21697_ftp

A Theoretical Analysis of a New Finite VolumeScheme for Second Order Hyperbolic Equationson General Nonconforming MultidimensionalSpatial MeshesAbdallah BradjiDepartment of Mathematics, University of Annaba, Annaba, Algeria

Received 27 March 2011; accepted 12 October 2011Published online 9 February 2012 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/num.21697

We consider the wave equation, on a multidimensional spatial domain. The discretization of the spatialdomain is performed using a general class of nonconforming meshes which has been recently studied forstationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMAJ Numer Anal 30 (2010),1009–1043). The discretization in time is performed using a uniform mesh. We derive a new implicit finitevolume scheme approximating the wave equation and we prove error estimates of the finite volume approx-imate solution in several norms which allow us to derive error estimates for the approximations of the exactsolution and its first derivatives. We prove in particular, when the discrete flux is calculated using a stabilizeddiscrete gradient, the convergence order is hD + k, where hD (resp. k) is the mesh size of the spatial (resp.time) discretization. This estimate is valid under the regularity assumption u ∈ C3([0, T ]; C2(�)) for theexact solution u. The proof of these error estimates is based essentially on a comparison between the finitevolume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc.Numer Methods Partial Differential Eq 29: 1–39, 2013

Keywords: discrete gradient; fully discretization scheme; implicit scheme; multidimensional spatial domain;nonconforming grid; second order hyperbolic equations; wave equation

I. MOTIVATION AND DESCRIPTION OF THE MAIN RESULTS

We consider the following wave problem, as a model for second order hyperbolic problems:

utt (x, t) − �u(x, t) = f (x, t), (x, t) ∈ � × (0, T ), (1)

where � is an open polygonal bounded subset in Rd and f is a given function.

Correspondence to: Abdallah Bradji, Department of Mathematics, University of Annaba, Annaba, Algeria (e-mail:[email protected])Contract grant sponsor: Algerian Ministry of Higher Education and Scientific Research; contract grant number:B01120090113Contract grant sponsor: PNR Project EMNDG (ANDRU)

© 2012 Wiley Periodicals, Inc.

2 BRADJI

Initial conditions are given by:

u(x, 0) = u0(x) and ut(x, 0) = u1(x) x ∈ �, (2)

where u0 and u1 are given functions defined on �.Homogeneous Dirichlet boundary conditions are given by

u(x, t) = 0, (x, t) ∈ ∂� × (0, T ). (3)

The wave equation is an important second-order linear partial differential equation for the descrip-tion of waves as they occur in physics such as sound waves, light waves and water waves. It arisesin fields like acoustics, electromagnetics, and fluid dynamics, see for instance [1, 13].

For this reason, substantial works in various numerical methods have already been carried outto approximate the wave equation, see for instance [2–14] and references therein.

The finite volume method is well established to approximate various types of conservationlaws used in many engineering fields, such as fluid mechanics, heat and mass transfer or petro-leum engineering. It can be applied in arbitrary geometries and is locally conservative, see [15]and the references therein. To yield a finite volume discretization, we integrate the equation tobe solved on the so called control volumes. We use then numerical fluxes to approximate, usingthe discrete unknowns, the continuous fluxes over the boundaries of the control volumes, whichappear after the integration by parts. A widely used definition of admissible finite volume meshesfor viscous conservation laws can be found in [15, Definition 9.1, Pages 762–763]. Among thefeatures of this definition is that control volumes are open polygonal convex sets. In addition tothis, for admissibility, the mesh should satisfy an orthogonality condition, that is, there exists afamily of points (xK)K∈T , such that for a given edge σKL, the line segment xKxL is orthogonal tothis edge. This condition is useful for approximating the fluxes over a given edge using two pointdifference quotients. The construction of such meshes in general geometries is possible in manyinteresting cases but still linked to a number of challenging problems [16]. Therefore in manycases it is useful to drop the orthogonality condition and to assume general polyhedral controlvolumes, where the boundary of each control volume is a finite union of subsets of hyperplanes,cf. [17] and the references therein. Recently [17], a large class of nonconforming meshes couldbe used in the approximation of stationary anisotropic heterogeneous diffusion equations, andsome error estimates have been provided. It is shown in [18] that the method introduced in [17]is efficient numerically.

The first aim of the present work is to derive a finite volume discretization scheme approxi-mating the wave problem (1)–(3) (as an important model for second order hyperbolic problems)using the new general class of spatial meshes introduced recently in [17]. The second aim is toprovide and to prove error estimates of our discretization scheme in possible different norms.

The general class of nonconforming multidimensional meshes introduced recently in [17] hasthe following advantages:

• The scheme can be applied on any type of grid: conforming or nonconforming, 2D and 3D,or more, made with control volumes which are only assumed to be polyhedral (the boundaryof each control volume is a finite union of subsets of hyperplanes).

• When the family of the discrete fluxes are satisfying some suitable conditions, the matricesof the generated linear systems are sparse, symmetric, positive and definite.

• A discrete gradient for the exact solution is formulated and converges to the gradient of theexact solution.

Numerical Methods for Partial Differential Equations DOI 10.1002/num

ERROR ESTIMATES FOR SECOND ORDER HYPERBOLIC EQUATIONS 3

Thanks to the basic ideas of the finite volume scheme developed in [17] to approximate stationaryproblems, we first shall derive the new finite volume scheme (42)–(44) in order to approximateproblem (1)–(3), see Section IV. Equation (44) is a discrete version for the weak formulation (7)of the wave Eq. (1) (with, of course, the boundary condition (3)). Whereas, the discrete initialconditions (42)–(43) of scheme (42)–(44) are discrete version for the orthogonal projections

a(u(0), v) = −(� u0, v)L2(�), ∀v ∈ H 10 (�), (4)

and

a(ut (0), v) = −(� u1, v)L2(�), ∀v ∈ H 10 (�), (5)

where

a(w, v) =∫

�

∇w(x) · ∇v(x)dx. (6)

The choices (42)–(43) for the discretization of initial conditions (2) are useful as explained inRemark 4.6.

Although, the scheme (42)–(44) stems from the finite volume ideas developed these last years(Would say, integration over the control volumes and then we approximate the fluxes arising afterintegration by parts by some suitable numerical ones.), its formulation seems a discrete versionfor the weak formulation (7) (given below) and (4)–(6). From this point of view, the scheme(42)–(44) presented in this work looks like a nonconforming finite element scheme for the waveproblem (1)–(3).

The scheme we present, that is (42)–(44), has the following advantages

• The scheme can be applied on any type of spatial grid: conforming or nonconforming, 2Dand 3D, or more, made with control volumes which are only assumed to be polyhedral (theboundary of each control volume is a finite union of subsets of hyperplanes).

• For each time level n, the scheme results in linear systems (42), (43), and (44) with a numberof unknowns being equal to card(M)+ card(H), the sum of the number of control volumesand the cardinality of a certain subset of the set of edges of the mesh equations.

• The matrix generated by the scheme (42)–(44) symmetric, positive and definite when thebilinear form 〈·, ·〉 is satisfying (56) and (57). The matrix is sparse for instance when(FK ,σ )K∈M,σ∈E is defined by (51)–(54).

• The scheme (42)–(44) is unconditionally convergent, i.e., the space and time discretizationsare independent of each other. Indeed, it is sometime needed in order to get the convergenceof schemes approximating hyperbolic equations, to assume some condition between the timeand space discretizations, e.g., see condition required for convergence stated in [15, Remark29.2, Page 953].

• For each level n ∈ [[ 0, N + 1]], the finite volume solution of (42)–(44) converges to u(·, tn)in the L2(�)–norm, see first and fourth item of Remark 4.4.

• Using the discrete gradient provided in [17] for the stationary case, suitable discrete deriv-atives of the finite volume solution of (42)–(44) can be formulated in order to approximatespatial first derivatives of the exact solution of problem (1)–(3), see second and fourth itemof Remark 4.4.

• A discrete time derivative is formulated in order to approximate the time derivative of theexact solution of (1)–(3), see third and fourth item of Remark 4.4.


4 BRADJI

The main results of the present work are stated in Theorem 4.1. It provides us with error estimatesof scheme (42)–(44) in several discrete norms, namely in those which allow us to get error esti-mates for the approximation of the exact solution of (1)–(3) and its first derivatives. We deriveerror estimates (62)–(64) in discrete norms L∞(0, T ; H 1

0 (�)) and W 1,∞(0, T ; L2(�)), and errorestimate for an approximation for the gradient, in a general framework in which the discretebilinear form involved in the discretization scheme (42)–(44) and given by (36) (or also (55)),where (FK ,σ )K∈M,σ∈E are a family of linear mappings from XD into R, is coercive, symmetric,and continuous in the sense of (56), (57), and (58). We prove in particular, see (65), when thediscrete flux is given by (51)–(54), that the convergence order is hD + k, where hD (resp. k) isthe mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularityassumption u ∈ C3([0, T ]; C2(�)) for the exact solution u.

The proof of Theorem 4.1 is based on the comparison between the solution of scheme (42)–(44) and the auxiliary solution defined by (91). As a first principal part of the proof of Theorem4.1, we prove Lemma 4.6 and as a second principal part, we prove Lemma 4.7, and then we proveresults of Theorem 4.1 using Lemmata 4.6 and 4.7 together with the technical Lemmata 4.1–4.5.

Lemma 4.6 provides us with some estimates on the error between the solution of (91) and theexact solution of (1)–(3), and its proof is based on the proof of [17, Theorem 4.8, Page 1033]with some special attention paid to determine the dependence of the constants, which appear inthe estimates, on the exact solution. This attention appears, essentially, in Lemmata 4.3 and 4.4(which are used to prove Lemma 4.6) and in their proofs. Lemma 4.7 is a preliminary result whichhelps us to conclude the proof of Theorem 4.1.

Lemmata 4.1–4.4 are some preliminary technical tools which are used in the proof of Lemmata4.6 and Theorem 4.1.

The organization of this article is as follows: in the second section, we state the weak formula-tion of the continuous problem and we recall some function spaces which will be used throughoutthis article. Third section is devoted to recall the definition of general nonconforming meshes aswell as some discrete spaces given in [17]. In the fourth section, we derive and present the finitevolume scheme (42)–(44) and the main result of our article, namely Theorem 4.1. The proof ofTheorem 4.1 is performed thanks to Lemmata 4.6 and 4.7. Among the tools used to prove Lemma4.6 and Theorem 4.1, we used some Lemmata and results from [17]. In fact, Lemmata 4.1, 4.2,and 4.5 are respectively the subject of [17, (4.6), Page 1026], [17, Lemma 4.2, Page 1026], and[17, (5.10), Lemma 5.4, Page 1038], when p = 2, and we recall them here for the sake of com-pleteness and to ease the reading of our article. Whereas, Lemmata 4.3 and 4.4 are the subjectof [17, Lemma 4.4, Page 1029] and [17, (4.20), Page 1031] in which the constants in estimates[17, (4.13), Lemma 4.4, Page 1029] and [17, (4.20), Page 1031] are depending on the functionunder consideration ϕ, whereas the constants which appear in estimate (70) of Lemma 4.3 andin estimate (82) of Lemma 4.4 are independent of the function under consideration ϕ. WritingLemmata 4.3 and 4.4 in which the constants are independent of the function under considerationϕ has at least two roles:

• The application of Lemmata 4.3 and 4.4 serves as to get constants independent of the exactsolution in the error estimates, whereas a straightforward application of [17, Lemma 4.4,Page 1029] and [17, (4.20), Page 1031] leads to constants, which appear in error estimates,depending on u(., tn) and consequently we obtain constants depending on the parameters ofthe time discretization.

• The required regularity in Lemmata 4.3 and 4.4 is ϕ ∈ C2(�). This regularity assumptiontogether with the regularity assumption in Lemma 4.6, and regularity required in the proofof Theorem 4.1 yields the regularity assumption u ∈ C3([0, T ]; C2(�)) in Theorem 4.1 on



the exact solution u of problem (1)–(3). So, we expect that this regularity may be weakenedto W 3,∞(0, T ; H 2(�)), see Remark 4.2.

So, some efforts have been devoted in order to determine the dependence of the constants whichappear in the estimates of [17, Theorem 4.8, Page 1033] on the exact solution.

The fifth section is devoted to present some computational results and, finally, we describe inthe sixth section some interesting tasks to work on in the future.

II. WEAK PROBLEM AND PRELIMINARIES

The following Theorem, see [19, Theorem 3, Page 384] and [19, Theorem 4, Page 385], gives asense for a weak solution for problem (1)–(3) (recall that H−1(�) is the dual of H 1

0 (�)):

Theorem 2.1 (Existence and uniqueness for the continuous problem (1)–(3), cf. [19]). Letf ∈ L2((0, T ) × �)), u0 ∈ H 1(�) and u1 ∈ L2(�). Then, there exists a unique weak solu-tion for (1)–(3) in the following sense: there exists a function u ∈ L2(0, T ; H 1

0 (�)) such thatut ∈ L2(0, T ; L2(�)) and utt ∈ L2(0, T ; H−1(�)) and:

i. For a.e. 0 ≤ t ≤ T

〈utt , v〉 +∫

�

∇u(x, t) · ∇v(x)dx =∫

�

f (x, t)v(x)dx, ∀v ∈ H 10 (�) (7)

ii.

u(0) = u0, (8)

iii.

ut(0) = u1. (9)

Remark 2.1 ((8) and (9) make sense). Assumptions of Theorem 2.1 imply that u ∈H 1(0, T ; L2(�)) and ut ∈ H 1(0, T ; H−1(�)). So, thanks to [19, Theorems 2, Pages 286], wededuce that u ∈ C([0, T ]; L2(�)) and ut ∈ C([0, T ]; H−1(�)). Thus (8) and (9) make sense.

The convergence of the finite volume scheme we want to present is analyzed using the spaceCm([0, T ]; Cl(�)), where m and l are integers, of m–times continuously differentiable mappingsof the interval [0, T ] with values in Cl(�), see [20, Pages 47–48]. The space Cm([0, T ]; Cl(�))

is equipped with the norm

‖ u‖Cm([0,T ]; Cl(�)) = maxj∈[[ 1,m]]

{sup

t∈[0,T ]

∥∥∥∥dju

dtj(t)

∥∥∥∥Cl(�)

}, (10)

where ‖ · ‖Cl(�) denotes the usual norm of Cl(�).


6 BRADJI

FIG. 1. Notations for two neighboring control volumes in the case d = 2.

III. MESHES AND DISCRETE SPACES

This article deals with a finite volume scheme approximating (1)–(3) on a general class of non-conforming meshes which include the admissible mesh of [15, Definition 9.1, Page 762]. Thisgeneral class of meshes is introduced in [17]. An example of two neighboring control volumesK and L is depicted in Fig. 1. For the sake of completeness, we recall the general finite volumesmesh given in [17].

Definition 1 (Definition of a large class of finite volume grids, cf. [17]). Let � be a polyhe-dral open bounded connected subset of R

d , where d ∈ IN\{0}, and ∂� = �\� its boundary. Adiscretization of �, denoted by D, is defined as the triplet D = (M, E , P), where:

1. M is a finite family of nonempty connected open disjoint subsets of � (the “control vol-umes”) such that � = ∪K∈MK . For any K ∈ M, let ∂K = K\K be the boundary of K;let m(K) > 0 denote the measure of K and hK denote the diameter of K .

2. E is a finite family of disjoint subsets of � (the “edges” of the mesh), such that, for allσ ∈ E , σ is a nonempty open subset of a hyperplane of R

d , whose (d − 1)–dimensionalmeasure is strictly positive. We also assume that, for all K ∈ M, there exists a subset EK

of E such that ∂ K = ∪σ∈EKσ . For any σ ∈ E , we denote by Mσ = {K , σ ∈ EK}. We

then assume that, for any σ ∈ E , either Mσ has exactly one element and then σ ⊂ ∂ �

(the set of these interfaces, called boundary interfaces, denoted by Eext) or Mσ has exactlytwo elements (the set of these interfaces, called interior interfaces, denoted by Eint). For allσ ∈ E , we denote by xσ the barycentre of σ . For all K ∈ M and σ ∈ EK , we denote bynK ,σ the unit vector normal to σ outward to K .

3. P is a family of points of � indexed by M, denoted by P = (xK)K∈M, such that for allK ∈ M, xK ∈ K and K is assumed to be xK–star-shaped, which means that for all x ∈ K ,the property [xK , x] ⊂ K holds. Denoting by dK ,σ the Euclidean distance between xK andthe hyperplane including σ , one assumes that dK ,σ > 0. We then denote by DK ,σ the conewith vertex xK and basis σ .

For our need, we use the discrete spaces and their norms of the following Definition:

Definition 2 (Discrete spaces and norms, cf. [17]). Let � be a polyhedral open bounded con-nected subset of Rd and D = (M, E , P) be a discretization in the sense of Definition 1. Throughoutthis article we use the following spaces and norms:



1. The space XD

XD = {v = ((vK)K∈M, (vσ )σ∈E), vK ∈ R, vσ ∈ R}. (11)

The space XD is equipped with the following semi–norm:

| v|2X =∑

K∈M

∑σ∈EK

m(σ )

dK ,σ(vσ − vK)2. (12)

2. The space XD,0

XD,0 = {v = ((vK)K∈M, (vσ )σ∈E) ∈ XD, vσ = 0, ∀σ ∈ Eext}. (13)

The seminorm | · |X given by (12) is a norm on the subspace XD,0 of XD.3. For a given family of real numbers {βK

σ , K ∈ M, σ ∈ Eint}, with βKσ �= 0 only for some

control volumes which are “close” to σ , and such that

1 =∑

K∈M

βKσ and xσ =

∑K∈M

βKσ xK . (14)

We define a space with dimension smaller than that of XD,0. This can be achieved byexpressing uσ , for all σ ∈ B, where B ⊂ Eint as a consistent barycentric combination ofthe values uK :

uσ =∑

K∈M

βKσ uK . (15)

We decompose then the set Eint of interfaces into two nonintersecting subsets, that is:Eint = B ∪ H and H = Eint\B. The interface unknowns associated with B will be com-puted by using the barycentric formula (15). The unknowns of the scheme will be then thequantities uK for K ∈ M and uσ for σ ∈ H. Consider then space XD,B ⊂ XD,0 given by

XD,B = {v ∈ XD,0 such that vσ satisfying (15), ∀σ ∈ B}. (16)

The semi–norm | · |X given by (12) is a norm on the subspace XD,B of XD,0.4. The subspace HM(�) of L2(�) defined by the function which are constant on each control

volume K ∈ M. We then denote, for all v ∈ HM(�) and for all σ ∈ Eint with Mσ = {K , L},Dσ v = | vK − vL| and dσ = dK ,σ + dL,σ , and for all σ ∈ Eext with Mσ = {K}, we denoteDσ v = | vK | and dσ = dK ,σ . We then define the following norm:

∀ v ∈ HM(�), ‖ v‖21,2,M =

∑K∈M

∑σ∈EK

m(σ )dK ,σ

(Dσ v

dσ

)2

=∑σ∈E

m(σ )(Dσ v)2

dσ

. (17)

We also need the following interpolation operators:

Definition 3 (Interpolation operators, cf. [17]). Let � be a polyhedral open bounded connectedsubset of R

d and D = (M, E , P) be a discretization in the sense of Definition 1. Throughout thisarticle we use the following interpolation operators:


8 BRADJI

1. For all v ∈ XD, we denote by �M v ∈ HM(�) the piecewise function from � to R definedby �M v(x) = vK , for a.e. x ∈ K , for all K ∈ M.

2. For all ϕ ∈ C(�), we denote by PDϕ ∈ XD the element defined by ((ϕ(xK))K∈M,(ϕ(xσ ))σ∈E).

3. For all ϕ ∈ C(�), we denote by PD,Bϕ ∈ XD,B the element v ∈ XD,B such that

vK = ϕ(xK), ∀K ∈ M, (18)

vσ = 0, ∀σ ∈ Eext, (19)

vσ =∑

K∈M

βKσ ϕ(xK), ∀σ ∈ B, (20)

and

vσ = ϕ(xσ ), ∀σ ∈ H. (21)

4. For all ϕ ∈ C(�), we denote by PMϕ ∈ HM(�) the element defined by PMϕ(x) = ϕ(xK),for a.e. x ∈ K , for all K ∈ M.

We need, to analyse the convergence, to consider the size of discretization D is defined by:

hD = sup{diam(K), K ∈ M}, (22)

and the regularity of the mesh is given by

θD = max

(max

σ∈Eint ,K ,L∈Mσ

dK ,σ

dL,σ, maxK∈M,σ∈EK

hK

dK ,σ

). (23)

For a given set B ⊂ Eint and for a given family (βKσ )K∈M,σ∈Eint satisfying property (14), we

introduce some measure of the resulting regularity with

θD,B = max

(θD, max

K∈M,σ∈EK∩B

∑L∈M

∣∣βLσ

∣∣ |xσ − xL|2h2

K

). (24)

IV. FORMULATION OF AN IMPLICIT VOLUME SCHEME AND STATEMENT OF THEMAIN RESULT

The discretization of the spatial domain � is performed using the mesh D = (M, E , P) givenin Definition 1, whereas the time discretization is performed with a constant time step k = T

N+1 ,where N ∈ IN, and we shall denote tn = nk, for n ∈ [[0, N + 1]]. The scheme we want to con-sider is to find an approximation for (1)–(3) by setting up, for each n ∈ [[0, N + 1]], a system ofequations for a family of values ((un

K)K∈M, (unσ )σ∈E) in the control volumes and on the interfaces.

Following the idea of finite volume method, we first integrate Eq. (1) over each control volumeK and on each interval (tn, tn+1), and then we use an integration by parts to get (recall that nK ,σ

is the unit vector normal to σ outward to K)∫K

(ut (x, tn+1) − ut(x, tn)) dx −∑σ∈EK

∫ tn+1

tn

∫σ

∇ u(x, t) · nK ,σ (x)dγ (x) dt

=∫ tn+1

tn

∫K

f (x, t)d x dt . (25)



The left hand side of the previous equation is the sum of two terms. We will then approximatethese two terms.

• The first term∫

K(ut (x, tn+1) − ut(x, tn)) dx can be approximated using the quadrature:

m(K)k∂2 u(xK , tn+1),

where ∂2 vn+1 denotes the value

∂2 vn+1 = vn+1 − 2vn + vn−1

k2, ∀ n ∈ [[1, N ]], (26)

• For each n ∈ [[0, N ]], the flux − ∫ tn+1tn

∫σ∇ u(x, t) · nK ,σ (x)dγ (x) dt is approximated by a

function kFK ,σ (un+1 ) of the values ((un+1K )K∈M, (un+1

σ )σ∈E) at the “centers” and the inter-faces of the control volumes (in all particular cases, FK ,σ (un+1) only depends on un+1

K and(un+1

σ ′ )σ ′∈EK), thus the the proposed scheme is implicit in time. The numerical flux FK ,σ (un+1)

satisfies the following conservativity:

FK ,σ (un+1 ) + FL,σ (un+1 ) = 0, ∀ σ ∈ Eint such that Mσ = {K , L}. (27)

Therefore, a discrete equation corresponding to (25) can be written asA discrete problem for (1) is then defined by, for n ∈ [[1, N ]]

m(K)∂2 un+1K +

∑σ∈EK

FK ,σ (un+1) = m(K)f nK , ∀K ∈ M, (28)

where

f nK = 1

km(K)

∫ tn+1

tn

∫K

f (x, t)d x dt . (29)

The discretization of initial conditions (2) is performed as:

∑σ∈EK

FK ,σ (u0) = −∫

K

�u0(x)dx, (30)

and

∑σ∈EK

FK ,σ (∂1u1) = −∫

K

�u1(x)dx, (31)

where

∂1 vn = vn − vn−1

k, ∀ n ∈ [[1, N + 1]], (32)

The Dirichlet boundary condition (3) can be approximated as, for n ∈ [[0, N + 1]]

unσ = 0, ∀σ ∈ Eext. (33)


10 BRADJI

Equation (28) could be written in some weak formulation; multiplying, for any v ∈ XD,0, bothsides of (28) by the value vK of v on the control volume, and summing over K ∈ M to get∑

K∈M

m(K)∂2 un+1K vK +

∑K∈M

∑σ∈EK

FK ,σ (un+1) vK =∑

K∈M

m(K)f nKvK . (34)

Using (27), (34) yields the following discrete weak formulation: for any n ∈ [[0, N ]], findun ∈ XD,0 such that∑

K∈M

m(K)∂2 un+1K vK + 〈 un+1, v〉F =

∑K∈M

m(K)f nKvK , ∀ v ∈ XD,0, (35)

where

〈 w, v〉F =∑

K∈M

∑σ∈EK

FK ,σ (w) (vK − vσ ). (36)

By the same way, (30)–(31) can be written in the following discrete weak forms:

〈 u0, v〉F = −∑

K∈M

vK

∫K

�u0(x)dx, ∀ v ∈ XD,0, (37)

and

〈 ∂1 u1, v〉F = −∑

K∈M

vK

∫K

�u1(x)dx, ∀ v ∈ XD,0. (38)

It is useful to mention that (35) is equivalent to ((27),(28)); indeed, set v ∈ XD,0 in (35) suchthat vK = 1 and vL = 0, for all L �= K , and vσ = 0, for all σ ∈ E , we get (28). Similarly,choosing v ∈ XD,0 such that vK = 0, for all K ∈ M, and vσ = 1 and vτ = 0 for any τ ∈ Eint,such that Mσ = {K , L}, leads to (27).

By the same way, we can justify that ((27),(30)) and ((27),(31)) are respectively equivalentto (37) and (38). This means that under the conservativity property (27), problem (28)–(33) isequivalent to problem (35)–(38).

We may also choose a space with dimension smaller than that of XD,0. This can be achievedby expressing uσ , for all σ ∈ Eint, as the consistent barycentric combination (15) of the values uK ,where {βK

σ ; K ∈ M, σ ∈ Eint} is a family of real numbers, with βKσ �= 0 only for some control

volumes which are “close” to σ , and satisfies (14).Hence the new scheme could be written as: for any n ∈ [[0, N ]], find un ∈ XD,0 such that

unσ = ∑

K∈M βKσ un

K , for all σ ∈ Eint∑K∈M

m(K)∂2 un+1K vK + 〈 un+1, v〉F

=∑

K∈M

m(K)f nKvK , ∀v ∈ XD,0 with vσ =

∑K∈M

βKσ vK , ∀σ ∈ Eint, (39)

and find u0 ∈ XD,0 and u1 ∈ XD,0 such that u0σ = ∑

K∈M βKσ u0

K and u1σ = ∑

K∈M βKσ u1

K , for allσ ∈ Eint

〈 u0, v〉F = −∑

K∈M

vK

∫K

�u0(x)dx, ∀ v ∈ XD,0, (40)



and

〈 ∂1u1, v〉F = −∑

K∈M

vK

∫K

�u1(x)dx, ∀ v ∈ XD,0. (41)

Let us decompose the set Eint of interfaces into two nonintersecting subsets, that is: Eint = B∪Hand H = Eint\B. The interface unknowns associated with B will be computed by using thebarycentric formula (15).

In terms of the space XD,B given by (16), we suggest the following finite volume scheme: findu0

D, u1D ∈ XD,B such that⟨

u0D, v

⟩F

= −(�u0, �M v)L2(�), ∀ v ∈ XD,B, (42)

and (recall that ∂1 vn is defined by (32))⟨∂1 u1

D, v⟩F

= −(�u1, �M v)L2(�), ∀ v ∈ XD,B, (43)

and for any n ∈ [[1, N ]], find unD ∈ XD,B such that (recall that ∂2 vn is defined by (26))

(∂2 �M un+1

D , �M v)L2(�)

+ ⟨un+1

D , v⟩F

=∑

K∈M

m(K)f nKvK , ∀v ∈ XD,B, (44)

where f nK is given by (29), and (·, ·)L2(�) denotes the L2 inner product, and �M v, for all v ∈ XD,

is the piecewise constant function from � to R defined by �M v(x) = vK , for a.e. x ∈ K , for allK ∈ M, see Definition 3.

A. Construction of the Numerical Flux Using the Discrete Gradient

We recall here the explicit expression for flux FK ,σ given in [17]. The idea is to derive the fluxfrom a discrete gradient and could be explained as follows:

〈u, v〉F =∑

K∈M

∑σ∈EK

FK ,σ (u)(vK − vσ ) =∫

�

∇D u(x) · ∇D v(x)d x, ∀u ∈ XD, ∀v ∈ XD.

(45)

Let us consider the discrete gradient given in [17]:

∇D u(x) = ∇K ,σ u, a.e. x ∈ DK ,σ , (46)

where DK ,σ is the cone with vertex xK and basis σ and

∇K ,σu = ∇Ku + RK ,σu nK ,σ , (47)

∇Ku = 1

m(K)

∑σ∈EK

m(σ )(uσ − uK)nK ,σ , (48)

and

RK ,σu =√

d

dK ,σ(uσ − uK − ∇uK · (xσ − xK)). (49)


12 BRADJI

Let us set

∇K ,σu =∑

σ ′∈EK

(uσ ′ − uK) yσσ ′, (50)

where

yσσ ′ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

m(σ )

m(K)nK ,σ +

√d

dK ,σ

(1 − m(σ )

m(K)nK ,σ · (xσ − xK)

)nK ,σ , σ ′ = σ

m(σ ′)m(K)

nK ,σ ′ −√

d

dK ,σ m(K)m(σ ′)nK ,σ ′ · (xσ − xK)nK ,σ , σ ′ �= σ .

(51)

Therefore∫�

∇D u(x) · ∇D v(x)d x =∑

K∈M

∑σ∈EK

∑σ ′∈EK

Aσσ ′(uσ − uK)(vσ ′ − vK), ∀u ∈ XD, ∀v ∈ XD,

(52)

where

Aσσ ′ =∑

σ ′′∈EK

K ,σ ′′ yσ ′′σ · yσ ′′σ ′and =

∫DK ,σ ′′

Id x. (53)

The identification, using (45) and (52), leads to

FK ,σ (u) =∑

σ ′∈EK

Aσσ ′(uK − uσ ′). (54)

Before stating the main result of this article, we need the following definition:

Definition 4 (Continuous, coercive, and symmetric families of fluxes). Let D = (M, E , P) bea discretization in the sense of Definition 1. Let (FK ,σ )K∈M,σ∈E be a family of linear mappingsfrom XD into R. We consider the bilinear form defined by

〈u, v〉F =∑

K∈M

∑σ∈EK

FK ,σ (u)(vK − vσ ), ∀ u, v ∈ XD. (55)

• The bilinear form 〈·, ·〉F is said to be coercive if there exists α > 0 such that

α| v |2X ≤ 〈v, v〉F , ∀ v ∈ XD. (56)

• The bilinear form 〈·, ·〉F is said to be symmetric on XD × XD if

〈u, v〉F = 〈v, u〉F , ∀ (u, v) ∈ X 2D. (57)

• The bilinear form 〈·, ·〉F is said to be continuous if there exists a positive constant M

independent of D such that

〈u, v〉F ≤ M| u|X | v|X , ∀ (u, v) ∈ X 2D. (58)



The following theorem provides us with the convergence analysis of the fully implicit finitevolume scheme (42)–(44):

Theorem 4.1 (Error estimates for the finite volume scheme (42)–(44)). Let � be a polyhedralopen bounded connected subset of R

d , where d ∈ IN\{0}, and ∂� = �\� its boundary. Assumethat the weak solution of (1)–(3) in the sense of Theorem 2.1 satisfies u ∈ C3([0, T ]; C2(�)). Letk = T

N+1 , with N ∈ IN, and denote by tn = nk, for n ∈ [[0, N + 1]]. Let D = (M, E , P) be adiscretization in the sense of Definition 1. Let B ⊂ Eint be given and let {βK

σ , σ ∈ B, K ∈ M} be asubset of R satisfying (14). Assume that θD,B, given by (24), satisfies θ ≥ θD,B. Let (FK ,σ )K∈M,σ∈Ebe a family of linear mappings from XD into R such that the bilinear form 〈·, ·〉F defined by (55)is coercive in the sense of Definition 4.

Then there exists a unique solution (unD)N+1

n=0 ∈ X N+2D,B for problem (42)–(44).

For a function u ∈ C1(�), we define the following expression:

ED(u) =⎛⎝ ∑

K∈M

∑σ∈EK

dK ,σ

m(σ )

(FK ,σ (PD,B(u)) +

∫σ

∇ u(x) · nK ,σ d γ (x)

)2⎞⎠

12

. (59)

We set for any u ∈ C([0, T ]; C1(�))

EkD(u) = max

j∈[[0,2]]max

n∈[[j ,N+1]]ED(∂j u(·, tn)), (60)

where ∂0vn = vn, and ∂1vn (resp. ∂2vn) is defined by (32) (resp. (26)).For any n ∈ [[0, N + 1]], we define the error en

M ∈ HM(�) in the level n by:

enM = PM u(·, tn) − �M un

D. (61)

Assume in addition that the bilinear form 〈·, ·〉F defined by (55) is symmetric and continuous inthe sense of Definition 4. Then, the following error estimates hold, for positive constants C1, C2,and C3 only depending on �, d , α, M , θ and T

• Discrete L∞(0, T ; H 10 (�))–estimate: for all n ∈ [[0, N + 1]]∥∥en

M∥∥

1,2,M ≤ C1

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

). (62)

• Discrete W 1,∞(0, T ; L2(�))–estimate: for all n ∈ [[1, N + 1]]∥∥∂1 enM

∥∥L2(�)

≤ C2

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

). (63)

• Error estimate in the gradient approximation: for all n ∈ [[0, N + 1]]∥∥∇D unD − ∇ u(·, tn)

∥∥L2(�)

≤ C3

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

). (64)

Moreover, in the particular case where (FK ,σ )K∈M,σ∈E is defined by (51)–(54), there exists aconstant C4 only depending on θ , �, and d such that

EkD(u) ≤ C4 hD‖u‖C2([0,T ];C2(�)) and ED(u1) ≤ C4 hD‖u‖C1([0,T ];C2(�)). (65)


14 BRADJI

Consequently, in the particular case where (FK ,σ )K∈M,σ∈E is defined by (51)–(54), the right handsides of estimates (62)–(64) can be estimated by (k + hD)‖ u‖C3([0,T ];C2(�)).

Remark 4.1 (Similar schemes to (44) and similar results to Theorem 4.1). Similar schemes tothat described in (44) with similar (or better ) error estimates to those provided in Theorem 4.1can be derived using for instance the Crank–Nicolson or Newmark finite difference schemes fordiscretization in time, see the case of the variational approximation in [21, Pages 205–216]. Thissubject will be detailed in future works in which we expect to provide schemes approximating(1)–(3) with higher order in time. For instance, we consider the following implicit scheme (whichmight be of interest) which is similar to (44) and the discretization in time is a particular case ofthe Newmark’s method (cf. [13, Page 314]), for any n ∈ [[1, N ]], find un

D ∈ XD,B such that

(∂2 �M un+1

D , �M v)L2(�)

+ 1

2

⟨un+1

D + un−1D , v

⟩F

=∑

K∈M

m(K)f nKvK , ∀v ∈ XD,B, (66)

with initial conditions (42)–(43) (or possibly some modified versions for these initial conditionsin order to get higher order in time, see [22, Pages 232–233]).

We can prove that the scheme (66) with (42)–(43) has similar estimates to those provided inTheorem 4.1. The proof of such estimates is based on the proof of a similar lemma to 4.7 givenlater for the scheme (66), see Remark 4.7, and similar techniques to that used in (142)–(177). Wecan also prove (the subject of [23]) that the convergence order with respect to k can be improvedto be two (instead of one), under suitable choices for the right hand side of (66) and for the initialconditions u0

D and u1D. The scheme (66) with (42)–(43) is useful for instance to fight the numerical

damping which maybe arises during the implementation of scheme (44), see Section V.

Remark 4.2 (Regularity required for the results of Theorem 4.1). It seems that the extensionof Theorem 4.1 to u ∈ W 3,∞(0, T ; H 2(�)) can be studied for the case d = 2 or d = 3 (see[17, Remark 4.9, Page 1033]). Indeed, e.g., in the case when the mesh is admissible in the senseof [15, Definition 9.1, Pages 762–763] and d = 2 or d = 3, it is may be possible to show thatresults of Lemma 4.6 and the proof of of Theorem 4.1 hold when only u ∈ W 3,∞(0, T ; H 2(�)),and the estimates will be obtained under some condition on the mesh, see [24, Theorem 3.2, Page1942].

Remark 4.3 (Sufficient conditions on the data to get the required regularity of Theorem 4.1).The required regularity assumption u ∈ C3([0, T ]; C2(�)) in Theorem 4.1 can be reached byassuming sufficient regularity for the data u0, u1, f , and � and some compatibility conditions,see for instance [1, Theorem X. 8, Page 214] and [19, Theorem 7, Page 393].

Remark 4.4 (Some applications of Theorem 4.1). Results of Theorem 4.1 are useful since theyallow us to get error estimates for approximations for the exact solution and its first derivativesof the exact solution, of order E

kD(u) + ED(u1) + (hD + k)‖ u‖C3([0,T ]; C2(�)); indeed

• Estimate (62) implies that using Lemma 4.5 below and the triangle inequality, forall n ∈ [[0, N + 1]], �Mun

D approximates u(·, tn) by order EkD(u) + ED(u1) +

(hD + k)‖ u‖C3([0,T ]; C2(�)), in L2(�)–norm.• Estimate (64) implies that, for all n ∈ [[0, N +1]], the i–th component of the discrete gradient

∇DunD, defined by (46)–(49) by replacing u with un

D, approximates the i–th component ofthe gradient ∇u(·, tn) by order E

kD(u)+ED(u1)+(hD+k)‖ u‖C3([0,T ]; C2(�)), in L2(�)–norm.



• Estimate (63) implies that (using the triangle inequality), for all n ∈ [[0, N ]], ∂1PM un+1D

approximates ut(·, tn) by order EkD(u)+ED(u1)+(hD+k)‖ u‖C3([0,T ]; C2(�)), in L2(�)–norm.

• In the particular case where (FK ,σ )K∈M,σ∈E is defined by (51)–(54), �MunD, the i–th com-

ponent of the discrete gradient ∇DunD, and ∂1PM un+1

D approximate respectively u(·, tn), thei–th component of the gradient ∇u(·, tn), and ut(·, tn) by order hD + k in L2(�)–norm.

Remark 4.5 (A semidiscretization scheme). The present work is devoted to the full discretiza-tion scheme (which is the more practical) (42)–(44), i.e., discretization in time and space, but theanalysis presented here can be extended also to a semi–discretization scheme, i.e. discretizationonly in space.

Remark 4.6 (Discretization (42)–(43) of initial conditions (2)). The choices (42)–(43) of thediscretization of the initial conditions (2) are useful to get the case when n = 0 of estimates (62)and (64) in Theorem 4.1 and also when comparing the finite volume solution of scheme (42)–(44)and the auxiliary approximation given by (91), see for instance (150)–(159) and (160)–(170).More precise:

• The choice of the discretization (42) of the first initial condition in (2), i.e., u(x, 0) = u0(x),for all x ∈ �, is useful in the proof of Theorem 4.1. Indeed, this choice implies the prop-erty (93) which means that the finite volume solution of scheme (42)–(44) starts from thesame value as the auxiliary solution of (91) which in turn approximates the exact solutionof (1)–(3) according to Lemma 4.6. The property (93) is used, for instance, to deduce thecase when n = 0 of estimates (62) and (64) in Theorem 4.1 from estimates (94) and (96) inLemma 4.6. The choice (42) together with (43) serve us to get (141) which yields estimate(161).

• The choice of the discretization (43) of the second initial condition in (2), i.e., ut(x, 0) =u1(x), for all x ∈ �, serves us for instance to get (156).

The proof of Theorem 4.1 is performed thanks to several technical lemmata. We will thenquote these lemmata and then we prove Theorem 4.1. We begin with the following lemma whichis concerned with some interpolatory relations and norm inequalities. Results of Lemma 4.1 aregiven in [17], see [17, (4.6), Page 1026], and we recall them here for the sake of completeness.

Lemma 4.1 (Some interpolatory relations and norm inequalities, cf. [17]). Let � be a polyhe-dral open bounded connected subset of R

d , where d ∈ IN and D = (M, E , P) be a discretizationin the sense of Definition 1. Let B ⊂ Eint be given and let {βK

σ ; σ ∈ B, K ∈ M} be a subset of R

satisfying (14).

1. Interpolatory relations: Let PM, PD, and PD,B be the interpolatory operators given inDefinition 3, and ϕ ∈ C(�). The following relation holds:

PMϕ = �MPDϕ = �MPD,Bϕ. (67)

2. Norm inequalities: let ‖ ·‖1,2,M and | · |X be the norm and the semi norm given in Definition2. Then, the following inequality holds:

‖�M v‖1,2,M ≤ | v|X , ∀ v ∈ XD,0. (68)


16 BRADJI

The following lemma, which is the subject of [17, Lemma 4.2, Page 1026], provides us withthe equivalence between the norm of the gradient, given in (46)–(49), and the norm | · |X , givenin (12). This lemma is useful since it allows us, for instance, to get the uniqueness (and then theexistence) of the solution un

D of (42)–(44) when (FK ,σ )K∈M,σ∈E is defined by (51)–(54), and alsoto prove error estimate (64) of Theorem 4.1, see for example (111)–(112).

Lemma 4.2 (Stability property for the discrete gradient, cf. [17]). Let � be a polyhedral openbounded connected subset of R

d , where d ∈ IN and D be a discretization of � in the sense ofDefinition 1, and let θ ≥ θD be given (where θD is defined by (23)). Then there exists C5 ≥ 1 onlydepending on θ and d such that:

C−15 |v|X ≤ ‖∇D v‖L2(�) ≤ C5|v|X , ∀v ∈ XD, (69)

where ∇D is the discrete gradient given in (46)–(49).

Lemmata 4.3 and 4.4, given below, provide us, respectively, with error estimate for the gradientapproximation and a consistency result. Lemma 4.3 (resp. 4.4) is the subject of [17, Lemma 4.4,Page 1029] (resp. [17, (4.20), Page 1031]) with some slight modification on the r.h.s. (right handside) of [17, (4.13), Lemma 4.4, 1029] (resp. [17, (4.20), Page 1031]) . Indeed, the constantswhich appear in [17, (4.13), Lemma 4.4, 1029] and [17, (4.20), Page 1031] are depending onthe function ϕ. So when we apply [17, (4.13), Lemma 4.4, 1029] (resp. [17, (4.20), Page 1031])directly, for instance, in (118)–(119) (resp. (114)–(115)), we get constants depending on u(x, tn)and then on n, whereas the application of Lemmata 4.3 and 4.4, given below, leads to constantsindependent of discretization parameters. In addition to this, the application of Lemmata 4.3 and4.4 below helps us that to see clearly which regularity is required to get results of Theorem 4.1,see also Remark 4.2.

Lemma 4.3 (Consistency result for the discrete gradient, see [17]). Let D be a discretizationof � in the sense of Definition 1, and let θ ≥ θD be given (where θD is defined by (23)). Then, forany function ϕ ∈ C2(�), the following estimate holds:

‖∇DPDϕ − ∇ ϕ‖(L∞(�))d ≤ C6hD max|α|=2

‖ Dαϕ‖C(�), (70)

where ∇D is the discrete gradient given in (46)–(49) and C6 = d3 θ + d72 θ 2 + d

52 θ + 1.

Proof. Using the triangle inequality, and the definitions (47) and (49), we get

|∇K ,σPDϕ − ∇ ϕ(xK)| ≤ |∇KPDϕ − ∇ ϕ(xK)| + |RK ,σPDϕ|. (71)

We then estimate each term on the r.h.s. of the previous inequality; thanks to (48) and the Taylorexpansion

∇KPD ϕ = 1

m(K)

∑σ∈EK

m(σ )(ϕ(xσ ) − ϕ (xK))nK ,σ

≤ 1

m(K)

∑σ∈EK

m(σ )

((xσ − xK)T ∇ϕ(xK) + d2h2

K max|α|=2

‖ Dαϕ‖C(�)

)nK ,σ , (72)

where (xσ − xK)T denotes the transpose of xσ − xK ∈ Rd .



We use the following geometrical relation, it is the subject of [17, (2.17), Page 1017]:∑σ∈EK

m(σ )nK ,σ (xσ − xK)T = m(K)I, ∀K ∈ M, (73)

where I is the d × d identity matrix (Recall that (xσ − xK)T is a 1 × d matrix and nK ,σ is a d × 1matrix, therefore the product nK ,σ (xσ − xK)t is meaningful, namely nK ,σ (xσ − xK)T is a d × d

matrix; consequently equality (73) makes sense.)Therefore (72) with (73), and the definition (23) of θD, yields that

|∇KPD ϕ − ∇ϕ(xK)| ≤ d2hKθD

m(K)max|α|=2

‖ Dαϕ‖C(�)

∑σ∈EK

m(σ )dK ,σ . (74)

Thanks to the assumption that K is xK -star-shaped, the following property holds, cf. [17, (4.3),Page 1025] ∑

σ∈EK

m(σ )dK ,σ = dm(K). (75)

Consequently, (74) with (75) and the fact that θD ≤ θ , implies that

|∇KPD ϕ − ∇ϕ(xK)| ≤ max|α|=2

‖ Dαϕ‖C(�) d3θ hK . (76)

Let us move to estimate the second term on the r.h.s. of (71); using definition (49) combined with(72) and (76), we get for some values ρK ,σ such that |ρK ,σ | ≤ d2 max|α|=2 ‖ Dαϕ‖C(�):

| RK ,σPD ϕ| =∣∣∣∣∣√

d

dK ,σ(ϕ(xσ ) − ϕ(xK) − ∇KPD ϕ · (xσ − xK))

∣∣∣∣∣=

∣∣∣∣∣√

d

dK ,σ((xσ − xK) · ∇ϕ(xK) + h2

K ρK ,σ − ∇KPD ϕ · (xσ − xK))

∣∣∣∣∣=

∣∣∣∣∣√

d

dK ,σ((xσ − xK) · (∇ϕ(xK) − ∇KPD ϕ) + h2

K ρK ,σ )

∣∣∣∣∣≤

√d

dK ,σ

(θdK ,σ max

|α|=2‖ Dαϕ‖C(�) d

3θ hK + d2hKθdK ,σ max|α|=2

‖ Dαϕ‖C(�)

)

= √dd2θ max

|α|=2‖ Dαϕ‖C(�)(θd + 1) hK , (77)

Combining then inequalities (71), (76), and (77), we get

|∇K ,σPDϕ − ∇ ϕ(xK)| ≤ max|α|=2

‖ Dαϕ‖C(�)

√dd2θ (

√d + θd + 1) hK . (78)

It is easily seen that, since | xK − x| ≤ hK , for all x ∈ DK ,σ , for all σ ∈ EK

|∇ ϕ(xK) − ∇ ϕ(x)|(L∞(DK ,σ ))d ≤ max|α|=2

‖ Dαϕ‖C(�) hK . (79)


18 BRADJI

Using the triangle inequality combined with (78)–(79), we get, for all σ ∈ EK , for all K ∈ M

‖∇K ,σPDϕ − ∇ ϕ(x)‖(L∞(DK ,σ ))d ≤ max|α|=2

‖ Dαϕ‖C(�)

(d3 θ + d

72 θ 2 + d

52 θ + 1

)hK . (80)

Which implies, since hK ≤ hD, for all K ∈ M

‖∇K ,σPDϕ − ∇ ϕ(x)‖(L∞(�))d ≤ max|α|=2

‖ Dαϕ‖C(�)

(d3 θ + d

72 θ 2 + d

52 θ + 1

)hD. (81)

This concludes the proof of the desired inequality (70).

Lemma 4.4 (Consistency result, cf.[17]). Let � be a polyhedral open bounded connected sub-set of R

d , where d ∈ IN and D = (M, E , P) be a discretization in the sense of Definition 1.Let B ⊂ Eint be given and let {βK

σ ; σ ∈ B, K ∈ M} be a subset of R satisfying (14). Let ϕ bea function satisfying ϕ ∈ C2(�). Then, the following estimate holds, for all K ∈ M and for allσ ∈ EK ∩ B:

|ϕ(xσ ) − ϕσ | ≤ d2 max|α|=2

‖ Dαϕ‖C(�)θD,B h2K , (82)

where θD,B is given by (24) and

ϕσ =∑L∈M

βLσ ϕ(xL). (83)

Proof. Thanks to a Taylor expansion, for ϕ ∈ C2(�), we have

ϕ(xL) = ϕ(xσ ) + ∇u(xσ ) · (xL − xσ ) +∫ 1

0H(ϕ)(txσ + (1 − t)xL)(xL − xσ ) · (xL − xσ )dt ,

(84)

where H(ϕ)(z) denotes the Hessian matrix of ϕ at the point z.This implies, using (14)

∑L∈M

βLσ ϕ(xL) =

∑L∈M

βLσ ϕ(xσ ) +

∑L∈M

βLσ ∇u(xσ ) · (xL − xσ ) + Lσ

= ϕ(xσ ) +(∑

L∈M

βLσ xL −

∑L∈M

βLσ xσ

)· ∇u(xσ ) + Lσ

= ϕ(xσ ) +((∑

L∈M

βLσ xL

)− xσ

)· ∇u(xσ ) + Lσ

= ϕ(xσ ) + Lσ (85)

where

Lσ =∑L∈M

βLσ

∫ 1

0H(ϕ)(txσ + (1 − t)xL)(xL − xσ ) · (xL − xσ )dt (86)



It is easily seen that

|Lσ | ≤ d2 max|α|=2

‖ Dαϕ‖C(�)

∑L∈M

|βLσ | xL − xσ |2. (87)

This with (24) implies that

|Lσ | ≤ d2 max|α|=2

‖ Dαϕ‖C(�)θD,B h2K . (88)

Which gives ∣∣∣∣∣ϕ(xσ ) −∑L∈M

βLσ ϕ(xL)

∣∣∣∣∣ ≤ d2 max|α|=2

‖ Dαϕ‖C(�)θD,B h2K . (89)

Which completes the proof of Lemma 4.4.

For the sake of completeness, we recall the discrete version for the Poincaré lemma given in[17, (5.10), Lemma 5.4, Page 1038], when p = 2.

Lemma 4.5 (Discrete Poincaré lemma, see [17]). Let � be a polyhedral open bounded con-nected subset of R

d , where d ∈ IN\{0}. Let D = (M, E , P) be a discretization in the sense ofDefinition 1. Assume that θD, given by (23), satisfies θ ≥ θD. Then, there exists a constant Cp

only depending on �, d and θ such that

‖ u‖L2(�) ≤ C p‖ u‖1,2,M, ∀v ∈ HM(�), (90)

where ‖u‖1,2,M is defined in (17).

To analyse the convergence of the finite volume scheme (42)–(44), we need to use the followingauxiliary scheme: for any n ∈ [[0, N + 1]], find un


⟨un

D, v⟩F

= −∑

K∈M

vK

∫K

� u(x, tn)dx, ∀v ∈ XD,B. (91)

Note that, taking n = 0 in (91) with (2) leads to

⟨u0

D, v⟩F

= −∑

K∈M

vK

∫K

� u0(x)dx, ∀v ∈ XD,B, (92)

which together with (42) implies, when the condition (56) is satisfied (and then the uniquenessof the solution of (92) holds)

u0D = u0

D. (93)

This previous property will be used later in the proof of Theorem 4.1, see proof of (62) and (64).The following Lemma concerns with the convergence of the auxiliary scheme (91), where the

results therein are based on the use of [17, Theorem 4.2] which concerns with the isotropic case.


20 BRADJI

Lemma 4.6 (Some error estimates for the auxiliary scheme (91)). Let � be a polyhedral openbounded connected subset of R

d , where d ∈ IN\{0}, and ∂� = �\� its boundary. Assume that theweak solution of (1)–(3) in the sense of Theorem 2.1 satisfies u ∈ C([0, T ]; C2(�)). Let k = T

N+1 ,with N ∈ IN, and denote by tn = nk, for n ∈ [[0, N + 1]]. Let D = (M, E , P) be a discretizationin the sense of Definition 1. Let B ⊂ Eint be given and let {βK

σ , σ ∈ B, K ∈ M} be a subset ofR satisfying (14). Assume that θD,B, given by (24), satisfies θ ≥ θD,B. Let (FK ,σ )K∈M,σ∈E be afamily of linear mappings from XD into R satisfying (56), where 〈 ·, ·〉F is defined by (55).

Then for each n ∈ [[0, N + 1]], there exits a unique solution unD for the auxiliary scheme (91).

For a function u ∈ C1(�), we define the expression ED(u) given by (59). For a functionu ∈ C([0, T ]; C1(�)), we define the quantity E

kD(u) given by (60).

The following error estimates hold:

• Discrete L∞(0, T ; H 10 (�))–error estimate: for all n ∈ [[0, N + 1]]

α∥∥PM u(·, tn) − �M un

D∥∥

1,2,M ≤ EkD(u). (94)

• Wj ,∞(0, T ; L2(�))–error estimate, for all j ∈ [[0, 2]]: for all n ∈ [[j , N + 1]]

α∥∥∂j

(PM u(·, tn) − �M un

D)∥∥

L2(�)≤ Cp E

kD(u), (95)

where we have denoted ∂0vn = vn, ∂1vn = vn−vn−1

k, and ∂2vn = 1

k(∂1vn − ∂1vn−1), and Cp

is the constant of inequality (90) of Lemma 4.5.• Error estimate in the gradient approximation: for a constant C7 only depending on θ , d, �,

and α such that, for all n ∈ [[0, N + 1]]∥∥∇D un

D − ∇ u(·, tn)∥∥

(L2(�))d≤ C7

(E

kD(u) + hD‖ u‖C([0,T ];C2(�))

). (96)

Moreover, in the particular case where (FK ,σ )K∈M,σ∈E is defined by (51)–(54), there exists aconstant C4 only depending on θ , �, and d such that, for all j ∈ [[0, 2]]

EkD(u) ≤ C4 hD‖u‖C2([0,T ];C2(�)). (97)

Proof. Let us first remark that, thanks to the regularity assumption u ∈ C3([0, T ]; C2(�)),the right hand side of equation (91) is meaningful.

The proof of Lemma 4.6 is based on [17, Theorem 4.2] and the proof of its results.

1. Proof of existence and uniqueness: For each n ∈ [[0, N + 1]], Eq. (91) is equivalentto a linear system of N unknowns, namely {(un

K , unσ ); K ∈ M, σ ∈ H}, and N equa-

tions, where N = card(M) + card(H) (recall that H = Eint\B and B ⊂ Eint such that{βK

σ , σ ∈ B, K ∈ M} is satisfying (14)).For a fixed n ∈ [[0, N + 1]], assume that the right hand side of (91) equals to zero, taking

vD = unD, and using (56) yields that un

D = 0. This uniqueness implies the existence.2. Proof of estimate (94): Using an integration by parts yields that

−∑

K∈M

vK

∫K

� u(x, tn)dx = −∑

K∈M

∑σ∈EK

vK

∫σ

∇ u(x, tn) · nK ,σ (x)d γ (x). (98)



Since vσ = 0 for all σ ∈ Eext and∫

σ∇ u(x, tn)·nK ,σ (x)d γ (x)+∫

σ∇ u(x, tn)·nL,σ (x)d γ (x),

for all σ ∈ E such that Mσ = {K , L} (it stems from the fact that nK ,σ = −nL,σ ), we have

−∑

K∈M

∑σ∈EK

vσ

∫σ

∇ u(x, tn) · nK ,σ (x)d γ (x) = 0. (99)

This with (98) leads to

−∑

K∈M

vK

∫K

� u(x, tn)dx = −∑

K∈M

∑σ∈EK

(vK − vσ )

∫σ

∇ u(x, tn) · nK ,σ (x)d γ (x).

(100)

Substituting this in (91) and multiplying both sides of the resulting equation by −1, we get,for all n ∈ [[0, N + 1]]

−⟨un

D, v⟩F

=∑

K∈M

∑σ∈EK

(vK − vσ )

∫σ

∇ u(x, tn) · nK ,σ (x)d γ (x), ∀v ∈ XD,B. (101)

Adding∑

K∈M∑

σ∈EKFK ,σ (PD,Bu(·, tn))(vK − vσ ) to both sides of the previous equality,

and using definition (55), we get, for all v ∈ XD,B

⟨PD,Bu(·, tn) − un

D, v⟩F

=∑

K∈M

∑σ∈EK

RK ,σ (u(·, tn))(vK − vσ ), (102)

where

RK ,σ (u) = FK ,σ (PD,B(u)) +∫

σ

∇ u(x) · nK ,σ d γ (x), (103)

Taking PD,Bu(·, tn) = v + unD ∈ XD,B (therefore v = PD,Bu(·, tn) − un

D) in the previousequality, we get, for all n ∈ [[0, N + 1]]

〈v, v〉F =∑

K∈M

∑σ∈EK

RK ,σ (u(·, tn))(vK − vσ ). (104)

The previous inequality with the coercivity (56) and the definitions (12), (59), and (60)leads to

α| v|X ≤ EkD(u). (105)

The previous inequality implies, since v = PD,Bu(·, tn) − unD

α∣∣PD,Bu(·, tn) − un

D∣∣X ≤ E

kD(u). (106)

Using now (68), (106) implies

α∥∥�MPD,B u(·, tn) − �M un

D∥∥

1,2,M ≤ EkD(u). (107)


22 BRADJI

This with (67) yields


D∥∥

1,2,M ≤ EkD(u), (108)

which implies the required estimate (94).3. Proof of estimate (95): Estimate (108) with the inequality (90) of Lemma 4.5 implies, since

PM u(·, tn) − �M unD ∈ HM(�) (see Definition 3), for all n ∈ [[0, N + 1]]


D∥∥

L2(�)≤ Cp E

kD(u), (109)

which is the required estimate (95) when j = 0.Using the definition of ∂j and (91) and the fact that 〈·, ·〉F is a bilinear form, to deduce

that for any n ∈ [[j , N + 1]], ∂j unD ∈ XD,B is the solution of the following problem

⟨∂j un

D, v⟩F

= −∑

K∈M

vK

∫K

� ∂j u(x, tn)dx, ∀v ∈ XD,B. (110)

Therefore, we can apply estimate (109) to get (95), for any j ∈ {1, 2}.4. Proof of estimate (96): Using the triangle inequality to get∥∥∇DPD u(·, tn) − ∇D un

D∥∥

L2(�)d≤ ‖∇DPD u(·, tn) − ∇DPD,Bu(·, tn)‖L2(�)d

+ ∥∥∇DPD,Bu(·, tn) − ∇D unD∥∥

L2(�)d. (111)

The second term on the r.h.s. of the previous inequality could be written as‖∇D(PD,Bu(·, tn) − un

D)‖L2(�)d ; gathering (69) of Lemma 4.2 and (106) leads to

∥∥∇DPD,Bu(·, tn) − ∇D unD∥∥

L2(�)d≤ C5

αE

kD(u). (112)

The first term on the r.h.s. of (111) could be written as ‖∇D(PD u(·, tn) −PD,Bu(·, tn))‖L2(�)d ; using then (69) to get

‖∇DPD u − ∇DPD,Bu‖L2(�)d ≤ C5‖ PD u(·, tn) − PD,Bu(·, tn)‖X , (113)

On the other hand, using the definition (12) of the norm | · |X , we get

|PD u(·, tn) − PD,Bu(·, tn)|2X =∑

K∈M

∑σ∈EK∩B

m(σ )

dK ,σ

(u(xσ , tn) − un

σ

)2. (114)

Recall that unσ = ∑

L∈M βLσ u(xL, tn) and using (82) of Lemma 4.4 yields, since θD,B ≤ θ

|PD u(·, tn) − PD,Bu(·, tn)|2X ≤ d4‖ u‖2C([0,T ];C2(�))

θ 2h2D

∑K∈M

∑σ∈EK∩B

m(σ )

dK ,σh2

K . (115)

Using (23) and (75), the previous inequality implies that

|PD u(·, tn) − PD,Bu(·, tn)|2X ≤ ‖ u‖2C([0,T ];C2(�))

θ 4d5m(�)h2D. (116)



This with (113) implies that

‖∇DPD u(·, tn) − ∇DPD,Bu(·, tn)‖L2(�)d ≤ C5‖ u‖C([0,T ];C2(�))θ2d

52√

m(�)hD. (117)

Gathering now (111), (112), and (117) yields that

∥∥∇DPD u(·, tn) − ∇D unD∥∥

L2(�)d≤ C5‖ u‖C([0,T ];C2(�))θ

2d52√

m(�)hD + C5

αE

kD(u).

(118)

This with (70) of Lemma 4.3 and the triangle inequality implies

∥∥∇ u(·, tn) − ∇D unD∥∥

L2(�)d≤ C5‖ u‖C([0,T ];C2(�))θ

2d52√

m(�)hD + C5

αE

kD(u)

+ C6hD‖ u‖C([0,T ];C2(�))

√m(�), (119)

which leads to (96).5. Proof of estimate (97): estimate (97) is given in [17, (4.27), Theorem 4.8, Page 1033] when

j = 0 but with a constant depending on u. Thanks to the proof of [17, (4.27), Theorem4.8, Page 1033] and the previous techniques, we prove that there exists a constant C4 onlydepending on θ , �, and d such that, for all j ∈ [[0, 2]]

ED(∂ju(·, tn)) ≤ C4 hD‖∂ju(·, tn)‖C2(�). (120)

On the other hand

‖∂ju(·, tn)‖C2(�) = max|α≤2

supx∈�

|Dα∂ju(x, tn)|

= max|α≤2

supx∈�

|∂j (Dα u(x, tn))|. (121)

For j = 0, the previous equality leads to

‖∂ju(·, tn)‖C2(�) ≤ ‖ u‖C([0,T ];C2(�)). (122)

For j = 1, we remark that

∂1(Dα u(x, tn)) = 1

k

∫ tn

tn−1

(Dα u)t (x, t)dt , (123)

which implies that

|∂1(Dα u(x, tn))| ≤ 1

k

∫ tn

tn−1

supx∈�

|(Dα u)t (x, t)| dt

≤ maxt∈[0,T ]

supx∈�

|(Dα u)t (x, t)|

= ‖ u‖C1([0,T ];C2(�)), ∀α ∈ INd satisfying |α| ≤ 2. (124)


24 BRADJI

For j = 2, we remark that

∂2(Dα u(x, tn)) = 1

k2

∫ tn

tn−1

∫ t

t−h

(Dα u)tt (x, t)dsdt , (125)

which yield, thanks to the technique used to prove (124), that

|∂2(Dα u(x, tn))| ≤ ‖ u‖C2([0,T ];C2(�)), ∀α ∈ INd satisfying |α| ≤ 2. (126)

Gathering now (120)–(126) to get the first part of estimate (65).

The following lemma will help us to prove Theorem 4.1

Lemma 4.7 (Useful lemma: an a priori estimate). Let � be a polyhedral open bounded con-nected subset of R

d , where d ∈ IN\{0}, and ∂� = �\� its boundary. Let k = T

N+1 , with N ∈ IN,and denote by tn = nk, for n ∈ [[0, N + 1]]. Let D = (M, E , P) be a discretization in the senseof Definition 1. Let B ⊂ Eint be given and let {βK

σ , σ ∈ B, K ∈ M} be a subset of R satisfying(14). Assume that θD,B, given by (24), satisfies θ ≥ θD,B. Let (FK ,σ )K∈M,σ∈E be a family of linearmappings from XD into R satisfying (56), where 〈·, ·〉F is defined by (55). Assume that the bilinearform 〈·, ·〉F is symmetric on XD × XD in the sense of (57) and is continuous in the sense (58).

Assume in addition that there exists (ηnD)N+1

n=0 ∈ X N+2D,B such that for any n ∈ [[1, N ]], for all

v ∈ XD,B

(�M ∂2ηn+1

D , �M v)L2(�)

+ ⟨ηn+1

D , v⟩F

=∑

K∈M

m(K)SnKvK , (127)

where SnK ∈ R, for all n ∈ [[1, N ]] and for all K ∈ M.

Then the following estimate holds, for some positive constant C8 only depending on M and T ,for all j ∈ [[1, N ]].

∥∥�M ∂1 ηj+1D

∥∥2

L2(�)+ 2α

∣∣ηj+1D

∣∣2

X ≤ C8

(∥∥�M ∂1 η1D∥∥2

L2(�)+ ∣∣η1

D∣∣2

X + (S)2), (128)

where

S = max

⎧⎨⎩( ∑

K∈M

m(K)(Sn

K

)2

) 12

, n ∈ [[1, N ]]⎫⎬⎭ . (129)

Proof. Taking v = ∂1 ηn+1D in (127) and summing the result over n ∈ [[1, j ]], where

j ∈ [[1, N ]], we get

j∑n=1

(�M ∂2 ηn+1

D , �M ∂1 ηn+1D

)L2(�)

+j∑

n=1

⟨ηn+1

D , ∂1 ηn+1D

⟩F

=j∑

n=1

∑K∈M

m(K)SnK∂1 ηn+1

K ,

(130)

where we have denoted ηn+1D = ((ηn+1

K )K∈M, (ηn+1σ )σ∈E).



The following rules will be useful to simplify the left hand side of the previous equality:

(�M ∂2 ηn+1

D , �M ∂1 ηn+1D

)L2(�)

= 1

2k

(αn+1

D − αnD, αn+1

D − αnD)L2(�)

+ 1

2k

{(αn+1

D , αn+1D

)L2(�)

− (αn

D, αnD)L2(�)

}, (131)

where

αnD = �M ∂1 ηn

D. (132)

A similar identity to that of (131) is, thanks to the symmetry (57)

⟨ηn+1

D , ∂1 ηn+1D

⟩F

= 1

2k

{⟨ηn+1

D − ηnD, ηn+1

D − ηnD⟩F

+ ⟨ηn+1

D , ηn+1D

⟩F

− ⟨ηn

D, ηnD⟩F

}. (133)

Identities (131)–(133) yield

j∑n=1

(�M ∂2 ηn+1

D , �M ∂1 ηn+1D

)L2(�)

+j∑

n=1

⟨ηn+1

D , ∂1 ηn+1D

⟩F

≥ 1

2k

((α

j+1D , αj+1

D)L2(�)

+ ⟨η

j+1D , ηj+1

D⟩F

)− 1

2k

((α1

D, α1D)L2(�)

+ ⟨η1

D, η1D⟩F

). (134)

This with (130), (56) and (58) implies

1

2k

(‖αj+1‖2L2(�)

+ α∣∣ηj+1

D∣∣2

X

) ≤j∑

n=1

∑K∈M

m(K)SnK∂1 ηn+1

K + 1

2k

(∥∥α1D∥∥2

L2(�)+ M

∣∣η1D∣∣2

X

).

(135)

Multiplying both sides of the previous inequality by 2k; using (132) and the Cauchy Schwarzinequality, we get

∥∥�M ∂1 ηj+1D

∥∥2

L2(�)+ α

∣∣ηj+1D

∣∣2

X ≤ 2kSj∑

n=1

∥∥�M ∂1 ηn+1D

∥∥L2(�)

+ ‖�M ∂1 η1D‖2

L2(�)+ M

∣∣η1D∣∣2

X ,

(136)

where S is given by (129).This with the inequality ab ≤ εa2 + b2/ε, for all ε > 0, (136) implies, for all j ∈ [[1, N ]]

(recall that k(N + 1) = T and k/T = 1/(N + 1) ≤ 1/2)

∥∥�M ∂1 ηj+1D

∥∥2

L2(�)+ 2α

∣∣ηj+1D

∣∣2

X ≤ 2k

T

j∑n=2

(‖�M ∂1 ηnD‖2

L2(�)+ α

∣∣ηnD∣∣2

X

)

+ 2‖�M ∂1 η1D‖2

L2(�)+ 2M

∣∣η1D∣∣2

X + 8T 2(S)2. (137)


26 BRADJI

Using the discrete version of the Gronwall’s lemma, the previous inequality with the fact that(N + 1)k = T implies the required estimate (128).

Remark 4.7 (An a priori estimate for the scheme (66)). As mentioned in Remark 4.1, we canprove a priori estimate for the scheme (66) similar to that provided in Lemma 4.7. Indeed, assumethat there exists (ηn

D)N+1n=0 ∈ X N+2

D,B such that for any n ∈ [[1, N ]], for all v ∈ XD,B

(�M ∂2 ηn+1

D , �M v)L2(�)

+ 1

2

⟨ηn+1

D + ηn−1D , v

⟩F

=∑

K∈M

m(K)SnKvK . (138)

We can prove that, by taking v = ηn+1D −ηn−1

Dk

in (138) and using the same techniques used in theproof of Lemma 4.7 with the fact v = ∂1ηn+1

D + ∂1ηnD

‖�M ∂1 ηj+1D ‖2

L2(�)+ 2α

∣∣ηj+1D

∣∣2

X ≤ C(‖�M ∂1 η1

D‖2L2(�)

+ ∣∣η1D∣∣2

X + ∣∣η0D∣∣2

X + (S)2). (139)

PROOF OF THEOREM 4.1.

1. Well posedness for (42)–(44):a. Existence and uniqueness for (42): Eq. (42) is equivalent to a linear system of N

unknowns, namely {(u0K , u0

σ ); K ∈ M, σ ∈ H}, and N equations, where N =card(M)+card(H) (recall thatH = Eint\B andB ⊂ Eint such that {βK

σ , σ ∈ B, K ∈ M}is satisfying (14)).

Assume that the right hand side of (42) equals to zero, taking vD = u0D, and using

(56) yields that u0D = 0. This uniqueness implies the existence.

b. Existence and uniqueness for (43): equation (43), since (42) is well posed, is equivalentto

⟨u1

D, v⟩F

= −(� u, �M v)L2(�), ∀ v ∈ XD,B. (140)

u(x) = u0(x) + ku1(x), ∀ x ∈ �. (141)

Therefore, existence and uniqueness for the solution of (43) can be justified as in theprevious item for (42).

c. Existence and uniqueness for (44): we justify now that (44) has a unique solution(un

D)N+1n=0 ∈ X N+2

D,B . To do so, we first prove that there exists a unique solution un+1D ∈ XD,B

for (44) for given unD, un−1

D and f nK . We then set, as usual to get the uniqueness for linear

systems, unD = un−1

D = 0 and f nK = 0 and taking v = un+1

D in (44) yields, thanks to (44),un+1

D = 0. The existence of un+1D follows immediately, since (44) is a finite dimensional

linear system with respect to the unknowns un+1D (with as many unknowns as many equa-

tions). This with the fact that u0D and u1

D exists and unique (thanks to the two previousitems), implies that, successively on n, un

D for each n ∈ [[0, N + 1]].2. Some primary estimates: writing (91) in the step n + 1 yields, for all n ∈ [[0, N ]]

⟨un+1

D , v⟩F

= −∑

K∈M

vK

∫K

� u(x, tn+1)dx, ∀v ∈ XD,B. (142)



Subtracting the previous equation from (44), we get

(�M ∂2 un+1

D , �M v)L2(�)

+ ⟨ηn+1

D , v⟩F

=∑

K∈M

(m(K)f n

K +∫

K

� u(x, tn+1)dx

)vKdx,

(143)

where ηnD is given by

ηnD = un

D − unD, ∀ n ∈ [[0, N + 1]]. (144)

Equation (143) is equivalent to the following one

(�M ∂2 ηn+1

D , �M v)L2(�)

+ ⟨ηn+1

D , v⟩F

=∑

K∈M

m(K)SnKvK , (145)

where SnK is given by, using the expression (29) of f n

K

SnK = 1

km(K)

∫ tn+1

tn

∫K

f (x, t)d x dt + 1

m(K)

∫K

� u(x, tn+1)dx − ∂2un+1K . (146)

One remarks that (ηnD)N+1

n=0 ∈ X N+2D,B satisfies (127), with Sn

K is given by (146), one can apply(128) of Lemma 4.7 to obtain for all n ∈ [[1, N ]].

∥∥�M ∂1 ηn+1D

∥∥2

L2(�)+ 2α

∣∣ηn+1D

∣∣2

X ≤ C8

(∥∥�M ∂1 η1D∥∥2

L2(�)+ ∣∣η1

D∣∣2

X + (S)2), (147)

with S is defined by (129) and (146).We now estimate the terms on the right hand side of the previous inequality. To do

so, we consider the following new discrete function which will help us to simplify ourcomputations, for all n ∈ [[0, N + 1]]

ξnD = un

D − PD u(·, tn), (148)

where the operator PD (resp. unD) is defined in (3) (resp. (91)). It is useful to remark that,

thanks to (144) and (148)

unD − PD u(·, tn) = ηn

D + ξnD. (149)

Let us set

ξnD = ((

ξnK

)K∈M,

(ξnσ

)σ∈E

).

a. Estimate of ‖�M ∂1 η1D‖L2(�): using (149), the triangle inequality and (67), we get (recall

that ut(·, 0) = u1(·))

∥∥�M ∂1η1D∥∥

L2(�)≤

4∑i=1

Ti , (150)


28 BRADJI

where

T1 = ∥∥�M ∂1 ξ 1D∥∥

L2(�), (151)

T2 = ∥∥�M ∂1 u1D − u1

∥∥L2(�)

, (152)

T3 = ‖ ut(·, 0) − ∂1 u(·, t1)‖L2(�), (153)

and

T4 = ‖ ∂1 u(·, t1) − PM∂1 u(·, t1)‖L2(�). (154)

Estimate (95), when j = 1, with (148) and (67) leads to

T1 ≤ Cp

αE

kD(u). (155)

Equation (43) with techniques used to prove the case j = 0 of (95) (We replaceu(·, tn) by u1 and un

D by ∂1 u1D in (98)–(109).), the triangle inequality, and the fact

that ‖PM u1 − u1‖L2(�) ≤ C9hD‖ u1‖C1(�) implies that, since u1(·) = ut(·, 0)

T2 ≤ C10(ED(u1) + hD‖ u‖C1([0,T ];C1(�))), (156)

where C10 is a positive constant only depending on α and θ .We also remark that, for some positive constantsC11 andC12 which are only depending

on �

T3 ≤ C11k‖ u‖C1([0,T ];C(�)), (157)

and, using techniques (123)–(124)

T4 ≤ C12hD‖ u‖C1([0,T ];C1(�)). (158)

Thanks to (150)–(158), the following estimate for the first term on the right hand sideof (147) holds, using the fact that ‖ u‖C1([0,T ];C(�)) ≤ ‖ u‖C1([0,T ];C1(�))∥∥�M ∂1 η1

D∥∥

L2(�)≤ C13

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C1([0,T ];C1(�))

), (159)

where C13 is a positive constant only depending on θ , �, d, and α.b. Estimate of |η1

D|X : thanks to (141), we expect that u1D is an approximation for u0 + ku1.

For this reason and in order to bound |η1D|X = | u1

D − u1D|X , we analyse as follows

using the triangle inequality∣∣η1D∣∣X ≤ ∣∣u1

D − PD,B (u0 + ku1)∣∣X + | PD,B (u0 + ku1) − PD (u0 + ku1)|X

+ | PD (u0 + ku1) − PD u(·, t1)|X + |PD u(·, t1) − PD,B u(·, t1)|X+ | PD,B u(·, t1) − u1

D|X . (160)

So, in order to estimate |η1D|X , it suffices to estimate the five terms on the right hand

side of the previous inequality.



Equation (141) with a similar reasoning to that used in (98)–(105) yields that

∣∣u1D − PD,B (u0 + ku1)

∣∣X ≤ 1

αED(u0 + ku1). (161)

One remarks that that ED is semi norm and the fact that k < T , one can deduce from(161), since ED(u0) ≤ E

kD(u)

∣∣u1D − PD,B (u0 + ku1)

∣∣X ≤ 1

α

(E

kD(u) + T ED(u1)

). (162)

Using now (12), Definition 3 of PD,B and PD, (82) (23), and (75), we get

|PD,B (u0 + ku1) − PD (u0 + ku1)|X ≤ C14hD‖ u0 + ku1‖C2(�), (163)

where C14 is a positive constant only depending on d, � and θ .A suitable version for (163) is, since k < T

|| PD,B (u0 + ku1) − PD (u0 + ku1)|X ≤ C14(1 + T )hD‖ u‖C1([0,T ];C2(�)). (164)

We need to use the following Taylor expansion in order to estimate |PD (u0 + ku1)−PD u(·, t1)|X (third term on the right hand side of (160))

u(·, t1) = u(·, t0) + kut (·, t0) +∫ t1

t0

utt (·, t)dt . (165)

this with the fact u(·, t0) = u(·, 0) = u0 and ut(·, t0) = ut(·, 0) = u1 yields

‖ u(x, t1) − (u0(x) + ku1(x))‖C1(�) ≤ k‖ u‖C2([0,T ];C1(�)). (166)

This with the fact |ϕ(xσ )−ϕ(xK)| ≤ hK maxx∈� |∇ϕ(x)|, for all σ ∈ EK , for all K ∈ M,and for all ϕ ∈ C1(�), (12), (23), and (75) the implies that, for some positive constantC15 only depending on d , � and θ

|PD (u0 + ku1) − PD u(·, t1)|X ≤ C15k‖ u‖C2([0,T ];C1(�)). (167)

As done in (163), the following estimate holds, since ‖ u(·, t1)‖C2(�) ≤ ‖ u‖C([0,T ];C2(�))

|PD u(·, t1) − PD,B u(·, t1)|X ≤ C14hD‖ u‖C([0,T ];C2(�)). (168)

As reasoned to get (161), we obtain

∣∣PD,B u(·, t1) − u1D∣∣X ≤ 1

αED(u(·, t1)). (169)

Gathering (160), (162), (164), (167), (168), and (169) yields∣∣η1D∣∣X ≤ C16

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C2([0,T ];C2(�))

), (170)

where C16 is only depending on T , α, d, � and θ .


30 BRADJI

c. Estimate of S given by (129) and (146): substituting f by utt − � u (it is the subject of(1)) in the expansion (146), we get

SnK = 1

km(K)

∫ tn+1

tn

∫K

utt (x, t)d x dt − 1

km(K)

∫ tn+1

tn

∫K

� u(x, t)d x dt

+ 1

m(K)

∫K

� u(x, tn+1)dx − ∂2un+1K . (171)

Thanks to the Taylor expansion and the triangle inequality, the previous expansion leadsto

∣∣SnK

∣∣ ≤ ∣∣∂2ξn+1K

∣∣ + C17(k + hD)‖ u‖C3([0,T ];C2(�)), (172)

where C17 is positive constant (which can be computed explicitly) and ξnD is given by

(148). Estimate (172) with the definition (129) of S and estimate (95), when j = 2,implies that

S ≤ C18

(E

kD(u) + (k + hD)‖ u‖C3([0,T ];C2(�))

), (173)

where C18 is a positive constant only depending on �, d, θ , and α.Gathering now (147), (159), (170), and (173), and using the facts ED(u0) = ED(u(·, t0))

and√

a + b + c ≤ √a + √

b + √c for all a, b, c ≥ 0 yields, for all n ∈ [[2, N + 1]]

∥∥�M∂1 ηnD∥∥

L2(�)≤ C19

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

), (174)

and

α∣∣ηn

D∣∣X ≤ C19

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

), (175)

where C19 is a positive constant only depending on M , T �, d, θ , and α.Thanks to the two previous estimates together with Lemma 4.6, we will derive the

estimates (62)–(64) of Theorem 4.1.3. Proof of estimate (62): estimate (175) with (68) implies, since ηn

D ∈ XD,0 (see (148)), forall n ∈ [[2, N + 1]]

α∥∥�M ηn

D∥∥

1,2,M ≤ C19

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

). (176)

This with (149), the fact that �M ξnD = �Mun

D − PM u(·, tn) (see (67)), estimate (94) ofLemma 4.6, and the triangle inequality implies estimate (62) for all n ∈ [[2, N + 1]].

The case n = 1 in estimate (62) can be proved by gathering (170), (68), and the casen = 1 in (94) of Lemma 4.6.

Thanks to (93), the case n = 0 of (62) can be deduced from the case n = 0 of (94) ofLemma 4.6.

4. Proof of estimate (63): the case when n ∈ [[2, N + 1]] of (63) can be proved by gathering(174), (95) of Lemma 4.6, and the triangle inequality.

The case n = 1 of (63) can be proved by gathering (159), the case n = 1 of (95) ofLemma 4.6, and the triangle inequality.



FIG. 2. Left Last three steps of finite difference scheme (178) with (181) and (182), right Last three stepsof finite difference scheme (202) with (181) and (182). The time step k = 1/700. The initial data are givenby u0(x) = exp (−40(x − 1)2) and u1(x) = sin(π x). [Color figure can be viewed in the online issue, whichis available at wileyonlinelibrary.com.]

5. Proof of estimate (64): thanks to estimate (69) of Lemma 4.2, estimates (170) and (175)implies that for all n ∈ [[1, N + 1]]∥∥∇Dηn

D∥∥

L2(�)≤ C20

(E

kD(u) + ED(u1) + (k + hD)‖ u‖C3([0,T ];C2(�))

), (177)

where C20 is a positive constant only depending on M , T �, d, θ , and α.Combining (177), (96), and the triangle inequality yields (64) for all n ∈ [[1, N + 1]].

The case n = 0 of (64) can be deduced directly from the case n = 0 of (96), thanks to (93).6. Proof of estimates (65): the first part of estimate (65) is the estimate (97) of Lemma 4.6,

whereas the second part of estimate (65) can be proved by the same way used to justify (97)in the proof of Lemma 4.6.

Which completes the proof of Theorem 4.1. �

V. SOME COMPUTATIONAL RESULTS

The present section is devoted to provide some computational results and it is divided into thefollowing two subsections. The first subsection is treating the so called numerical damping whichappears during the implementation of the main scheme presented in this article, that is (44).Indeed, in the one dimensional case and on uniform meshes with f ≡ 0, we will remark that thescheme (44) is damped whereas the scheme (66) is not damped, see Figs. 2 and 3. So, to fightthe numerical damping of the scheme(44) with (42)–(43), we instead use the scheme (66) with(42)–(43) (recall that this scheme, i.e. (66) with (42)–(43), has similar estimates to those providedin Theorem 4.1, see Remarks 4.1 and 4.7). The aim of the second subsection, i.e. Subsection VB,is to present a numerical example supporting the theoretical results quoted in Theorem 4.1.

However, our contribution concerning the numerical damping, that is the subject of subsectionA, is just an attempt in order to provide some simple information on the numerical damping inthe one dimensional case and maybe it (numerical damping) needs an extensive study which isbeyond the scope of this present study, see for instance [13, Page 320] and references therein. Wewill explain that the scheme (44) is damped in the one dimensional case and on uniform meshesbecause the absolute value of its amplification factor is less than one, whereas the absolute value of


32 BRADJI

FIG. 3. Left Last three steps of the Newmark finite difference scheme (179) with (181) and (182), right Lastthree steps of the Newmark finite difference scheme (199) with (181) and (182). The time step k = 1/700.The initial data are given by u0(x) = exp (−40(x − 1)2) and u1(x) = sin(π x). [Color figure can be viewedin the online issue, which is available at wileyonlinelibrary.com.]

the amplification factor of (66), in the one dimensional case and on uniform meshes, is one. Suchexplanation will be supported by other two schemes provided in the paragraph. More precise, weconsider the two schemes (198) (in which the absolute value of the amplification factor is one)and (201) (in which the absolute value of the amplification factor is less than one) and we willsee that in fact the scheme (198) is not damped, whereas the scheme (201) is damped, in the onedimensional case and on uniform meshes.

A. Numerical Damping in One Dimension

The scheme (66) with (42)–(43) (as mentioned in Remarks 4.1 and 4.7, this scheme has alsosimilar estimates to those provided in Theorem 4.1) is useful for instance to fight the numericaldamping which maybe arises during the implementation of scheme (44). More precise, let usconsider respectively the one dimensional case of schemes (44) and (66) with uniform meshesand f ≡ 0 (recall that ∂2vn+1 is given by (26)):

∂2un+1i = δ2 un+1

i , (178)

and

∂2un+1i = δ2u

n+1i + δ2 un−1

i

2, (179)

where δ2vi denotes the divided difference

δ2vi = vi+1 − 2vi + vi−1

h2, (180)

with the following usual discretization for initial conditions (for the sake of simplicity)

u0i = u0(xi), (181)

and

u1i = ku1(xi) + u0(xi), (182)

with, of course, the Dirichlet boundary conditions un0 = un

M−1 = 0, for all n.



(Where we have denoted the spatial mesh points by xi = ih, i ∈ [[0, M−1]] with h = 1/(M−1)

and the temporal mesh points by tn = nk, n ∈ [[0, N + 1]] with k = 1/(N + 1).)The scheme (178) with (181)–(182) can be written in the following way which helps us to

compute successively the solution uni

(I + rA)Un+1 = 2Un − Un−1, (183)

where

U 0 = (u0(x1), u0(x2), . . . , u0(xM−2))

T , (184)

U 1 = (ku1(x1) + u0(x1), . . . , ku1(xM) + u0(xM−2))T , (185)

r =(

k

h

)2

, (186)

I is the identity matrix and A is the matrix

A =

⎛⎜⎜⎜⎝

2 −1 0 . . . 0−1 2 −1 . . . 0

......

0 . . . 0 −1 2

⎞⎟⎟⎟⎠ (187)

Therefore (183) can be written as

LUn+1 = 2Un − Un−1, (188)

where L is given by

L =

⎛⎜⎜⎜⎝

1 + 2r −r 0 . . . 0−r 1 + 2r −r . . . 0...

...0 . . . 0 −r 1 + 2r

⎞⎟⎟⎟⎠ (189)

From (188), we have

Un+1 = 2L−1 Un − L−1 Un−1. (190)

From (190), it is clear that L−1 (with all eigenvalues 0 < ρ < 1) damps the overall right handside. The numerical tests on the left of Fig. 2 show this damping.

To be more clear, we use the concept of the amplification factor (in fact the amplificationfactor is in some sense the growth of the solution from a level to another) as a tool to analyse thisphenomenon of numerical damping. Indeed, when computing the amplification factor of scheme(178) with (181)–(182), we find that (by assuming that the solution of (178) with (181)–(182) canbe written as un

i = λnk

√2 sin (kiθ), where k ∈ [[1, M − 2]] and θ = π/(M − 1), see for instance

[22, Pages 224–226]) the amplification factor λk has two values λk,1 and λk,2

λk,1 = 1

1 + 2r sin(

kθ

2

)j, (191)

and

λk,2 = 1

1 − 2r sin(

kθ

2

)j, (192)


34 BRADJI

where

j2 = −1, (193)

and

θ = π

M − 1. (194)

Since

kθ

2= kπ

2(M − 1)∈[

π

2(M − 1),(M − 2)π

2(M − 1)

]⊂]0,

π

2[ (195)

then sin( kθ

2 ) �= 0 and therefore

0 < |λk,1| = |λk,2| < 1. (196)

We turn now to the scheme (179) with (181)–(182). Numerical tests presented on the left ofFig. 3 show that scheme (179) with (181)–(182) is not damped. By some computations, we findthat the amplification factor λk has two values λk,1 and λk,2 with λk,1 = λk,2. One remarks that

λk,1λk,2 = 1+2r2 sin2( kθ2 )

1+2r2 sin2( kθ2 )

= 1, one deduces

| λk,1|2 = | λk,2|2 = λk,1λk,2 = 1. (197)

This shows that the absolute value of the amplification factor of scheme (179) is one.

Numerical Damping of Some Other Schemes In this present paragraph, we quote twoschemes which have similar error estimates to those provided in Theorem 4.1. Numerically in theone dimensional case on uniform meshes, the first one (that is (198)) is not damped, whereas thesecond one (that is (201)) is damped. We will check as mentioned above that the absolute valueof the amplification factor of the scheme (199) (resp. (202)) is one (resp. less than one) (recallthat (199) and (202) are respectively the schemes (198) and (201) in the one dimensional case onuniform meshes with f ≡ 0).

1. A Newmark scheme: we consider the following Newmark scheme, for n ∈ [[1, N ]], findun


(∂2 �M un+1

D , �M v)L2(�)

+ 1

4

⟨un+1

D + 2unD + un−1

D , v⟩F

=∑

K∈M

m(K)f nKvK , ∀v ∈ XD,B.

(198)

We can prove that the scheme (198) with (42)–(43) has similar estimates to those providedin Theorem 4.1, see Remarks 4.1 and 4.7.

Let us consider the one dimensional case of (198) on uniform meshes and f ≡ 0 (see[11, Pages 263–264]), for n ∈ [[1, N ]]

∂2un+1i = δ2un+1

i + 2δ2uni + δ2un−1

i

4. (199)



Numerical tests presented on the right of Fig. 3 show that scheme (199) with (181)–(182)is not damped. By some computations, we find that the amplification factor is given by

λk = 1 ∓ r sin(

kθ

2

)j

1 ± r sin(

kθ

2

)j. (200)

This shows that the absolute value of the amplification factor of scheme (199) is one, i.e.,no damping.

2. A Crank Nicolson scheme: we consider the following Crank Nicolson scheme, for anyn ∈ [[1, N ]], find un


(∂2 �M un+1

D , �M v)L2(�)

+ 1

2

⟨un+1

D + unD, v

⟩F

=∑

K∈M

m(K)f nKvK , ∀v ∈ XD,B. (201)

We can prove that the scheme (201) with (42)–(43) has similar estimates to those providedin Theorem 4.1, see Remarks 4.1 and 4.7.

Let us consider the one dimensional case of (201) on uniform meshes and f ≡ 0, forn ∈ [[1, N ]]

∂2un+1i = δ2un+1

i + δ2uni

2h2. (202)

Numerical tests presented on the right of Fig. 2 show that scheme (202) with (181)–(182) isdamped. By some computations, we show that in fact the absolute value of the amplificationfactor is less than one.

B. A Numerical Example Supporting Theorem 4.1

The present subsection is devoted to provide a numerical test to justify theoretical results pro-vided in Theorem 4.1 of our article in two dimensions. We consider � = (0, 1)2 meshed with therectangular meshes described as in [15, Pages 756–758] (which is a particular case of the meshintroduced recently in [17]), with uniform meshes with mesh size h. We will consider the purehybrid finite volume scheme, that is B = ∅. For the sake of simplicity, we only considered thediscrete gradient described in [25, (211)–(212), Page 333] (which is also a particular case of thediscrete gradient introduced recently in [17]). The exact solution is given by

u(x, y, t) = sin(π x) sin(π y) cos(√

2π t), (x, y, t) ∈ (0, 1)2 × (0, 1). (203)

By this way

u0(x, y) = sin(π x) sin(π y), (x, y) ∈ (0, 1)2, (204)

u1 ≡ 0, (205)

and

f ≡ 0. (206)


36 BRADJI

By some computations, we find that the discrete gradient is given by, for Kij =](i − 1)h, ih[×](j − 1)h, jh[

∇DuD|Kij=

(ui+1,j − ui−1,j

2h,ui,j+1 − ui,j−1

2h

), Kij ∩ ∂� = ∅, (207)

and with slightly modification when Kij ∩ ∂� �= ∅The following results are obtained using a Scilab programme with k = h (k is the time step):

1. Error in W 1,∞(L2)

1/h| Error|

W1,∞(L2)

h

30 17.12268935 17.64855940 18.05297250 18.63383360 19.030728

2. Error in L∞(H 1)

1/h| Error|

L∞(H1)

h

30 20.51129435 20.86519340 21.13657250 21.51155555 21.64628560 21.7578370 21.940252

3. Error in the gradient approximation

1/h| Gradient of error|

L∞(L2)

h

35 21.4444340 21.64843445 21.80409450 21.92658955 22.02539660 22.10671870 22.24295880 22.34917390 22.427915



From the previous tests, we remark that the error has the same behavior as h which supportsour theoretical results quoted in Theorem 4.1.

VI. CONCLUSIONS AND PERSPECTIVES

We provided a new finite volume scheme approximating the wave equation (as a model for secondorder hyperbolic equations). The finite volume scheme can be applied on any type of spatial grid:conforming or nonconforming, 2D and 3D, or more, made with control volumes which are onlyassumed to be polyhedral. The matrix generated by this scheme is sparse, symmetric, positiveand definite when the bilinear form devoted the Laplace operator is satisfying some suitable con-ditions. Although the numerical scheme stems from the finite volume methods, its formulationseems a discrete version for a weak formulation for the wave problem. From this point of view, thediscretization scheme can be viewed as a nonconforming finite element method. The discretiza-tion of the initial conditions are performed as approximations for orthogonal projection of theinitial data functions. This study can be extended to the general second order hyperbolic equationutt − ∇ · (κ∇ u) − bu = f where κ (resp. b) is a smooth function from � into Md(R) (resp. R),where Md(R denotes the set of d × d matrices. However, nice path to be followed in the futureis to consider the general case of second order hyperbolic equation when κ and b are dependingon the time t . We think that techniques provided in the present contribution are not so helpful inthis case and we will then write our problem as a system of first order equations. We write then(1)–(3) as the system of first order hyperbolic equations wt − A w = F , where w = (v, u)T ,

A =(

0 �

I 0

), and F = (f , 0)T , and then the approximation of wt − A w = F . This will be

another subject to work on in the near future.As explained in Remark 4.4, results of Theorem 4.1 are useful since they allow us to get error

estimates for approximations for the exact solution and its first derivatives of the exact solution.These approximations for the first derivatives of the exact solution of (1)–(2) allow us, using ourapproach [26] to get higher order finite volume approximation for problem (1)–(2) without makeappeal to schemes of higher order. This task will be the subject of another future work. Anotherinteresting path to be followed is to work on the case when the data are not smooth.

The author thanks the referees for their valuables advices that helped to improve the article. Hewishes to express his deepest gratitude to the referee who attracted his attention to the so-callednumerical damping. The advices of the referee helped the author to add Remarks 4.1 and 4.7 andthe fifth section. The author also thanks Prof. T. Gallouët, Prof. E. Süli, Prof. S. Tomar, Prof. A.Baeza, and Prof. F. R. de Boer for the useful discussions.

References

1. H. Brezis, Analyse fonctionnelle: théorie et applications, Dunod, Paris, 1999.

2. M. Ainsworth, P. Monk, and W. Muniz, Dispersive and dissipative properties of the discontinuousGalerkin finite element methods for second-order wave equation, J Sci Comput 27 (2006), 5–40.

3. C. Bernardi and E. Süli, Time and space adaptivity for the second-order wave equation, Math ModelsMethods Appl Sci 15 (2005), 199–225.

4. G. C. Cohen, Higher–Order numerical methods for transient wave equations, Springer-Verlag, Berlin,2002.


38 BRADJI

5. B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time dependent problems and difference methods, Pure andApplied Mathematics, A Wiley-Interscience Series of Texts, Monographs and Tracts, Wiley, New York,1995.

6. S. Kumar, N. Nataraj, and A. K. Pani, Finite volume element method for second order hyperbolicequations, Int J Numer Anal Model 5 (2008), 132–151.

7. T. Lähivaara and T. Huttunen, A non-uniform basis order for the discontinuous Galerkin method of theacoustic and elastic wave equations, Appl Numer Math 61 (2011), 473–486.

8. M. Lukácová-Medvid’ováa, G. Warneckeb, and Y. Zahaykahb, Finite volume evolution Galerkin(FVEG) methods for three-dimensional wave equation system, Appl Numer Math 57 (2007), 1050–1064.

9. S. Petersen, C. Farhat, and R. Tezaur, A space-time discontinuous Galerkin method for the solution ofthe wave equation in the time domain, Int J Numer Methods Eng 78 (2009), 275–295.

10. M. Picasso, Numerical study of an anisotropic error estimator in the L2(H 1)–norm for the finite elementdiscretization of the wave equation, SIAM J Sci Comput 32 (2010), 2213–2234.

11. R. D. Richtmyer and K. W. Morton, Difference methods for initial value problems, Reprint of the 2nded., 1967, Krieger Publishing Company, Melbourne, FL, 1994.

12. M. Yang, Two-time level ADI finite volume method for a class of second–order hyperbolic problems,Appl Math Comput 215 (2010), 3239–3248.

13. E. Zampieri and L. F. Pavarino, Approximation of acoustic waves by explicit Newmark’s schemes andspectral element methods, J Comput Appl Math 185 (2006), 308–325.

14. E. Zampieri and A. Tagliani, Numerical approximation of elastic waves equations by implicit spectralmethods, Comput Methods Appl Mech Engrg 144 (1997), 33–50.

15. R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, P. G. Ciarlet and J. L. Lions, editors,Elsevier, North–Holland, Amsterdam, 2000, pp. 723–1020.

16. H. Si, K. Gärtner, and J. Fuhrmann, Boundary conforming Delaunay mesh generation, Comput MathMath Phys 50 (2010), 38–53.

17. R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion prob-lems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces,IMA J Numer Anal 30 (2010), 1009–1043.

18. R. Eymard, T. Gallouët, and R. Herbin, Benchmark on anisotropic problems, SUSHI: a scheme usingstabilization and hybrid interfaces for anisotropic heterogeneous diffusion problems, R. Eymard andJ.-M. Hérard, editors, Finite Volumes for Complex Applications V, Proceedings of the 5th Inter-national Symposium on Finite Volume for Complex Applications, Wiley, London, 2008, pp. 801–814.

19. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American MathematicalSociety, Vol. 19, Providence, RI, 1998.

20. M. Feistauer, J. Felcman, and I. Straskraba, Mathematical and computational methods for compressibleflow, Oxford Science Publications, Oxford University Press, Oxford, 2004.

21. P. A. Raviart and J. M. Thomas, Introduction à l’analyse numérique des equations aux dérivées partielles,Masson, Paris, 1983.

22. S. Godounov and V. Riabenki, Schémas aux différences, Editions Mir, Moscow, (French), 1977.

23. A. Bradji, Convergence analysis of some high–order time accurate schemes for a finite volume methodfor second order hyperbolic equations on general nonconforming multidimensional spatial meshes,Numer Methods Partial Differential Eq, to appear.

24. T. Gallouët, R. Herbin, and M. H. Vignal, Error estimates for the approximate finite volume solutionof convection diffusion equations with general boundary conditions, SIAM J Numer Anal 37 (2000),1935–1972.



25. R. Eymard, T. Gallouët, and R. Herbin, A cell–centred finite–volume approximation for anisotropicdiffusion operators on unstructured meshes in any space dimension, IMA J Num Anal 26 (2006),326–353.

26. A. Bradji, An approach to improve the convergence order in finite volume and finite element meth-ods, Numerical analysis and applied mathematics. International conference on numerical analysis andapplied mathematics, Rethymno, Crete, Greece, September 18–22, 2009, AIP Conference Proceedings,1168, Theodore E. Simos, George Psihoyios, and Ch. Tsitouras, editors, Vol. 2, American Institute ofPhysics, Melville, 2009, pp. 1162–1165.


21697_ftp

Documents