � � C�n1

�t1/2 �tn

tn�1 �K

�Du�2 dxdt 1/2

,

which leads to



n � �Kn �� C�n�t1/2�e��vK

n � vKe

n �CK,en �

tn

tn�1 �K

�Du�2 dxdt 1/2

.

If we apply Cauchy-Schwarz inequality, we obtain


�Rh2�� CQh

1/2�� n��

�K��h

�e��K �

�n�e� �tn

tn�1 �K

�Du�2 dxdt1/2

,

which together with the a priori estimate (4.17) yields

�Rh2�� Ch1/2Qh

1/2��u0�H1 � � f �L2��0,T �;H1��.

Estimate for �h3 � 2Rh

3. It is easy to check out that

�h3�� 2Rh

3�� T ��d

�uh�x, 0� � u0�x��2 dx. (4.19)

As a consequence of (4.6) we have

�K

�u0�x� � uh�x, 0��2 dx � �K

�u0�x� �1

�K� �K

u0�y�dy� 2

dx � Ch2 �K

�Du0�2 dx.

Then from (4.19) one obtains

��h3�� 2Rh

3�� Ch2 ��d

�Du0�2 dx � Ch2�u0�H12 .

Estimate for Rh4. Due to the definition of AK,e

n in (3.5), Assumption 2.1 (i), and Lemma 4.4,we have

� �e��K �

�e��K� AK,e

n �en �

1

�t�K� �tn

tn�1 �K

�i�1

d

�Ai��xidxdt� � � 1

�t�K� �e��K �

�tn

tn�1 �e

� �AK,en � �

i�1

d

nK,ei Ai� �� K

n � d�dt� �Ch

�t�K� �e��K �

�tn

tn�1 �e

�� Kn � d�dt

� C1

�t1/2 �tn

tn�1 �K

��u

�t� 2

� �Du�2 � �u�2 dxdt 1/2

.

The Cauchy-Schwarz inequality together with the estimate above yields

�Rh4�� Ch �

n��

�K��h

�t�K��vKn �21/2�

0

T ��d

��u

�t�2

� �Du�2 � �u�2 dxdt1/2

.


Thus, by the a priori estimates (3.13) for the discrete solution and (4.16), (4.17) for the weaksolution, we obtain

�Rh4�� Ch��u0�H1 � �f�L2��0,T �;H1��

2.

Estimates for Rh5, Rh

6. Lemma 4.4 yields

� 1

�t�K� �tn

tn�1 �K

�div A�� dxdt � �div A�Kn �K

n� � � 1

�t�K� �tn

tn�1 �K

�div A�� Kn � dxdt�

�Ch

��t�K��1/2 �tn

tn�1 �K��u

�t�2

� �Du�2 � �u�2 dxdt1/2

.

Consequently, we get

�Rh5�� Ch �

n��

�K��h

�t�K��vKn �21/2�

0

T ��d

��u

�t�2

� �Du�2 � �u�2 dxdt1/2

,

and therefore,

�Rh5�� Ch��u0�H1 � �f�L2��0,T �;H1��

2.

In a similar way one obtains the same estimate for Rh6.

Estimate for Rh7. From the estimate

� 1

�t�K� �tn

tn�1 �K

f � � dxdt � fKn � �K

n � � � 1

�t�K� �tn

tn�1 �K

f � �� Kn � dxdt�

�Ch

�t�K� �tn

tn�1 �K

�f�2 dxdt 1/2�tn

tn�1 �K��u

�t� 2

� �Du�2 � �u�2 dxdt 1/2

and the Cauchy-Schwarz inequality we have

�Rh7�� Ch�f�L2��0,T �;L2��u0�H1 � �f�L2��0,T �;H1�� Ch��u0�H1 � �f�L2��0,T �;H1��

2.

Estimates for �h4, �h

5, �h6. From

�vKn � � �

e��K �

�e��K� �nAK,e

n �1

�t�K� �tn

tn�1 �K

�i�1

d

� �Ai�xidxdt�vK

n �� 1

�t�K� vKn � �

e��K �

�tn

tn�1 �e

��n � ��AK,en � �

i�1

d

nK,ei Ai d�dtvK

n � � C�th�e�

�K� �vKn �2,


and the a priori estimate (3.13) for the discrete solution one obtains

��h4�� Ch��u0�2 � �f�L2��0,T �;L2��

2.

The same estimates can be obtained for �h5(� ) and �h

6(� ) using a similar procedure as forRh

5(�) and Rh7(�), respectively.

Estimate for �h7. By the Cauchy-Schwarz inequality we have

��h7�� C�t1/2 �

n��

�K��h


n �2�n 1/2

� �n��

�K��h

�n�t�K��BKn � �div A�K

n �vKn � fK

n �21/2

.

As a consequence of (3.13), (3.15), and Assumption 2.1 one obtains

��h7�� C�t1/2Qh

1/2�� t1/2��u0�L2 � �f�L2��0,T �;L2��u0�L2 � �f�L2��0,T �;L2�

� Ch1/2Qh1/2��u0�L2 � �f�L2��0,T �;L2�� Ch��u0�L2 � �f�L2��0,T �;L2��

2.

Taking into account the basic estimate (4.2) and the estimates for Rhl , �h

l , l � 1, . . . , 7, we have

�u � uh�L2��d� �0,T ��2 � �Qh�� Ch1/2�u � uh�L2��d� �0,T ��u0�H1 � �f�L2��0,T �;H1��

� Ch1/2Qh1/2��u0�H1 � �f�L2��0,T �;H1�� Ch��u0�H1 � �f�L2��0,T �;H1��

2.

Appropriate application of Young’s inequality on the right-hand side of the last estimateleads us to

1

2 �0

T ��d

e�t�u � uh�2 dxdt �1

2�Qh�� Ch��u0�H1 � �f�L2��0,T �;H1��

2. (4.20)

Thus we have proven the theorem for smooth data and a classical solution but in terms of weakernorms.

Complete estimate for the weak solution. Now, consider the sequences {u0k}k�� and

{ f k}k�� with u0k � [C0

�(�d)]m and f k � [C0�(�d � [0, T])]m for k � �. Assume that we have

limk3�

�u0k � u0�H1 � lim

k3�

�f k � f �L2��0,T �;H1� � 0.

Such sequences can be constructed using mollifiers. Thanks to the linearity of the initial valueproblem for the Friedrichs system (1.1) and thanks to the a priori estimate (2.4) in the case s � 1we conclude uk 3 u in [C([0, T]; H1(�d))]m. Here uk denotes the weak solution of (1.1), (1.2)with initial function u0

k and source term f k. Let uhk denote the corresponding finite-volume


approximation from Definition 3.1. We have uhk3 uh in [L�((0, T ); L2(�d))]m. This results from

the linearity of the scheme (3.7) and the discrete a priori estimate (3.13). According to (4.20)one obtains

�0

T ��d

�uk � uhk�2 dxdt � Ch��u0

k�H1 � �fk�L2��0,T �;H1��2. (4.21)

Passing to the limit in the inequality above, one concludes the proof. y

B. Error Estimate for Weak Solutions in [C([0, T], Hs(�d))]m, s � (0, 1]

We complete our convergence estimates in this section by considering the intermediate casesu � [C([0, T], Hs(�d))]m for s � (0, 1]. We obtain the convergence rate hs/2. Thus the rate tendsto zero for s 3 0. However note that the scheme converges for u � [C([0, T], L2(�d))]m, butno algebraic order of convergence is available ([5]). The main result of this section is as follows.

Theorem 4.5. Let Assumption 2.1 hold and, for s � (0, 1], assume u0 � [Hs(�d)]m andf � [L2((0, T); Hs(�d))]m. Consider the finite-volume approximation from Definition 3.1 andsuppose that the conditions (3.8), (3.9), (3.10), (3.11), and (3.12) hold.

Then there exists a constant C � C(T, �, Ai, B) such that the error estimate

�u � uh�L2��d� �0,T �� Chs/2��u0�Hs � �f�L2��0,T �;Hs�� (4.22)

holds. u � [C([0, T]; Hs(�d))]m is the weak solution to the initial problem (1.1), (1.2).Proof. From the obvious relation

�0

T ��d

�u � uh�2 dxdt � 2 �0

T ��d

�u�2 dxdt � 2 �0

T ��d

�uh�2 dxdt

and the a priori estimates (4.16) and (3.13) one obtains the low regularity estimate

�u � uh�L2��d� �0,T �� C�T ��u0�2 � �f�L2��0,T �;L2��. (4.23)

In order to conclude the proof, we apply the K-method of real interpolation (see [16]) on thehigh regularity estimate (4.1) and (4.23).

Let A1, A2, B1, B2 be Banach spaces with continuous embeddings A1 � A2, B1 � B2. Then,if R : A23 B2 is a bounded linear operator such that its restriction R : A13 B1 is bounded, thenthe restrictions R : (A1, A2)�,2 3 (B1, B2)�,2 are also bounded and the interpolation inequality,

�R��A1,A2��,23�B1,B2��,2 � �R�A13B1

1�� R�A23B2

� , (4.24)

holds for 0 � � � 1. (A1, A2)�,q denotes the interpolation space.Assume f � 0. Define R1u0 :� u � uh and let A1 � [H1(�d)]m, A2 � [L2(�d)]m, B1 � B2

� [L2(�d � (0, T ))]m. With this definitions we have (A1, A2)�,2 � [Hs(�d)]m and (B1, B2)�,2


� [L2(�d � (0, T ))]m. For s � 1 � �, the estimates (4.1) and (4.23) together with theinterpolation inequality (4.24) yield

�u � uh�L2��d� �0,T �� R1u0�2 � Chs/2�u0�Hs. (4.25)

Assume now u0 � 0 and define R2f :� u � uh. We let A1 � [L2((0, T ); H1(�d))]m,A2 � [L2((0, T ); L2(�d)]m, B1 � B2 � [L2(�d � (0, T ))]m. Similarly as above we get

�u � uh�L2��d� �0,T �� R2f�2 � Chs/2�f�L2��0,T �;Hs�. (4.26)

By the linearity of the exact and discrete solution operators we have u � uh � R1u0 � R2 f inthe general case. Now the conclusion follows from (4.25) and (4.26). y

Remark 4.3.

(i) Discontinuous functions are elements of the space Hs(�d) for s � [0, 1/2). To see this weshow, for example, that v � Hs(�2) for s � [0, 1/2), where v(x1, x2) � w(x1)w(x2) and

w�x� � �1, x � �0, 1�,0, elsewhere.

For the proof we use the definition of fractional Sobolev spaces via Fourier transform (see, i.e.,[17]). According to it,

v � Hs��2� N v � L2��2� and ��2

�1 � �y�2�s�v �y��2 dy � �,

where

v �y� �1

2� ��2

v �x1, x2�e�i�x1y1�x2y2� dx1dx2.

Obviously, it is enough to prove that

��2

�yi�2s�v �y��2 dy � � �i � 1, 2�, (4.27)

since (a � b)s � as � bs for s � [0, 1] and a, b � 0. We see that

w �z� :�1

2� ��

e�izxw �x� dx �1

2� �0

1

e�izx dx �1

iz 2��1 � e�iz�.

Clearly, there exists a number c � �, such that


�w �z�� c

1 � �z� �z � ��.

Further, it is

�v �y1, y2�� w �y1��w �y2�� c2

�1 � �y1��1 � �y2��.

Hence,

��2

�y1�2s�v �y��2 dy � c4 ��2

�y1�2s

�1 � �y1��2�1 � �y2��2 dy1dy2 � c4 ��2

�y1�2s

�1 � y12��1 � y2

2�dy1dy2

� c4 ��

�y1�2s

1 � y12 dy1 �

�

1

1 � y22 dy2 � �,

for s � [0, 12

). Analogously, one obtains (4.27) for i � 2. )

(ii ) The convergence results that we obtain in Theorem 4.5 are not optimal. For instance,numerical experiments show that one should expect the order 1/2 for a discontinuousfunction of bounded variation like w in (i ).

V. THE A POSTERIORI ERROR ESTIMATE

Our starting points for the a posteriori analysis are again Theorem 2.4 and Proposition 3.3.However, instead of (2.8) we use (2.7) in order to localize the error. From the practical point ofview all constants appearing in the a posteriori error estimate should be known explicitly. Theconstants depend in particular on the dimension and the chosen form of cell volumes. Here wechoose d � 2 and the cell volumes to be triangles. Together with a proper choice of the testfunctions we then exploit carefully the specialized inequalities in Lemma 4.3.

For �, R � 0 and x0 � �2 and �(x0, R) from (2.3) we introduce the notation

Dn � K � �h�K � �tn, tn�1� � ��x0, R � � � � A,

Theorem 5.1. Let Assumption 2.1 hold and assume u0 � [H1(�2)]m, f � [C([0, T]; H1(�2))]m.Let u � [C([0, T]; H1(�2))]m be the weak solution of (1.1), (1.2). Consider the finite-volumeapproximation from Definition 3.1 and suppose that the condition (3.8) holds.

Let �, R � 0, x0 � �2 be given and denote by � the ball around x0 with radius R � �T � �.Then, the a posteriori error estimate

��x0,R�

e�t�u � uh�2 dxdt � �l�1

7

�2�hl � Eh

l �� T ��

�uh�x, 0� � u0�x��2 dx � �Qh��

(5.1)


holds. For � from (2.9) the quantities Eh1(� ), . . . , Eh

7(� ), Qh(� ) are defined in Proposition 3.3and �h

1, . . . , �h7 by

�h1 � L1� �

n��

�K�Dn

�tn�K��vKn�1 � vK

n �2� 1/2

,

�h2 � C1L2� �

n��

�K�Dn

�tn�K

4

�K� �e��K �

�CK,en �vK

n � vKe

n ��2� 1/2

, �h3 � 0,

�h4 � 3C1L2� �

n��

�K�Dn

�tnsK,n2

�K4

K�vK

n �2� 1/2

� 3L1� �n��

�K�Dn

��tn�3sK,n2

�K2

�K� �vKn �2� 1/2

,

�h5 � C2L2�div A��

n��

�K�Dn

�tn�K��K2 �vK

n �2�1/2

� L1�div A�� n��

�K�Dn

��tn�3�K��vKn �2�1/2

,

�h6 � C2L2�B��

n��

�K�Dn

�tn�K��K2 �vK

n �2�1/2

� L1�B�� n��

�K�Dn

��tn�3�K��vKn �2�1/2

,

�h7 � C2L2� �

n��

�K�Dn

�K2 �

tn

tn�1 �K

�f�2 dxdt�1/2

� L1� �n��

�K�Dn

��tn�2 �tn

tn�1 �K

�f�2 dxdt�1/2

,

with

L1 � 2 max�A1��, A2��G1�T � supt��0,T�

e��1t�Du� � , t��2

� G0�T ��B � div A��

� � F�T � supt��0,T�

e��0t�u� � , t��2, ��x0,R��

� �t�2�f�2 dxdt1/2

L2 �G0�T �

�sup

t��0,T�

e��0t�u� � , t��2 � G1�T � supt��0,T�

e��1t�Du� � , t��2, (5.2)

and

C1 � 64 6 � 120 2

3�2 �

2 6

3 1/2

, C2 �8 2

3�,

Gi�T � � T supt��0,T�

e��i��t�T � t�, F�T � � T supt��0,T�

e��0��t��T � t� � 1�,

sK,n � maxe��K �

��AK,en � �

i�1

2

nK,ei Ai�

�

�.

Finally �K2 � ¥e��(K ) �e�2, � is given by (2.9), and �0, �1 are from estimates (2.5), (2.6).


Remark 5.4.

(i ) Note that the suprema in the definition of L1, L2 in (5.2) can be estimated in terms of u0

and f by use of (2.5) and (2.6).(ii ) The term Qh(� ) in the a posteriori error estimate is nonnegative, thus does not increase

the error estimate.(iii ) The quantities �h

1 and �h2 are associated with the time derivative and the conservative

part of the flux in (1.1). In [18, 20] a posteriori estimates for the scalar and weaklycoupled conservation laws are presented. It contains corresponding quantities, how-ever, with respect to the L1-norm.

Proof of Theorem 5.1. With the assumptions of the theorem it can be proven that the weaksolution u satisfies u � [C1([0, T]; L2(�2))]m � [C([0, T]; H1(�2))]m (see [19, Chapter 7,Proposition 7.1]). Let {u0

k}k�� and { f k}k�� be sequences of functions in [C0�(�2)]m and

[C0�(�2 � [0, T])]m such that u0

k 3 u0 in [H1(�2)]m and f k3 f in [L2((0, T ); H1(�2))]m. Let uk

be the associated classical solution. According to the a priori estimate (2.4) one sees that theassociated classical solution uk converges to u in [C([0, T]; H1(�2))]m and consequently �tu

k3�tu in [L2((0, T ); L2(�2))]m. Consider now the inequality (2.7) with the choices u � uk, u0 � u0

k,f � f k, v � [L2(�2 � (0, T ))]m arbitrary and denote the corresponding measures by �v

k, vk:

�C�x0,R�

e�t�uk � v�2 dxdt � 2� �vk, �� v

k, �uk��.

After performing the limit k 3 � we have

�C�x0,R�

e�t�u � v�2 dxdt � 2� �v, �� v, �u��.

For the choice v � uh we can apply Proposition 3.3 with � � �u, � � � and obtain

�C�x0,R�

e�t�u � uh�2 dxdt � �l�1

7

�Ehl �� 2Rh

l ��u�� Qh�� ,

where Ehl , Rh

l for l � 1, . . . , 7 and Qh are defined as in Proposition 3.3. Note that Proposition3.3 holds also for less regular data and nonuniform time step (Remark 3.1).

The terms Ehl (�), Qh(� ) don’t contain the weak solution u and therefore further estimations

of these terms are not necessary. It remains to estimate the terms Rh1(�u), . . . , Rh

7(�u). This canbe done similar as in Section 4. We restrict ourselves here to the terms Rh

2 and Rh3.

From the inequality (4.7) and C� 2 � [(32�2 � 20�6)/3]�2 � �2/3 being the constant inthat inequality one obtains for � � �u


��en � �K

n � �1

�tn�e�1/2 �tn

tn�1 �e

��, t� �1

�K� �K

��x, t� dx� 2

d� 1/2

� C��K

3/2

��tn�e��K��1/2 �tn

tn�1 �K

�D��2 dxdt 1/2

,

which together with �¥e��K� �e��1/ 2 � 43�K1/ 2 implies that

� �e��K �

�tn�e��vKn � vKe


n � �Kn ��

� C��K

3/2

��tn�K��1/2 �tn

tn�1 �K

�D��x, t��2 dxdt 1/2 �e��K �

�CK,en �vK

n � vKe

n ��2 1/2 �e��K �

�e� 1/2

� 4 3 C��K

2

��tn�K��1/2 �tn

tn�1 �K

�D��x, t��2 dxdt 1/2 �e��K �

�CK,en �vK

n � vKe

n ��2 1/2

.

So we have

�Rh2�� 43 C� �

n��

�K�Dn

�tn�K

4

�K� �e��K �

�CK,en �vK

n � vKe

n ��21/2�0

T ��2

�D��x, t��2 dxdt1/2

.

By using �D��2 � 2[(�/� )2�u�2 � �2�Du�2], after some elementary calculations we obtain

�0

T ��2

�D��x, t��2 dxdt 1/2

� 2L2.

Hence, �Rh2(�u)� � �h

2.For what concerns case Rh

3, we have that

Eh3�� 2Rh

3��u� � T ��

�uh�x, 0� � u0�x��2 dx.

Thus (5.1) holds with �h3 � 0. y

References

1. R. Eymard, T. Gallouet, and R. Herbin, Finite volume methods, P. G. Ciarlet, et al., editors, Handbookof numerical analysis. Vol. 7: Solution of equations in Rn (Part 3), Techniques of scientific computing(Part 3), Amsterdam, 2000.

2. D. Kroner, Numerical schemes for conservation laws, Wiley, 1997.


3. K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Commun Pure Appl Math 11(1958), 333–418.

4. J. P. Vila and P. Villedieu, Convergence de la methode des volumes finis pour les systemes deFriedrichs, CR Acad Sci Paris Ser I Math 325(6) (1997), 671–676.

5. J. P. Vila and P. Villedieu, Friedrichs’ systems with measure source terms and convergence of thefinite volume method, Publication du laboratoire MIP, Universite Paul Sabatier, 1998.

6. R. Eymard, T. Gallouet, M. Ghilani, and R. Herbin, Error estimates for the approximate solutions ofa nonlinear hyperbolic equation given by finite volume schemes, IMA J Numer Anal 18(4) (1998),563–594.

7. P. Brenner, The Cauchy problem for the symmetric hyperbolic systems in Lp, Math Scand 19 (1966),27–37.

8. J. Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimension greater than one,Comm Math Phys 106 (1986), 481–484.

9. S. Kruzkov, First-order quasilinear equations with several space variables, Math USSR Sbornik 10(1970), 217–273.

10. P. Houston, J. A. Mackenzie, E. Suli, and G. Warnecke, A posteriori error analysis for numericalapproximation of Friedrichs systems, Numer Math 82 (1999), 433–470.

11. Y. Coudiere, J. P. Vila, P. Villedieu, Convergence d’un schema volumes finis explicite en temps pourles systemes hyperboliques lineaires symetriques en domaines bornes, CR Acad Sci Paris Ser I Math331(1) (2000), 95–100.

12. B. Achchab, A. Agouzal, J. Baranger, and J. F. Maitre, A posteriori error estimates in finite elementmethods for general Friedrichs systems, Comput Methods Appl Mech Eng 184(1) (2000), 39–47.

13. J. P. Vila and P. Villedieu, Convergence of an explicit finite volume scheme for first order symmetricsystems, Numer Math 94(3) (2003), 573–602.

14. S. Alinhac and P. Gerard, Operateurs pseudo-differentiels et theoreme de Nash-Moser, Inter Edition/Edition du CNRS (1991).

15. T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math Comp 34(1980), 441–463.

16. J. Berg and J. Lofstrom, Interpolation spaces, Springer, 1976.

17. J. L. Lions and E. Magenes, Non homogeneous boundary value problems and applications, Springer,1972.

18. D. Kroner and M. Ohlberger, A posteriori error estimates for upwind finite volume schemes fornonlinear conservation laws in multi dimensions, Math Comput 69(229) (2000), 25–39.

19. M. E. Taylor, Partial differential equations II, Springer, 1996.

20. M. Ohlberger and C. Rohde, Adaptive Finite Volume Approximations for Weakly Coupled Convec-tion Dominated Parabolic Systems, IMA J Numer Anal 22(2) (2002), 253–280.


Documents

Finite-volume schemes for Friedrichs systems in multiple space dimensions: A priori and a posteriori error estimates