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CONVERGENCE OF FINITE VOLUME APPROXIMATIONSFOR A NONLINEAR ELLIPTIC-PARABOLIC PROBLEM: A

“CONTINUOUS” APPROACH∗

BORIS A. ANDREIANOV† , MICHAEL GUTNIC‡ , AND PETRA WITTBOLD‡

SIAM J. NUMER. ANAL. c© 2004 Society for Industrial and Applied MathematicsVol. 42, No. 1, pp. 228–251

Abstract. We study the approximation by finite volume methods of the model parabolic-elliptic problem b(v)t = div (|Dv|p−2Dv) on (0, T ) × Ω ⊂ R × R

d with an initial condition and thehomogeneous Dirichlet boundary condition. Because of the nonlinearity in the elliptic term, a carefulchoice of the gradient approximation is needed. We prove the convergence of discrete solutions to thesolution of the continuous problem as the discretization step h tends to 0, under the main hypothesesthat the approximation of the operator div (|Dv|p−2Dv) provided by the finite volume scheme is stillmonotone and coercive, and that the gradient approximation is exact on the affine functions of x ∈ Ω.An example of such a scheme is given for a class of two-dimensional meshes dual to triangular meshes,in particular for structured rectangular and hexagonal meshes. The proof uses the rewriting of thediscrete problem under a “continuous” form. This permits us to directly apply the Alt–Luckhausvariational techniques which are known for the continuous case.

Key words. doubly nonlinear elliptic-parabolic equations, finite volume methods, convergenceof approximate solutions, continuous approach

AMS subject classifications. 35J60, 35K55, 35K65, 35M10, 65M12, 76M12

DOI. 10.1137/S00361429014000062

1. Introduction. Let Ω be an open bounded polygonal domain in Rd, d ≥ 1,

and T > 0. We consider the initial boundary value problem for a system of nonlinearelliptic-parabolic equations: b(v)t = div ap(Dv) on Q = (0, T )× Ω,

v = 0 on Σ = (0, T )× ∂Ω,b(v)(0, ·) = u0 on Ω,

(1.1)

where 1 < p < ∞ and div ap(Dv) = div (|Dv|p−2Dv) is the N -dimensional p-Laplacian, i.e.,

ap : ξ = (ξ1, . . . , ξN ) ∈ (Rd)N → |ξ|p−2ξ =

(∑i,j

|ξji |2)p/2−1

(ξ1, . . . , ξN ) ∈ (Rd)N .

We assume thatb : R

N → RN is continuous cyclically monotone; i.e.,

there exists a convex differentiable function Φ : RN → R s.t. b = ∇Φ,

(1.2)

normalized by b(0) = 0 and Φ(0) = 0. Moreover, we assume

u0 ∈ L1(Ω)N with Ψ(u0) ∈ L1(Ω),(1.3)

∗Received by the editors December 21, 2001; accepted for publication (in revised form) May 30,2003; published electronically January 28, 2004.

http://www.siam.org/journals/sinum/42-1/40006.html†LATP, CMI, Universite de Provence, Technopole de Chateau-Gombert, 39, rue Frederic Joliot-

Curie, 13453 Marseille Cedex 13, France ([email protected]).‡IRMA, Universite Louis Pasteur, 7, rue Rene Descartes, 67084 Strasbourg Cedex, France

([email protected], [email protected]).

228

FINITE VOLUME APPROXIMATIONS: A “CONTINUOUS” APPROACH 229

where Ψ is the Legendre transform of Φ given by

Ψ : z ∈ RN → sup

σ∈RN

∫ 1

0

(z − b(sσ))σ ds = supσ∈RN

(σz − Φ(σ)).

Equations of elliptic-parabolic type (1.1) arise as models of the flow of fluidsthrough porous media (cf., e.g., [6, 12]). They have already been studied extensivelyin the literature in the last decade from a theoretical point of view (cf., e.g., [1, 21,22, 12, 7, 26, 8, 10, 2]). Existence of weak solutions of general systems of elliptic-parabolic equations has been proved in [1], using Galerkin approximations and time-discretization. Similar results have been obtained later by other authors using differentmethods (e.g., using a semigroup approach as in [7, 8] in the case N = 1).

In particular, it is known that in the case of the system (1.1), for any u0 satisfying(1.3), there exists a weak solution of (1.1), where the weak solution is defined asfollows. Denote by E the Banach space Lp(0, T ;W 1,p

0 (Ω))N and by E′ its dual; E′ =Lp

′(0, T ;W−1,p′

(Ω))N , where p′ = p/(p − 1) is the conjugate exponent of p. Denoteby 〈·, ·〉E′,E the duality pairing between E′ and E.

Definition 1.1. A function v ∈ E is a weak solution of the problem (1.1) ifb(v) ∈ L∞(0, T ;L1(Ω))N and b(v)t ∈ D′(Q)N can be extended to a functional χ on Esatisfying

〈χ, φ〉E′,E +

∫∫Q

ap(Dv) ·Dφ = 0 for all φ ∈ E,(1.4)

〈χ, ξ〉E′,E = −∫∫

Q

b(v) ξt −∫

Ω

u0(·) ξ(0, ·) for all ξ ∈ E withξt ∈ L∞(Q)N , ξ(T, ·) = 0.

(1.5)

Note that if v is a weak solution of (1.1), then, by the “chain rule” lemma of [1],one has

B(v) ∈ L∞(0, T ;L1(Ω))N , where

B : z ∈ RN → b(z)z − Φ(z) ≡

∫ 1

0

(b(z)− b(sz))zds ≡ Ψ(b(z)) ∈ R.(1.6)

From the results of [26, 10] it also follows that, in the scalar case N = 1, there isuniqueness of a weak solution of (1.1). To our knowledge, the question of uniquenessis open in the case N ≥ 2.

In this paper we study the convergence of time-implicit approximations by finitevolume numerical schemes for the model nonlinear elliptic-parabolic problem (1.1).Finite volume methods are well suited for numerical simulation of processes whereextensive quantities are conserved, and they are popular methods among engineers inhydrology where equations of this type arise. Therefore justification of convergenceof this numerical approximation process is of particular interest. In [17] the finitevolume method has been studied and convergence of this approximation procedurehas been proved for problem (1.1) in the particular case p = 2, N = 1. The samemethod has also been studied for this equation (i.e., p = 2, N = 1) in the presenceof an additional convection term (cf. [18, 14]), and for a nonlinear diffusion problemin [16]. To our knowledge, in the case p = 2, only the convergence of finite elementmethods has been studied (cf. [19, 11, 5, 20] and their references).

230 B. A. ANDREIANOV, M. GUTNIC, AND P. WITTBOLD

Let us emphasize that our main object is not only to prove the convergence ofsome finite volume methods for (1.1), but also to develop a “continuous” approach forthis proof. The main idea of this adaptation is to rewrite the discrete finite volumescheme under an equivalent continuous form and to apply known stability techniquesfor the continuous equation (cf. [1] and [2, Chap. V] for the version we use) in orderto get convergence of discrete solutions to a solution of the continuous problem. The“continuous” approach and the convergence result have already been presented in [3].

In section 2, we describe the finite volume schemes and in particular the admissi-ble flux approximations we use. We show the existence and uniqueness of the solutionof a finite volume scheme and give some a priori estimates on discrete solutions. Thenwe state the convergence result. In section 3, we show in Proposition 3.3 that thesolution of a finite volume scheme, originally satisfying a discrete system of algebraicequations, also verifies a “continuous” formulation similar to (1.4), (1.5). This repre-sentation makes clear in which sense finite volume schemes approximate the ellipticoperator in (1.1); we prove that this approximation is consistent. In section 4 weprove the convergence theorem, passing to the limit in the “continuous” formulationof Proposition 3.3. In section 5, we analyze the two admissibility conditions imposedin section 2. For d = 2, we propose a scheme on meshes dual to triangular meshes thatenters into our framework; in particular, we have the convergence result on structuredrectangular and hexagonal meshes.

We consider the p-Laplacian as a prototype of a class of the so-called Leray–Lions-type operators; in [4], we discuss the extension of the techniques presented above toa particular case of the p-Laplacian operator with convection, studied in [12].

In order to simplify the notation, we restrict the exposition to the scalar equation(N = 1). The proofs of the auxiliary results used in section 4 can be found in [4].

2. The numerical method. In order to construct approximate solutions tothe problem (1.1), we will use the implicit discretization in time and a finite volumescheme in space.

2.1. Finite volume meshes, discrete gradients and finite volume sche-mes for the problem (1.1). Let Ω be an open bounded polygonal subset of R

d. Afinite volume mesh T of Ω is given by a family of open polygonal convex subsets ofΩ with positive measure, called “control volumes,” a family of subsets of Ω containedin hyperplanes of R

d, with positive (d−1)-measure (these are the interfaces betweencontrol volumes), and a family of points of Ω, one per control volume (these arethe “centers” of the volumes). For a volume K with center xK ∈ K, the interfacescontained in ∂Ω are considered as additional “boundary” volumes, unless xK ∈ ∂Ω.

For the sake of simplicity, we shall denote by T the family (K)K∈T of controlvolumes; (xK)K∈T denotes the family of their centers. The set of all volumes K suchthat xK ∈ ∂Ω is denoted by Text, and the set of all volumes K with xK ∈ Ω is denotedby Tint. The set of interfaces K|L such that K or L or both belong to Tint is denotedby E , and K|L denotes the interface between two neighbors K, L ∈ T . For all K|L, KL

denotes the “diamond” over K|L, i.e., the smallest convex set of Rd containing K|L, xK

and xL. Whenever we use K, K|L, or n to index objects and make summations, wemean that K ∈ T , K|L ∈ E , and n ∈ 1, . . . , [T/k]+1, where k is the time step of thescheme.

Following [15], we give the following definition.Definition 2.1. We say that T is a finite volume mesh of Ω if the following

hold:(2.1 i) The closure of the union of all control volumes is Ω.


(2.1 ii) For any (K, L) ∈ (T )2 with K = L, either the (d−1)-dimensional measure ofK∩L is 0 or K∩L = σ for some σ ∈ E (in which case we denote σ = K|L = L|K).

(2.1 iii) For any K ∈ T , there exists a subset EK of E such that ∂K = ∪σ∈EKσ.

Furthermore, E = ∪K∈T EK. We will denote by NK the set of volumes adjacentto K; i.e., NK = L ∈ T , K|L ∈ EK.

(2.1 iv) The family of points (xK)K is such that xK ∈ K for all K ∈ T , and it isassumed that the straight line joining xK and xL is orthogonal to K|L wheneverL ∈ NK.

Denote by m(K) and d(K) the d-dimensional measure and the diameter of K ∈ T ,respectively; and denote by m(K|L) the (d−1)-dimensional measure of K|L ∈ E . A meshT is characterized, in particular, by the following numbers:

size(T ) = maxK

d(K), ζ∗(T ) = minK

minσ∈EK

dist (xK , σ)

d(K),

M(T ) = maxK

card(EK), ζ∗(T ) =minK minσ∈EK

dist (xK , σ)

size(T ).

A finite volume method for (1.1) requires a family((T h, kh)

)h

of meshes and

corresponding time steps kh > 0 such that both the size of the mesh and the timestep go to zero. We will assume in our notation that the family is parametrized withh in some subset of (0, 1) whose closure contains zero, and size(T h) + kh ≤ h. Acouple (T h, kh) will be called a space-time grid.

In relation to a family((T h, kh)

)h, we define the numbers

M = suphM(T h) ∈ N, ζ∗ = inf

hζ∗(T h) ∈ R

+, and ζ∗ = infhζ∗(T h) ∈ R

+.(2.1)

Definition 2.2. We say that the family of meshes (T h)h is weakly proportional ifM <∞ and ζ∗ > 0. We say that the family of meshes (T h)h is strongly proportionalif, in addition, ζ∗ > 0.

Weak proportionality is standard (cf. [18]). Strong proportionality is a technicalassumption which ensures that (T h)h has the interpolation property (cf. sections 2.5and 5.2).

Given a grid (T h, kh), to each time-space volume QnK = I

n×K, In = (kh(n −

1), khn) one associates an unknown value vnK ∈ RN . In order to obtain a finite volume

scheme for (1.1), one “integrates” the equation in (1.1) over each grid volume QnK .

The time derivative in the left-hand side is approximated by the corresponding finitedifference. On the right-hand side, one uses the Green formula and then needs toreplace the flux on the lateral boundary of Qn

K by some function of the unknowns(vnK)K,n. For problem (1.1), this amounts to finding a substitution for Dv in theexpression

∫∫In×K|Lap(Dv) · νK,L (where νK,L is the unit normal vector to K|L pointing

from K into L). We will assume that this substitution is in Lp on each interfaceIn×K|L, typically constant in time and piecewise constant in space. We therefore

consider “discrete gradient” operators Dh of the form Dh : (vnK)K,n → (DnK|L)K|L,n,

DnK|L ∈ Lp(In×K|L) for all K|L, n.

(2.2)

It seems natural, though not necessary, to require that Dh be a linear operator.A finite volume scheme for (1.1) is defined by a grid (T h, kh) and a discrete

gradient Dh associated with the grid. Finally, a finite volume method for (1.1) is


given by a family((T h, kh,Dh)

)h

of grids and associated discrete gradient operators

Dh. In sections 2.3 and 2.5 we state the admissibility conditions for such methods.Now we are able to write the equations for a scheme (T h, kh,Dh):

m(K)b(vnK)− b(vn−1

K )

kh=

∑L∈NK

∫K|Lap(D

nK|L(x))dx · νK,L for all K ∈ T h

int,

for all n ∈ 1, . . . , [T/kh]+1.(2.3)

The homogeneous Dirichlet boundary condition is taken into account by assigning

vnK = 0 for all K ∈ T hext, for all n ∈ 1, . . . , [T/kh]+1.(2.4)

The initial condition is given by any values v0K ∈ b−1(u0K), where

u0K =1

m(K)

∫K

u0 for all K ∈ T hint.(2.5)

We denote by uh0 the piecewise constant initial function∑

Ku0K1lK , where 1lK is the

characteristic function of the set K. Other choices of u0K are possible, provided onehas uh0 → u0 a.e. on Ω and Ψ(uh0 ) → Ψ(u0) in L1(Ω) as h→ 0, where Ψ is defined inthe introduction. These properties hold for u0K given by (2.5), due to the convexityof Ψ.

We denote by (Sh) the system (2.3), (2.4), (2.5) corresponding to a given finitevolume scheme (T h, kh,Dh).

2.2. Memento on notation. In this section we collect the most used notationrelated to the finite volume schemes.

T : a finite volume mesh;

Text, Tint: the set of exterior,interior control volumes;

E : the set of interfaces between control volumes;

K,L: control volumes of T ;K|L : the interface between the two neighbors K and L;

EK : the set of all interfaces surrounding K;

NK : the set of all neighbors of K;

xK : the “center” of K;

dK,L: the distance between xK and xL, dK,L = |xK − xL|;dK,K|L: the distance between xK and K|L; one has dK,K|L + dL,K|L = dK,L;

νK,L: the unit normal vector to K|L pointing from K to L;

KL: the smallest convex set of Rd containing K|L, xK , and xL;

d(K),m(K): the diameter and the d-dimensional measure of K, respectively;

size(T ): the size of the mesh T , size(T ) = maxK d(K);

m(K|L): the (d−1)-dimensional measure of K|L;|R|: the (d+1)-dimensional measure of a set R ⊂ R

+ × Rd;

In: the time interval, I

n = ((n−1)k, nk);

QnK : the time-space grid element, Qn

K = In×K;

ΣnK : the lateral boundary of Qn

K , ΣnK = I

n×∂K;

Υς(K): the union of all control volumes of (T ) that are separated from K

by at most (ς − 1) other control volumes;


1lA: the characteristic function of a set A;

(T h, kh,Dh): a finite volume scheme (mesh, time step, discrete gradient);

(Sh): the corresponding system of equations (2.3),(2.4),(2.5);

h: the discretization parameter, h ≥ size(T h) + kh;

M, ζ∗: the weak proportionality bounds for (T h)h,

M = suph maxK card(NK), and ζ∗ = infh minKminL∈NK

dK,K|Ld(K) ;

ζ∗: the strong proportionality bound for (T h)h,

ζ∗ = infhminK,L∈NK

dK,K|Lsize(T ) ;

vnK : the unknown of the scheme (Sh) corresponding to the volume QnK ;

vh: a discrete solution for the scheme (T h, kh,Dh), vh =∑

K,n vnK1lQn

K;

uh0 : the discrete initial data, uh0 =∑

Ku0K1lK ;

DnK|L: the discrete gradient values on I

n×K|L, DnK|L ∈ Lp(In×K|L);

Dh: the discrete gradient operator, Dh : (vnK)K,n → (DnK|L)K|L,n;

Dn⊥,K|L: the value Dn

⊥,K|L=vnL−vnKdK,L

, featuring in the “discrete Lp(0, T ;W 1,p(Ω))

norm” of vh;Dh

⊥: the corresponding operator, Dh⊥ : (vnK)K,n → (Dn

⊥,K|L)K|L,n.

It is convenient to extend Dh (as well as Dh⊥) to an operator acting from E into Lp(Q).

Let Ph be the operator from Ω to⋃

K|L which projects x ∈ K on ∂K along the rayjoining xK to x. We define the appropriate lifting operator Lh and averaging operatorMh by

Lh[(Dn

K|L)K|L,n

](t, x) =

∑K|L,n

DnK|L(Ph(x)) 1l

In×KL(t, x),

Mh : η ∈ L1(Q) → Mh [η] = (ηnK)K,n ⊂ RN , ηnK =

1

|QnK |∫∫

QnK

η.

We will abusively write Dh for the operators Dh, Lh Dh, and Lh Dh Mh; and thesame for Dh

⊥.The following notations, specific to the “continuous” approach, are introduced in

sections 2.5 and 3.1.

uh: the continuous in t interpolation of b(vh), affine on each time interval In;

vh: an interpolated solution in E for vh (cf. Definition 2.8);

Gh: the interpolated gradient operator produced by (T h, kh,Dh) (cf. Defini-tion 3.2);

A: the elliptic operator in (1.1), A : η ∈ E → −div ap(Dη) ∈ E′;Ah: the finite volume approximation of A produced by the scheme

(T h, kh,Dh), given by Ah : η ∈ E → −div ap(Gh[η]) ∈ E′.

2.3. Admissible flux approximations. For simplicity, we consider only thegradients that yield fully implicit schemes; in this case Dh,Dh

⊥ act independently oneach set

(vnK

)K, and the dependence on n does not matter for their definition.

Let us introduce the operator Dh⊥, which appears naturally in the a priori esti-

mates of section 2.4:

Dh⊥ : (vK)K → (D⊥,K|L)K|L, D⊥,K|L =

vL − vKdK,L

∈ R.(2.6)


For ς ∈ N, denote by Υς(K) the union of all control volumes of T that are separatedfrom K by at most (ς − 1) other control volumes; for instance, Υ1(K) =

⋃L∈NK

L.The choice of ς corresponds to the choice of control volumes that are really involvedin the construction of Dh on ∂K.

Now we can make precise the assumptions on discrete gradient operators of theform (2.2).

Definition 2.3. Let (T h,Dh)h be a family of finite volume meshes and corre-sponding discrete gradient operators. The gradient approximation provided by Dh isadmissible if the following hold.

(2.3 i) Dh is linear and injective;(2.3 ii) Dh provides a strictly monotone scheme; i.e., for all (vK)K , (vK)K ⊂ (Rd)N

that do not coincide,

1

d

∑K|L

((vL−vK)− (vL−vK)

)∫K|L

(ap(DK|L(x))−ap(DK|L(x))

)dx ·νK,L > 0,

where (DK|L)K|L = Dh[(vK)K

], (DK|L)K|L = Dh

[(vK)K

];

(2.3 iii) Dh provides a scheme coercive at zero; i.e., there exists a constant C∗ > 0, in-dependent of h, such that for all (vK)K ⊂ (Rd)N and (DK|L)K|L = Dh

[(vK)K

],

one has

1

d

∑K|L

(vL − vK

)∫K|Lap(DK|L(x))dx · νK,L ≥ C∗

∥∥∥Dh⊥[(vK)K ]

∥∥∥pLp(Ω)

;

and there exists ς ∈ N, independent of h, such that the following hold.

(2.3 iv) For each h, Dh is consistent with affine functions. More exactly, assumethat, for K ∈ T h given, there exists an affine function w on Ω such that vL =

1m(L)

∫Lw whenever L ⊂ Υς(K). Then DK|L(x) = Dw = const for all x ∈

K|L for all L ∈ NK .

(2.3 v) There exists a constant C∗, independent of h, such that, for all K ∈ T h andall sets of values (vK)K of R

N ,∫K

∣∣∣Dh[(vK)K ]∣∣∣p ≤ C∗

∫Υς(K)

∣∣∣Dh⊥[(vK)K ]

∣∣∣p.Conditions (2.3 ii) and (2.3 iv) imply strong restrictions on the gradient approxi-

mation. We provide some examples of methods with admissible gradient approxima-tion in section 5.1.

2.4. Discrete solutions. Recall that we consider as unknowns the values vnKon K ∈ Tint, assigning vnK to be zero in K ∈ Text. We will repeatedly use the following“summation by parts” formula (cf., e.g., [15]).

Remark 2.4. Let T be a finite volume mesh of Ω in the sense of Definition 2.1.Let (vK)K∈T ⊂ R

N , (FK,L)(K,L)∈T 2 ⊂ RN . Assume vK = 0 for all K ∈ Text and

FK,L = −FL,K for all K|L ∈ E . Then∑K

vK∑

L∈NK

FK,L =∑K|L

(vK − vL)FK,L.


If (vnK)K,n verifies (Sh), we say that the function vh =∑

K,n vnK1lQn

Kis the corre-

sponding discrete solution. We prove the discrete version of the Lp(0, T ;W 1,p0 (Ω))- a

priori estimate on vh (which is exactly the estimate on Dh⊥[vh] in Lp), and the discrete

version of (1.6).Proposition 2.5. Let

((T h, kh)

)h

be a family of finite volume grids and let(Dh)h

be a family of corresponding discrete gradient operators satisfying property

(2.3 iii) of Definition 2.3. Then, for any solution vh of the discrete problem (Sh),there exists a constant C which depends only on p, d,Ω, T , on C∗ in (2.3 iii), and on‖Ψ(u0)‖L1(Ω) such that

(i)∥∥∥Dh

⊥[vh]∥∥∥pLp(Q)

=1

d

∑K|L,n

m(K|L)dK,L

∣∣∣vnL − vnKdK,L

∣∣∣p ≤ C;

(ii)∥∥∥B(vh)

∥∥∥L∞(0,T ;L1(Ω))

= supn∈1,...,[T/kh]+1

∑K

m(K)B(vnK) ≤ C.

Proof. Take i ∈ 1, . . . , [T/kh]+1 and multiply each term in (2.3) by viK . By(2.3 iii), using Remark 2.4 and (2.4), one gets∑

K

m(K)(b(viK)− b(vi−1K )) viK + C∗khd

∫Ω

∣∣Dh⊥[vh]

∣∣p ≤ 0.

By the convexity of Φ, one has (b(viK) − b(vi−1K )) viK ≥ B(viK) − B(vi−1

K ). Summingover i from 1 to n ∈ 1, . . . , [T/kh]+1 and taking into account the convexity of Ψ,we infer∑

K

m(K)B(vnK) + C∗d∫ nkh

0

∫Ω

∣∣Dh⊥[vh]

∣∣p≤∑K

m(K)Ψ(u0K) =∑K

m(K)Ψ

(1

m(K)

∫K

u0)

≤∫

Ω

Ψ(u0).

Next, let us prove the discrete version of the Poincare inequality and of the com-pact embedding of W 1,p(Ω) in L1(Ω). Note that we do not need any proportionalityassumptions on the mesh.

Lemma 2.6. Let Ω ⊂ Rd be a polygonal domain of diameter d(Ω), and let T be a

finite volume mesh of Ω. Let vh =∑

KvK1lK such that (vK)K∈T ⊂ R and vK = 0 for

all K ∈ Text. Then there exists a constant C which depends only on p and d such that

(i) ‖vh‖Lp(Ω) ≤ C d(Ω)∥∥∥Dh

⊥[vh]∥∥∥Lp(Ω)

;

(ii) for all ∆ > 0, sup|∆x|≤∆

∫Rd

|vh(x+∆x)− vh(x)| dx ≤ ∆ ×∥∥∥Dh

⊥[vh]∥∥∥L1(Ω)

.

Proof. (i) For x ∈ Ω, set ψK|L(x) = 1 in the case that the orthogonal projection ofK|L on the hyperplane x1 = 0 contains (0, x2, . . . , xd), and set ψK|L(x) = 0 otherwise.One has

|vh(x)|p ≤ 1

2

∑K|L

ψK|L(x)∣∣∣|vL|p − |vK |p

∣∣∣≤ C

∑K|L

ψK|L(x) dK,L|vL − vK |dK,L

(|vK |p−1 + |vL|p−1

).


Since∫

ΩψK|L(x) dx ≤ m(K|L) d(Ω), one has by the Holder inequality∫

Ω

|vh(x)|p dx

≤ Cd(Ω)

(∑K|L

1

dm(K|L) dK,L

∣∣∣vL − vKdK,L

∣∣∣p) 1p(∑

K|L

1

dm(K|L) dK,L

(|vK |p + |vL|p)) p−1

p

.

Denote h = size(T ). Assertion (i) will follow by the Young inequality if we show that∑K|L

1

dm(K|L) dK,L

(|vK |p + |vL|p)

≤ (1 + 2p)∑K

m(K) |vK |p + 2(2h)p∑K|L

1

dm(K|L) dK,L


∣∣∣p,(2.7)

since h ≤ d(Ω). Denote by R the left-hand side of (2.7). We have dK,L= dK,K|L+dL,K|L;thus

R =∑K

m(K)|vK |p +∑K|L

1

dm(K|L)(|vK |pdL,K|L + |vL|pdK,K|L).

Note that

|vK |pdL,K|L ≤

2p|vL|pdL,K|L if |vK | ≤ 2|vL|,(2h)p

∣∣∣ vL−vKdK,L

∣∣∣pdK,L otherwise.

Indeed, if |vK | > 2|vL|, one has |vL − vK | > 12 |vK | so that

|vK |pdL,K|L ≤ |vK |pdK,L ≤ 2p|vL − vK |pdK,L ≤ 2php∣∣∣vL − vKdK,L

∣∣∣pdK,L.

Using the same argument for |vL|pdK,K|L, we obtain the desired estimate (2.7).

(ii) Now for x ∈ Rd, set ψK|L(x) = 1 in the case where the segment [x, x + ∆x]

crosses K|L, and set ψK|L(x) = 0 otherwise. Note that∫

RdψK|L(x) dx ≤ m(K|L)∆; hence(ii) follows, since∫

Rd

|vh(x)− vh(x+∆x)| dx ≤∫

Rd

∑K|L

ψK|L(x)|vL − vK | dx

≤ ∆∑K|L

m(K|L)dK,L


∣∣∣.Now we can state the result for existence and uniqueness of a discrete solution.

Theorem 2.7. Let T h be a finite volume mesh of Ω, kh > 0, and let Dh bea discrete gradient associated to T h. If Dh satisfies (2.3 iii), there exists a solution(vnK)K,n to the discrete problem (Sh). If Dh satisfies (2.3 ii), the solution is unique.

Proof of Theorem 2.7. Using Remark 2.4 and the coercivity of the scheme, weapply the Brouwer fixed point theorem and get existence. Uniqueness follows fromthe monotonicity of b(·) and the strict monotonicity of the scheme. See [3] for moredetailed proofs.


2.5. Interpolation property and main result. Consider a family of finitevolume schemes

((T h, kh,Dh)

)h

such that h tends to 0. Let (vnK)K,n be a solution to

the scheme (Sh) and vh the corresponding discrete solution.We require the existence of what will be called “interpolated solutions” for vh,

denoted by vh, such that vh ∈ E; these should be close to vh (asymptotically as h→ 0)and satisfy the a priori estimate in E analogous to the estimate of Proposition 2.5(i)on vh. Moreover, the values (vnK)K,n should be recoverable from vh. To this end, werequire Mh[vh] = (vnK)K,n.

Definition 2.8. A family of grids(T h, kh

)h

has the interpolation property in E

if, for any family (vh)h of functions such that vh|QnK

= vnK ≡ const for each K ∈ T h,

n ∈ 1, . . . , [T/kh]+1, with vnK = 0 for K ∈ T hext and with

∥∥Dh⊥[vh]

∥∥Lp(Q)

≤ C for all

h, there exists a family (vh)h ⊂ E such that

‖vh − vh‖Lp(Q) → 0 as h→ 0,(2.8)

Mh[vh] = (vnK)K,n,(2.9)

‖vh‖E ≤ I(C) with some function I : R+ → R

+ independent of h.(2.10)

If vh is a solution to a finite volume scheme, we say that vh is an interpolated solutionfor vh.

The interpolation property is the main technical assumption required by the “con-tinuous” approach. In section 5.2 we give two conditions ensuring this property. Nowlet us state the main result of this paper.

Theorem 2.9. Let((T hm , khm ,Dhm)

)m∈N

be a sequence of finite volume sche-

mes, where khm +size(T hm) ≤ hm → 0 as m→ ∞. Assume that the family of meshesis weakly proportional, the gradient approximation is admissible, and the interpolationproperty holds (cf. Definitions 2.2, 2.3, and 2.8).

For m ∈ N, let vhm be a discrete solution of (Shm). Then there exists a subse-quence (hml

)l∈N such that vhml v in Lp(Q) as l → ∞, where v is a weak solutionof the problem (1.1).

Note that it suffices to strengthen slightly assumption (2.3 ii) of Definition 2.3in order to get the strong convergence of vhml to v in Lp(Q) (cf. [4, Corollary 1]).Moreover, in the case when N = 1 the whole sequence converges to the unique solutionof (1.1). In this case error estimates can be proved (cf., e.g., [15] for the linear case),but this is not the purpose of the present paper.

In what follows, we write k instead of kh and omit subscripts in sequences (hm)and(hml

), simply writing that h tends to zero.

3. The “continuous” approach. Take the discrete solution vh =∑

K,n vnK1lQn

K

produced by the finite volume scheme (Sh). Let vh ∈ E be a corresponding interpo-lated solution. We will show that there exist functions uh ∈ L1(Q) and Gh ∈ Lp(Q)such that uh(0, ·) = uh0 (·) and uht = div ap(G

h) in the weak sense of Definition 1.1,and the functions uh, Gh can be recovered from the interpolated solution. More ex-actly, we prove in Proposition 3.3 below that uht ∈ D′ can be extended to χh ∈ E′

and

〈χh, φ〉E′,E +

∫∫Q

ap(Gh[vh]) ·Dφ = 0 for all φ ∈ E,(3.1)


〈χh, ξ〉E′,E = −∫∫

Q

uh ξt −∫

Ω

uh0 (·) ξ(0, ·) for all ξ ∈ E withξt ∈ L∞(Q)N , ξ(T, ·) = 0,

(3.2)

with an operator Gh : E → Lp(Q) to be defined.The analogy of (3.1), (3.2) with (1.4), (1.5) in Definition 1.1 plays the key role in

the proof of the convergence result of Theorem 2.9.

3.1. Interpolated gradient and the “continuous” form of the scheme.First define uh as the piecewise affine in t interpolation of b(vh):

uh =∑K,n

(b(vnK) +

t− knk

(b(vnK)− b(vn−1K ))

)1lQn

K.(3.3)

Then (3.2) holds, since uh(0, ·) = uh0 (·) and the piecewise constant function uht ex-tends to χh ∈ E′ by

〈χh, φ〉E′,E =

∫∫Q

uht φ for all φ ∈ E.(3.4)

Next, note that in (Sh) the numerical flux is prescribed on the boundary ofeach control volume; we will extend it to Q as follows. For given K ∈ T h

int,n and afunction Fn

K : ∂K → R, consider the following Neumann problem in the factor spaceW(K) =W 1,p(K)/R: div ap(Dw) =

1

m(K)

∑K∈NK

∫K|LFn

K on K,

ap(Dw) · νK |∂K = FnK ,

(3.5)

where νK is the exterior unit normal vector to ∂K. For K ∈ T hext with m(K) > 0, we

drop in (3.5) the Neumann boundary condition on ∂K ∩ ∂Ω and seek w ∈ W 1,p(K)with w|∂K∩∂Ω = 0.

Lemma 3.1. Let FnK ∈ Lp′

(∂K) (FnK ∈ Lp′

(∂K \∂Ω), if K ∈ T hext). Then (3.5)

admits a unique solution.The proof is standard, using the coercivity and monotonicity argument [25, Chap.

2, Th. 2.1]. Now we can introduce the interpolated gradient operator.Definition 3.2. The interpolated gradient operator Gh : E → Lp(Q) maps η ∈ E

into Gh[η] given by

Gh[η] =∑

K,nDηnK1lQn

K, where ηnK ∈ W(K) solves

−∫K

ap(DηnK) ·Dϕ+

∑L∈NK

∫K|Lϕap(D

nK|L) · νK =

1

m(K)

∫K

ϕ∑

L∈NK

∫K|Lap(D

nK|L) · νK

for all ϕ ∈W 1,p(K) (for all ϕ ∈W 1,p(K) with ϕ|∂K∩∂Ω = 0, in case K ∈ T hext)

and the values DnK|L(x) are given by (Dn

K|L)K|L,n = Dh[η].

If vh solves (Sh), we set Gh = Gh[vh]. We remark that Gh = Gh[vh] by property(2.9) of interpolated solutions vh. We show that (3.1) follows from (3.4) and theconservation of fluxes.

Proposition 3.3. Assume that (vnK)K,n is a solution of (Sh). Let vh be a corre-sponding interpolated solution, let uh and χh be defined by (3.3) and (3.4), respectively,


and let Gh be the interpolated gradient operator of Definition 3.2. Then (3.1), (3.2)hold.

Proof. It remains to check (3.1). By (3.3), for all K ∈ T h and n, we have

uht − div ap(Gh) =

b(vnK)− b(vn−1K )

k− 1

m(K)

∑L∈NK

∫K|Lap(D

nK|L(x))dx · νK,L = 0

everywhere on QnK because of (2.3). Therefore, using (3.4) and integrating by parts

in each QnK , we have

〈χh, φ〉E′,E+

∫Q

ap(Gh[vh]) ·Dφ∫∫

Q

uhtφ+ ap(Gh) ·Dφ

=∑K,n

∫∫Qn

K

(uht − div ap(Gh))φ +

∑K,n

∑L∈NK

∫∫In×K|L

φ ap(DnK|L) · νK,L

= 0 +∑K|L,n

∫∫In×K|L

φ ap(DnK|L) · (νK,L + νL,K) = 0.

3.2. Properties of the interpolated gradient and consistency. In view of(3.1) and (1.4), it is natural to compare the elliptic operator in (1.1),

A : η ∈ E → −div ap(Dη) ∈ E′,(3.6)

with the operators

Ah : η ∈ E → −div ap(Gh[η]) ∈ E′.(3.7)

Indeed, Ah can be considered as the finite volume approximation of A, whence thefollowing definition.

Definition 3.4. Let((T h, kh,Dh)

)h

be a family of finite volume schemes for

the problem (1.1), with size(T h) + kh ≤ h → 0. We say that the approximation of(1.1) by these schemes is consistent if, for all η ∈ E, one has Ah[η] → A[η] in E′ ash→ 0.

In this section we prove the following result.Theorem 3.5. Let

((T h, kh,Dh)

)h

be a family of finite volume schemes with aweakly proportional family of meshes and an admissible gradient approximation (cf.Definitions 2.2 and 2.3). Then it provides a consistent approximation of (1.1), in thesense of Definition 3.4.

The proof of Theorem 3.5 is based upon the following properties of the interpo-lated gradient operator Gh.

Proposition 3.6. Let((T h, kh,Dh)

)h

be a family of finite volume schemes withadmissible gradient approximation and weakly proportional family of meshes.

(i) There exists a constant C such that for all η ∈ E and H ⊂ Q such thatH =

⋃mi=1Q

niKi

, ∫∫H

∣∣∣Gh[η]∣∣∣p ≤ C

∫∫Υς+1(H)

|Dη|p,

where Υς+1(H) =⋃m

i=1 Ini ×Υς+1(Ki). In particular,

(Gh)h

are uniformlybounded on E and ∥∥Gh[η]

∥∥Lp(Q)

≤ C ‖η‖E .(3.8)


(ii) The operators (Gh)h are locally equicontinuous on E. More exactly, there ex-ists a constant C(R) such that, whenever ‖η‖E ≤ R and ‖µ‖E ≤ R,∥∥Gh[η]− Gh[µ]

∥∥Lp(Q)

≤ C(R)(‖η − µ‖E)minp−1, 1

p−1,(3.9) ∥∥ap(Gh[η])− ap(Gh[µ])∥∥Lp′ (Q)

≤ C(R)(‖η − µ‖E)min(p−1)2, 1

p−1.(3.10)

In the statement above and in the rest of this section, C denotes a generic constantthat depends only on p, d,Ω, onM, ζ∗ of (2.1), and on C∗, C∗, ς of Definition 2.3, unlessthe additional dependence on R is specified. The proof uses the standard propertiesof the function ap(·) (cf. [19, 12]): for all y1, y2 ∈ R

d,|ap(y1)− ap(y2)|p′ ≤ C |y1 − y2|p, 1 < p ≤ 2;

|ap(y1)− ap(y2)|p′ ≤ C |y1 − y2|p′(|y1|p + |y2|p

) p−2p−1

, p ≥ 2;(3.11)

|y1 − y2|p ≤ C[(ap(y1)− ap(y2)) · (y1 − y2)

]p2[|y1|p + |y2|p

]2−p2

, 1 < p ≤ 2;

|y1 − y2|p ≤ C (ap(y1)− ap(y2)) · (y1 − y2), p ≥ 2.(3.12)

Before turning to the proofs of Proposition 3.6 and Theorem 3.5, note the follow-ing three lemmas.

Lemma 3.7. Let K ⊂ Rd be a bounded convex domain of R

d of diameter d(K) andd-dimensional measure m(K). Assume that K contains a ball of radius ζ∗d(K) > 0.Then there exists a constant C such that, assigning w = 1

m(K)

∫Kw, one has∫

∂K

|w − w|p ≤ C (d(K))p−1

∫K

|Dw|p

for all w ∈W 1,p(K), where w|∂K is understood in the sense of traces.Proof. Applying, e.g., the proofs of [13, Theorems 59, 60, and 76] with p = 2

replaced by a general p ∈ (1,+∞), we obtain the claim of the lemma with C dependingon p, d, and the Lipschitz continuity of ∂K. Due to the convexity of K, C actuallydepends only on p, d, and ζ∗.

Lemma 3.8. Let (T h)h be a weakly proportional family of meshes, and let (Dh⊥)h

be the operators defined by (2.6). Then there exists a constant C such that for all K, nfor all η ∈ E, ∫∫

QnK

∣∣∣Dh⊥[η]

∣∣∣p ≤ C∫∫

In×Υ1(K)

|Dη|p.

Proof. Let(ηnK

)K,n

= Mh[η] and ηnK|L = 1km(K|L)

∫∫In×K|Lη in the sense of traces.

By definition,∫∫Qn

K

∣∣∣Dh⊥[η]

∣∣∣p =∑K|L

1

dkm(K|L) dK,K|L

∣∣∣∣ηnL − ηnKdK,L

∣∣∣∣p≤ C

∑K|L

1

dkm(K|L) dK,K|L

(∣∣ηnK|L − ηnK∣∣p

(dK,K|L)p +

∣∣ηnK|L − ηnL∣∣p

dK,K|L (dL,K|L)p−1

)≤ C

∑K|L

1

dkm(K|L) dK,K|L

∣∣∣∣ηnK|L − ηnKdK,K|L

∣∣∣∣p + C∑K|L

1

dkm(K|L) dL,K|L

∣∣∣∣ηnK|L − ηnLdL,K|L

∣∣∣∣p .


By the convexity of the function z → |z|p and Lemma 3.7,

km(K|L) |ηnK|L − ηnK |p ≤∫∫

In×K|L|η − ηnK |p ≤ C d(K)

p−1

∫∫Qn

K

|Dη|p,

and the same holds if K and L are exchanged. Hence by (2.1) we have∫∫Qn

K

∣∣∣Dh⊥[η]

∣∣∣p ≤ C∑

L∈NK

(∫∫Qn

K

|Dη|p +

∫∫Qn

L

|Dη|p)

≤ C∫∫

In×Υ1(K)

|Dη|p.

Lemma 3.9. Let((T h, kh,Dh)

)h

be a family of finite volume schemes with aweakly proportional family of meshes and an admissible gradient approximation. Thenthe following hold:

(i) For all R > 0 there exists a constant C(R) such that, whenever ‖η‖E ≤ Rand ‖µ‖E ≤ R,∑

K,n

d(K)

∫∫Σn

K

∣∣∣ap(Dh[η])− ap(Dh[µ])∣∣∣p′

≤ C(R)(‖η − µ‖E

)minp,p′.

(ii) There exists a constant C such that for all η ∈ E and H,Υς+1(H) as inProposition 3.6 one has

m∑i=1

d(Ki)

∫∫ΣniKi

∣∣∣Dh[η]∣∣∣p ≤ C

∫∫Υς+1(H)

|Dη|p.

Proof. First take K and consider ϕK =∣∣ap(Dh[η])−ap(Dh[µ])

∣∣p′∈ L1(∂K). Recall

that the values of Dh have been extended from ∂K inside K by means of the projectionoperator Ph (cf. section 2.2). Hence by (2.1) we have

d(K)

∫∂K

|ϕK | = d(K)

∑L∈NK

∫K|L

|ϕK |

= d∑

L∈NK

d(K)

dK,K|L

∫KL∩K

|ϕK Ph| ≤ d

ζ∗

∫K

|ϕK Ph|.(3.13)

If 1 < p ≤ 2, (3.13) and (3.11) yield∑K,n

d(K)

∫∫Σn

K

∣∣∣ap(Dh[η])− ap(Dh[µ])∣∣∣p′

≤ C∑K,n

∫∫Qn

K

∣∣∣Dh[η]−Dh[µ]∣∣∣p.

In turn, (2.3 i), (2.3 iv), Lemma 3.8, and (2.1) imply that∑K,n

∫∫Qn

K

∣∣∣Dh[η]−Dh[µ]∣∣∣p ≤ C

∑K,n

∫∫In×Υς(K)

∣∣∣Dh⊥[η − µ]

∣∣∣p≤ C

∑K,n

∫∫In×Υς+1(K)

∣∣∣D(η − µ)∣∣∣p ≤ C

∫∫Q

|Dη −Dµ|p = C(‖η − µ‖E

)p

,(3.14)

which was the claim of (i) for 1 < p ≤ 2. Furthermore, we remark that (3.14) alsoholds for p ≥ 2, in particular with η = 0 or µ = 0. Therefore for p ≥ 2, using (3.13),and then (3.11) and the Holder inequality, we get

∑K,n

d(K)

∫∫Σn

K

∣∣∣ap(Dh[η])−ap(Dh[µ])∣∣∣p′

≤ C(∑

K,n

∫∫Qn

K

∣∣∣Dh[η]−Dh[µ]∣∣∣p)

p′p

×(Rp

) p−2p−1

.


Thus, in the case p ≥ 2, (i) also follows from (3.14).The proof of (ii) is similar, using the identity |ap(y)|p′

= |y|p instead of inequalities(3.11).

Proof of Proposition 3.6. Recalling Definition 3.2, for all K ∈ Tint and n, denoteby ηnK (respectively, by µnK) a function in W 1,p(K) that solves (3.5) with Fn

K(x) =ap(D

nK|L(x))·νK for x ∈ K|L ∈ EK , where (Dn

K|L)K|L,n = Dh[η] (respectively, (DnK|L)K|L,n =

Dh[µ]). In other words, each of ηnK , µnK verifies the integral identity corresponding to

(3.5) with all test functions in W 1,p(K). Taking for the test function (ηnK −µnK),subtracting the two identities, and integrating in t ∈ I

n, we obtain∫∫Qn

K

(ap(Dη

nK)− ap(DµnK)

)·(DηnK −DµnK

)=

∫In

∫∂K

(ηnK−µnK − ηnK−µnK

) (ap(Dh[η])− ap(Dh[µ])

)· νK ,

(3.15)

where ηnK−µnK = 1m(K)

∫KηnK−µnK for a.a. t ∈ I

n. Summing over K, n, using the Holder

inequality and Lemmas 3.7 and 3.9(i), we have from (3.15)∫∫Q

(ap(Gh[η])− ap(Gh[µ])

)·(Gh[η]− Gh[µ]

)≤

∑K,n

∫In

∫∂K

d(K)−1

p′ | ηnK − µnK − ηnK − µnK |

× d(K)1p′∣∣∣ap(Dh[η])− ap(Dh[µ])

∣∣∣≤

(∑K,n

∫In

d(K)1−p

∫∂K

| ηnK − µnK − ηnK − µnK |p) 1

p

×(∑

K,n

d(K)

∫∫Σn

K

∣∣∣ap(Dh[η])− ap(Dh[µ])∣∣∣p′

) 1p′

≤(∑

K,n

∫∫Qn

K

|DηnK −DµnK |p) 1

p

‖η − µ‖minp/p′,1E

=∥∥∥Gh[η]− Gh[µ]

∥∥∥Lp(Q)

‖η − µ‖minp/p′,1E .

(3.16)

In the same manner, taking µ = 0 and using Lemma 3.9(ii), we get

∫∫H

∣∣∣Gh[η]∣∣∣pC (∫∫

H

∣∣∣Gh[η]∣∣∣p) 1

p

(∫∫Υς+1(H)

|Dη|p) 1

p′

,

which proves (i).Now if 1 < p ≤ 2, (3.12), (3.16), and the Holder inequality yield∥∥∥Gh[η]− Gh[µ]

∥∥∥pLp(Q)

≤ C(∥∥∥Gh[η]− Gh[µ]

∥∥∥Lp(Q)

‖η − µ‖p

p′E

) p2(∥∥∥Gh[η]

∥∥∥pLp(Q)

+∥∥∥Gh[µ]

∥∥∥pLp(Q)

) 2−p2

.

Using (3.8), we obtain (3.9). Now (3.10) follows by (3.11).If p ≥ 2, (3.12) and (3.16) readily yield (3.9); hence (3.10) follows by (3.11).


Proof of Theorem 3.5. We have to prove that∥∥ap(Dη)− ap(Gh[η])

∥∥Lp′ (Q)

→ 0 as

h→ 0.Let us first prove the theorem for the case of η ∈ E that is piecewise constant in

t and piecewise affine in x. Let J ⊂ Q be the set of discontinuities of Dη. Clearly, Jis of finite d-dimensional Hausdorff measure Hd(J).

For ς given in Definition 2.3, let us introduce Hh =⋃

K,n | In×Υς(K)∩J =ØQnK .

Note that |Hh| ≤ (ς + 1)hHd(J) → 0 as h → 0; likewise, |Υς+1(Hh)| → 0 as h → 0.

Therefore, by Proposition 3.6(i), we have∫∫Hh

∣∣∣ap(Dη)− ap(Gh[η])∣∣∣p′

≤ C(∫∫

Hh

|Dη|p +

∫∫Υς+1(Hh)

|Dη|p)

→ 0

as h → 0. Moreover, for all QnK such that Qn

K ∩ Hh = Ø, we have Gh[η] ≡ Dη onQn

K . Indeed, we have D[η] ≡ const on Υς+1(QnK). Therefore Dh[η]|Qn

K≡ Dη = const

by property (2.3 iv) of admissible gradient approximations. Hence Dw = Dη satisfiesthe boundary condition in (3.5); the equation is also satisfied, since div ap(Dη) ≡ 0on K and 1

m(K)

∫∂Kap(Dh[η]) · νK = ap(Dη) ·

∫∂KνK = 0.

It follows that∥∥ap(Dη)− ap(Gh[η])

∥∥Lp′ (Q)

→ 0 as h→ 0, which was our claim.

Now let us approximate an arbitrary function η in E by functions ηl that arepiecewise constant in t and piecewise affine in x. Note that we can always choosethis sequence ηl in E such that ηl → η in E and a.e. on Q as l → ∞, and |Dηl|p aredominated by an L1(Q) function independent of l. We have∥∥∥ap(Dη)− ap(Gh[η])

∥∥∥Lp′ (Q)

≤∥∥∥ap(Dη)− ap(Dηl)∥∥∥

Lp′ (Q)

+∥∥∥ap(Dηl)− ap(Gh[ηl])

∥∥∥Lp′ (Q)

+∥∥∥ap(Gh[ηl])− ap(Gh[η])

∥∥∥Lp′ (Q)

.(3.17)

As l → 0, the first term in the right-hand side of (3.17) converges to zero by theLebesgue dominated convergence theorem, independently of h. The second one con-verges to zero as h → 0 for all l fixed. Finally, by Proposition 3.6(ii), the third oneconverges to zero as l→ ∞ uniformly in h. Hence the result follows.

4. Proof of Theorem 2.9. In the context of continuous dependence upon thedata of weak solutions to “general” elliptic-parabolic problems (cf. [2, Chap.V]), theproof of convergence of weak solutions of approximating problems is based upon thethree essential arguments (A), (B), and (C) below.

(A) A priori estimates, using (1.2) and the Alt–Luckhaus chain rule lemma (cf.[1, 26, 10]).

(B) Strong compactness in the parabolic term, using a variant of the Kruzhkovlemma (cf. [23]):

Lemma 4.1 (cf. [4], [2, Chap. V]). Let Ω be an open domain in Rd, Q =

(0, T )×Ω, and let the families of functions (uh)h, (Fhα )h,α be bounded in L1(Q)

and satisfy ∂∂tu

h =∑

|α|≤mDαFh

α in D′(Q). Assume that uh can be extendedby zero outside Q, and one has

sup|∆x|≤∆

∫∫Rd+1

|uh(t, x+ ∆x)− uh(t, x)| dxdt ≤ ω(∆), with lim∆→0

ω(∆) = 0,(4.1)

where ω(·) does not depend on h. Then (uh)h is relatively compact in L1(Q).


(C) Convergence in the elliptic term, using a variant of the Minty–Browder argu-ment (cf., e.g., [25]).

Lemma 4.2 (cf. [4], [2, Chap. V]). Let E be a Banach space, E′ its dual and〈·, ·〉E′,E denote the duality product of elements of E′ and E. Let (vh)h ⊂ Eand vh v as h → 0. Let Ah be a sequence of monotone operators from Eto E′ such that Ah[vh]

∗ −χ for some χ ∈ E′. Assume that Ah converge

pointwise to some operator A, and A is hemicontinuous (i.e., continuous inthe weak-∗ topology of E′ along each direction). Assume that

lim infh→0

〈Ah[vh], vh〉E′,E ≤ 〈−χ, v〉E′,E .(4.2)

Then χ+A[v] = 0, and (4.2) necessarily holds with equality.Taking advantage of the “continuous” form (3.1), (3.2) of the discrete problem

(Sh), we can prove the convergence of finite volume approximate solutions in thesame way, using the discrete a priori estimates shown in Propositions 2.5 and 3.6(i),using next Lemma 4.1, and then using finally Lemma 4.2 together with the essentialconsistency result of Theorem 3.5.

Proof of Theorem 2.9. Let vh be the solution of (Sh). Let vh be a correspondinginterpolated solution, and let Ah be the finite volume approximate of the operator Ain (1.1) (cf. (3.6), (3.7)). Note that all the convergences we state below take place upto extraction of a subsequence.

(A) By Proposition 2.5(i),∥∥Dh

⊥[vh]∥∥Lp(Q)

≤ const uniformly in h so that the

family (vh)h is bounded in E, by (2.10). Hence there exists a function v ∈ E suchthat vh v in E as h→ 0. By (2.8), one also has vh v in Lp(Q).

(B) We claim that the family (uh)h given by (3.3) is relatively compact in L1(Q).Indeed, let us check the assumptions of Lemma 4.1. We have uht = div ap(Gh[vh]) in

D′(Q) by (3.1), (3.2), and the family(ap(Gh[vh])

)h

is bounded in Lp′(Q) by Propo-

sition 2.5(i), equation (2.10), and Proposition 3.6(i) (note that(Ah[vh]

)h

is thusbounded in E′). Furthermore, (3.3) yields

‖uh‖L1(Q) ≤ 2

∫∫Q

|b(vh)|+ kh∑K

m(K)|u0K |,

and one has |b(z)| ≤ δB(z) + sup|ζ|≤1/δ |b(ζ)| for all δ > 0 (cf., e.g., [1]). By Propo-

sition 2.5(ii) and since uh0 =∑

Km(K)|u0K | → u0 in L1(Ω) as h → 0, it follows that

(uh)h is bounded in L1(Q).Finally, by Proposition 2.5(i) and Lemma 2.6(ii), we obtain (4.1) with uh replaced

by vh. Hence the estimate (4.1) for uh follows by (3.3), as in the continuous case (cf.[1]); see [4] for the detailed proof.

Thus the claim of (B) follows, and there exists a function u ∈ L1(Q) such thatuh → u in L1(Q) and a.e. on Q. In addition, we claim that u = b(v), where v isthe weak limit of vh in E. It suffices to show that vh v in L1(Q) and b(vh) → uin L1(Q), and then apply the monotonicity argument of [9]; see [4] for the detailedproof.

(C) By (A), we have vh v in E. We claim that v is a weak solution of (1.1).By Proposition 3.3, χh + Ah[vh] = 0 in E′ and the initial condition (3.2) is

verified for all h. The family(Ah[vh]

)h

is bounded in E′ (cf. (B)), thus (χh)h isweak-∗ relatively compact in E′. By (3.4), (B), and Definition 1.1, we also have

χh = uht → b(v)t = χ in D′(Q). Hence Ah[vh] = −χh ∗ −χ in E′.


Moreover, passing to the limit in (3.2), using (B) and the convergence of uh0 tou0 in L1(Ω), we get (1.5). Consequently, by the chain rule argument [1, Lemma 1.5]we have

〈−χ, v〉E′,E = −∫

Ω

Ψ(b(v(T, ·))) +

∫Ω

Ψ(u0).(4.3)

On the other hand, by (3.4), (3.3), (2.9), and the monotonicity of b(·), we have

〈−χh, vh〉E′,E = −1

k

∑K,n

(b(vnK)− b(vn−1K ))

∫∫Qn

K

vh

= −∑K,n

m(K)(b(vnK)− b(vn−1K )) vnK

≤ −∑K

m(K)Ψ(b(v[T/kh]+1K )) +

∑K

m(K)Ψ(u0K)

= −∫

Ω

Ψ(b(vh(T, ·))) +

∫Ω

Ψ(uh0 ).

Recall that Ψ(uh0 ) → Ψ(u0) in L1(Ω). Without loss of generality, we can assume thatvh(T, ·) → v(T, ·) a.e. on Ω; hence by the Fatou lemma and (4.3) we get (4.2).

Next, the operators Ah are monotone. Indeed, take ϕ ∈ E and (ϕnK)K,n = Mh[ϕ].Arguing as in the proof of Proposition 3.3, integrating by parts in Qn

K , and cancellingthe boundary terms, we get

〈Ah[η], ϕ〉E′,E =

∫∫Q

ap(Gh[η]) ·Dϕ = −∑K,n

∫∫Qn

K

ϕ div ap(Gh[vh])

= −∑K,n

∫∫Qn

K

ϕ× 1

m(K)

∑L∈NK

∫K|Lap(D

nK|L(x))dx · νK,L

= − k∑K,n

ϕnK∑

L∈NK

∫K|Lap(D

nK|L(x))dx · νK,L.

(4.4)

Substituting (4.4) and applying Remark 2.4, we infer by property (2.3 ii) of Defini-tion 2.3 that

〈Ah[η]−Ah[η], η − η〉E′,E

=1

dk∑K|L,n

((ηnL−ηnK)− (ηnL−ηnK)

)∫K|L

(ap(D

nK|L(x))− ap(Dn

K|L(x)))dx · νK,L ≥ 0,

where (ηnK)K,n = Mh[η], (DnK|L)K|L,n = Dh[η], and the same for η.

Finally, by Theorem 3.5, Ah converge pointwise to the hemicontinuous operatorA · = −div ap(D ·). By Lemma 4.2 we conclude that χ + A[v] = 0 in E′. Thus (1.4)holds and v is a weak solution of (1.1).

5. Examples of admissible methods. By an admissible method, we mean amethod which provides an admissible gradient approximation and weakly proportionalmeshes satisfying the interpolation property. Recall that in this case, we have theconvergence of the finite volume approximation (cf. Theorem 2.9). In this section, weprove that such admissible methods exist.


5.1. On discrete gradients. In this section we construct an admissible gradientfor a family (T h)h of finite volume meshes of the Voronoı kind dual to a family (T h)hof triangular meshes.

Let us introduce some notation. We use O to denote a triangle of the mesh T h; forall O ∈ T h, there exist K, L,M ∈ T h such that O = ∆xKxLxM (the triangle with thecorners xK , xL, xM). The three interfaces K|L, L|M,M|K intersect at point x

O, which is

the center of the circumscribed circle of triangle O. We require it to be inside O. Let usdenote by S

O, SK,L, SL,M , and SM,K the surfaces of ∆xKxLxM ,∆xO

xKxL,∆xOxLxM ,

and ∆xOxMxK , respectively. One has S

O= SK,L+SL,M +SM,K .

Recall that νK,L = −−−→xKxL/dK,L, νL,M = −−−→xLxM/dL,M , νM,K = −−−→xMxK/dM,K . Note thefollowing elementary lemma.

Lemma 5.1. Let O = ∆xKxLxM be a triangle in R2, let x

Obe the center of its

circumscribed circle, and let xO∈ ∆xKxLxM . With the above notation, for all r in

R2, we have

r =2

SO

SK,L(r · νK,L)νK,L + SL,M(r · νL,M)νL,M + SM,K(r · νM,K)νM,K

.

This property can be generalized to any polygon in R2 which admits the circum-

scribed circle.Furthermore, for O ∈ T h such that O = ∆xKxLxM , let vh,0

O: R

2 → RN be the

affine function that takes the values vK , vL, vM at the points xK , xL, xM , respectively.The discrete gradient operator Dh,0 = Lh Dh,0 is defined by

Dh,0 : (vK)K →∑

O∈T h

Dvh,0O(x)1l

O(x).

In the case of structured hexagonal meshes, as well as that of structured rectangularones, the family (Dh,0)h is admissible (this will be proved in Proposition 5.2, as aparticular case). In general, this construction does not work. Indeed, if the points xK

are not the barycenters of K ∈ T h, property (2.3 iv) fails.This can be overcome, for instance, in the following way. For all K ∈ T h, let yK

be the barycenter of K and set σK = xK − yK . For O ∈ T h such that O = ∆xKxLxM ,let vh

O: R

2 → RN be the affine function that takes the values vK , vL, vM at the points

yK , yL, yM , respectively. The discrete gradient operator Dh = Lh Dh is defined by

Dh : (vK)K →∑

O∈T h

DvhO(x)1l

O(x);(5.1)

i.e., the affine interpolation over the triangle ∆yKyLyM is actually used in the triangle∆xKxLxM .

We will take advantage of considering Dh as a perturbation of Dh,0. For allO ∈ T h, let us define the correction operators

RO

: r ∈ R2 → 2

SO

SK,L

(r · σL − σK

dK,L

)νK,L + SL,M

(r · σM − σL

dL,M

)νL,M

+ SM,K

(r · σK − σM

dM,K

)νM,K

,


with the notation introduced above. We need to guarantee that the Euclidiean normof R

Ois less than minp− 1, 1/(p− 1) for all O ∈ T h.

Proposition 5.2. Assume that (T h)h is a family of meshes dual to a family of

meshes (T h)h such that all O ∈ T h are triangles with angles less than or equal to π/2.

Assume that for all h, for all O ∈ T h,

2

SO

SK,L

|σL − σK |dK,L

+ SL,M|σM − σL|dL,M

+ SM,K|σK − σM |dM,K

< minp− 1, 1/(p− 1),

where σK is the difference between the “center” xK of the volume K and its barycenter,etc.

Then the family of discrete gradient operators (Dh)h on (T h)h defined by (5.1) isadmissible in the sense of Definition 2.3.

Proof. Since, for any affine function w on K, one has 1m(K)

∫Kw(x)dx = w(yK),

where yK is the barycenter of K, property (2.3 iv) holds for Dh (with ς = 1, byconstruction). Next, (2.3 i) is clear.

Let us establish the relation between Dh,0 and Dh. Denote by D0

O, D

Othe values

on O of Dh,0[(vK)K ] and Dh[(vK)K ], respectively. Let us show that for all O ∈ T h,

D0

O= (I −R

O)D

O.(5.2)

Indeed, if O = ∆xKxLxM , one has

DO· νK,L =

vhO(xL)− vh

O(xK)

dK,L=

(vL +DO· σL)− (vK +D

O· σK)

dK,L

=vL − vKdK,L

+DO· σL − σK

dK,L= D0

O· νK,L +D

O· σL − σK

dK,L.

Writing the same relation for L,M and M,K, from Lemma 5.1, we get DO− D0

O=

ROD

O, whence (5.2) follows.

By Lemma 5.1 and the definition of Dh⊥ = Lh Dh

⊥ for all O ∈ T h such thatO = ∆xKxLxM we have∫

O

∣∣∣Dh,0[(vK)K ]∣∣∣p = S

O

∣∣∣Dh,0[(vK)K ]∣∣∣p

≤ C∗SK,L


∣∣∣p + SL,M

∣∣∣vM − vLdL,M

∣∣∣p + SM,K

∣∣∣vK − vMdM,K

∣∣∣p= C∗

∫O

∣∣∣Dh⊥[(vK)K ]

∣∣∣p(5.3)

with a constant C∗ that depends only on p. Since for given K ∈ T h and O ∈ T h wehave K ∩ O = Ø if and only if O ∈ Υ1(K), it follows that property (2.3 v) holds for thediscrete gradient Dh,0, with ς = 1. Now set θ

O= ‖R

O‖. We have θ

O< 1. One has

|DO| ≤ ‖(I −R

O)−1‖ |D0

O| ≤ 1

1−θO

|D0

O|; therefore (2.3 v) also holds for Dh.

Next, each term in the sum in (2.3 iii) splits into two terms corresponding to

the two parts of the interface K|L included in different triangles O1, O2 ∈ T h. Let uswrite down all the terms corresponding to the same triangle O ∈ T h, O = ∆xKxLxM ,


combine them using Lemma 5.1, and estimate using (5.2):

SK,L(ap(DO) · νK,L)(D

0

O· νK,L) + SL,M(ap(DO

) · νL,M)(D0

O· νL,M)

+ SM,K(ap(DO) · νM,K)(D0

O· νM,K) =

SO

2ap(DO

) ·D0

O

=S

O

2ap(DO

) · (I −RO)D

O≥ 1− θ

O

2S

O|D

O|p ≥ 1− θ

O

2(1 + θO)pS

O|D0

O|p.

Property (2.3 iii) for Dh follows, because one has |D0| ≥ |D0

O·νK,L| = |vL−vK |

dK,L= Dh

⊥,K|Lso that ∑

O∈T h

SO|D0

O|p =

∥∥∥Dh,0[(vK)K ]∥∥∥pLp(Ω)

≥∥∥∥Dh

⊥[(vK)]∥∥∥pLp(Ω)

.

The proof of (2.3 ii) is similar. Denoting the values D0

O, D

Oof Dh,0[(vK)K ] and

Dh[(vK)K ], respectively, on O, one can rewrite the sum in (2.3 ii) as∑O∈T h

SK,L

((ap(DO

)− ap(DO)) · νK,L

)((D0

O− D0

O) · νK,L

)+ SL,M

((ap(DO

)− ap(DO)) · νL,M

)((D0

O− D0

O) · νL,M

)+ SM,K

((ap(DO

)− ap(DO)) · νM,K

)((D0

O− D0

O) · νM,K

)=

1

2

∑O∈T h

SO(ap(DO

)− ap(DO)) · (D0

O− D0

O).

Using (5.2) and denoting by H the Hessian matrix of the function x ∈ R2 → 1

p |x|p,we get

(ap(DO)−ap(DO

)) · (D0

O−D0

O)

= (DO−D

O)t[∫ 1

0

H(DO+τ(D

O−D

O)) dτ (I−R

O)

](D

O−D

O).

For all x ∈ R2, x = 0, H(x) is a symmetric matrix with positive eigenvalues λ1, λ2

such that λ1/λ2 = p − 1. Thus the condition ‖RO‖ < minp − 1, 1/(p − 1) ensures

that, for all τ ∈ [0, 1],

rt[H(D

O+ τ(D

O− D

O)) (I −R

O)]r ≥ a rt

[H(D

O+ τ(D

O− D

O))]r > 0

for all r ∈ R2, r = 0, with some constant a > 0. Now (2.3 ii) follows.

5.2. On interpolated solutions. First note that it is sufficient to prove theinterpolation property in W 1,p

0 (Ω) if we require, in addition to the time-independentanalogs of (2.8), (2.9) (referred to as (2.8′), (2.9′)), that

(2.10 ′) ‖vh‖W 1,p0 (Ω) ≤ c×

∥∥∥Dh⊥[vh]

∥∥∥Lp(Ω)

with a constant c independent of h. We obtain the interpolation property in E withthe function I : C → c × C by taking vh constant on each I

n and summing inn ∈ 1, . . . , [T/kh]+1.


Lemma 5.3. Let (T h)h be a strongly proportional family of finite volume meshesof Ω ⊂ R

d. Then it has the interpolation property in W 1,p0 (Ω).

In order to prove the lemma, we first show that the strong proportionality al-lows us to majorate the Lp norm of the translates of the discrete solutions vh inLemma 2.6(ii) by const∆ (h + ∆)p−1. Then we convolute vh with the appropriatemollifier; finally, we restore the average over each mesh volume as in Lemma 5.4below. The complete proof is given in [4].

Note that the interpolation property can fail on weakly proportional meshes, atleast for p > 2.

Indeed, consider Ω = (0, 1)2. For s ≥ 2, let T s be the finite volume mesh of Ωsuch that T s

int = Ks, Ls, where Ks = (x, y) ∈ Ω | x+ y < 1/s with xKs = ( 1

4s ,14s ),

and Ls is the interior of the complementary of K

s with xLs = ( 12 ,

12 ). Take vs such that

vs ≡ s1/p on Ks and vs ≡ 0 on L

s. Then∫

Ω

∣∣Dh⊥[vs]

∣∣p ≤ const uniformly in s. If there

exist vs ∈W 1,p0 (Ω) interpolated solutions for vs, we have ‖vs‖W 1,p

0 (Ω) ≤ const. Hence

by the standard embedding theorem, vs are uniformly bounded. This contradicts thefact that 1

m(Ks)

∫Ksvs = s1/p → +∞ as s→ +∞.

Nevertheless, we have the following result in the situation close to that of Propo-sition 5.2.

Lemma 5.4. Assume that (T h)h is a weakly proportional family of meshes of

Ω ⊂ R2 dual to a family of meshes (T h)h such that all O ∈ T h are triangles with angles

less than or equal to π/2. Then (T h)h has the interpolation property in W 1,p0 (Ω).

Proof. Take discrete solutions vh =∑

KvK1lK on each of T h such that, for all h∥∥Dh

⊥[vh]∥∥Lp(Q)

≤ C. Denote by c the generic constant that depends only on p and ζ∗.

Let vh,0 be the continuous piecewise affine function on Ω that interpolates the valuesvK , vL, vM at the points xK , xL, xM over O for all O ∈ T h (we use the construction andnotation of section 5.1). We have vh,0 ∈ W 1,p

0 (Ω) and Dvh,0 ≡ Dh,0[vh] so that (5.3)yields ‖Dvh,0‖Lp(Ω) ≤ c× C. Note that for all x ∈ K ∈ T h,

|vh,0(x)− vh(x)| = |vh,0(x)− vh,0(xK)| ≤ d(K) |Dvh,0(x)|.(5.4)

Hence ‖vh,0 − vh‖Lp(Ω) ≤ h ‖Dvh,0‖Lp(Ω) → 0 as h → 0. Now take a continuouslydifferentiable function π : R

2 → R+ such that suppπ = x ∈ R

2 | |x| ≤ 1 and∫R2 π = 1. For all K ∈ T h

int set ϕK = m(K)(ζ∗d(K))2 π

(x−xK

ζ∗d(K)

)(for boundary volumes K

of nonzero measure; i.e., if xK ∈ ∂Ω, an easy modification is needed in order to keepthe trace on ∂Ω equal to zero). Set

vh = vh,0 +∑K

αKϕK , with αK = vK − 1

m(K)

∫K

vh,0.

Since suppϕK ⊂ K, by the choice of αK , the family (vh)h verifies (2.9′). Moreover,

since m(K)

d(K)2≤ c for all K ∈ T h, for all h, by the Holder inequality we get∫

Ω

|vh − vh,0|p =∑K

|αK |p∫K

|ϕK |p

≤ c∑K

1

m(K)p

∣∣∣∣∣∫K

vh −∫K

vh,0

∣∣∣∣∣p

m(K)

(m(K)

d(K)2

)p

≤ c∑K

1

m(K)pm(K)

p′/p∫K

|vh − vh,0|pm(K) = c

∫Ω

|vh − vh,0|p → 0


as h→ 0. Thus (vh)h satisfies (2.8′). In the same manner, using (5.4) we have∫Ω

|Dvh −Dvh,0|p =∑K

|αK |p∫K

|DϕK |p

≤∑K

c∑K

1

m(K)pm(K)

p′/p∫K

|vh − vh,0|p × m(K)

(m(K)

d(K)3

)p

≤ c∑K

1

d(K)p

∫K

|vh − vh,0|p ≤ c∑K

1

d(K)p d(K)

p

∫K

|Dvh,0|p = c

∫Ω

|Dvh,0|p.

Hence ‖vh‖W 1,p0 (Ω) ≤ c ‖vh,0‖W 1,p

0 (Ω) ≤ c × C, so (vh)h satisfies (2.10′). Thus (vh)h

can be chosen as interpolated solutions for (vh)h.

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