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Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes
Boris Andreianov, Franck Boyer, Florence Hubert
To cite this version:
Boris Andreianov, Franck Boyer, Florence Hubert. Discrete duality finite volume schemes for Leray- Lions type elliptic problems on general 2D meshes. Numerical Methods for Partial Differential Equa- tions, Wiley, 2007, 23, pp 145-195. �10.1002/num.20170�. �hal-00005779v2�
DISCRETE DUALITY FINITE VOLUME SCHEMES FOR LERAY-LIONS TYPE ELLIPTIC PROBLEMS ON GENERAL 2D MESHES
BORIS ANDREIANOV∗, FRANCK BOYER†AND FLORENCE HUBERT†
Abstract. Discrete duality finite volume schemes on general meshes, introduced by Hermeline in [22] and Domelevo &
Omn`es in [11] for the Laplace equation, are proposed for nonlinear diffusion problems with non homogeneous Dirichlet boundary condition.
This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of non-conformal locally refined meshes.
We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W−1,p0(Ω) source term and W1−1p,p(∂Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient respectively in appropriate Lebesgue spaces.
Finally, error estimates are given in the case where the solution is assumed to be inW2,p(Ω). Numerical examples are given, including those on locally refined meshes.
Keywords. Finite-volume methods, Error estimates, Leray-Lions operators.
AMS Subject Classification. 35J65 - 65N15 - 74S10 1. Introduction.
1.1. Nonlinear elliptic equations. In this paper, we are interested in the study of a finite volume approximation of solutions to the nonlinear diffusion problem with non homogeneous Dirichlet boundary conditions:
(−div ϕ(z,∇ue(z))
=f(z), in Ω,
ue=g, on∂Ω, (1.1)
where Ω is a given bounded polygonal domain inR2.
We first recall the usual functional framework ensuring that the problem above is well-posed. Let p∈]1,∞[ andp0 = p−1p . The fluxϕ: Ω×R2 →R2 in equation (1.1) is supposed to be a Caratheodory function which is strictly monotonic with respect toξ∈R2:
(ϕ(z, ξ)−ϕ(z, η), ξ−η)>0,for allξ6=η, fora.e. z∈Ω. (H1) We also assume that there existC1, C2>0,b1∈L1(Ω), b2∈Lp0(Ω) such that
(ϕ(z, ξ), ξ)≥C1|ξ|p−b1(z),for allξ∈R2, a.e. z∈Ω, (H2)
|ϕ(z, ξ)| ≤C2|ξ|p−1+b2(z),for allξ∈R2, a.e. z∈Ω. (H3) These assumptions ensure thatu7→ −div (ϕ(·,∇u)) is a Leray-Lions operator, and in particular
the mapG∈(Lp(Ω))d7→ϕ(·, G(·))∈(Lp0(Ω))d is continuous. (1.2) Theorem 1.1. Under assumptions (H1),(H2),(H3), for anyf ∈W−1,p0(Ω)andg∈W1−1p,p(∂Ω)the problem (1.1)has a unique solutionue∈W1,p(Ω).
∗Laboratoire de Math´ematiques, Universit´e de Franche-Comt´e, 16 route de Gray, 25030 Besan¸con Cedex, France
†Laboratoire d’Analyse, Topologie et Probabilit´es, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France 1
Remark 1.1. Note that, in view of the numerical approximation of the source term, it is necessary to suppose thatf ∈Lp0(Ω). This is not a restriction of our approach. Indeed, in order to treat the case of general source termf ∈W−1,p0(Ω), it is possible to writef =f0+ divf1 with f0∈Lp0(Ω) andf1 ∈
Lp0(Ω)2
, so that the problem (1.1)is equivalent to
−div ( ˜ϕ(z,∇ue)) =f0(z),
whereϕ˜: (z, ξ)→ϕ(z, ξ) +f1(z). It is easily seen that, ifϕsatisfies (H1)-(H3), so does the new fluxϕ. It is˜ worth noticing that the couple(f0, f1) is not unique and each choice will lead to a different approximation of the original equation (see [13]). As a consequence, from now on we always assume at least thatf ∈Lp0(Ω).
1.2. Examples. Our framework includes classical elliptic operators like the linear anisotropic Laplace equation
−div (A(z)∇ue) =f, (1.3)
A(z) being a uniformly coercive matrix-valued map, or the p-laplacian
−div (|∇ue|p−2∇ue) =f. (1.4)
One can also encounter, for instance in the modelling of non-newtonian fluids flows in a porous medium, equations like
−div
k(z)|F(z) +∇ue|p−2(F(z) +∇ue)
=f, (1.5)
where F is a vector-valued map and k a positive scalar map bounded from below. Notice that F is not necessarily a gradient, so that this problem may not reduce to the p-laplacian (1.4) through a change of variables. The models presented in [10] are even more general, since the fluxϕdepends also on the unknown ue. In [20], the authors also propose such nonlinear elliptic problem for the study of glacier flows. In this case, the fluxϕdepends only on ξ but in an implicit way.
We recall the key technical lemma which imply the monotonicity and continuity properties of the two non-linear model problems above (see [5]).
Lemma 1.2. For anyp∈]1,+∞[andδ≥0, there existsC, C >0such that for any n∈Nwe have (|ξ|p−2ξ− |η|p−2η, ξ−η)≥C|ξ−η|2+δ(|ξ|+|η|)p−2−δ, ∀ξ, η∈Rn,
|ξ|p−2ξ− |η|p−2η≤C|ξ−η|1−δ(|ξ|+|η|)p−2+δ, ∀ξ, η∈Rn.
1.3. Sobolev spaces on the boundary of polygonal domains. We need to recall briefly the defini- tions and main properties of the Sobolev spaces defined on∂Ω and related trace theorems. A complete study of these topics can be found, for instance, in [21].
Definition 1.3. Letα∈]0,1[, andp∈[1,+∞[. We defineWα,p(∂Ω), to be the space of functionsg∈Lp(∂Ω) such that
kgkpWα,p(∂Ω)
def=kgkpLp(∂Ω)+ Z
∂Ω
Z
∂Ω
g(x)−g(y)
|x−y|α
p dλ(x)dλ(y)
|x−y| <∞, wheredλ is the natural length measure which can be defined on the boundary∂Ω(see [25]).
We recall that the trace operatorγis continuous fromW1,p(Ω) ontoW1−p1,p(∂Ω) and that there exists a linear continuous lift operatorR:W1−1p,p(∂Ω)→W1,p(Ω) such that
γ◦R= Id∂Ω, kR(g)kW1,p ≤CkgkW1−1p,p
(∂Ω), (1.6)
2
whereCdepends only on Ω andp. For anyg∈W1−1p(∂Ω) we denote byWg1,p(Ω) the closed subset ofW1,p(Ω) of all functions whose trace on∂Ω is equal tog.
Let us denote by Γ1, . . . ,Γk the sides of Ω. Since each Γi is a segment we can define naturally the spaces W1+α,p(Γi) by
h∈W1+α,p(Γi)⇔h∈W1,p(Γi), and Z
Γi
Z
Γi
∇Th(x)− ∇Th(y)
|x−y|α
p dλ(x)dλ(y)
|x−y| <∞,
where ∇Thstands for the derivative ofhin the direction of∂Ω that we call “tangential gradient”. Since∂Ω is not smooth enough, it is not possible to define the spaceW1+α,p(∂Ω) but we can introduce the following space
Wf1+α,p(∂Ω) =
g∈W1,p(∂Ω), g|Γi ∈W1+α,p(Γi) ,
endowed with its natural norm. We recall that the trace operatorγ is continuous fromW2,p(Ω) onto a finite codimensional subset of Wf2−1p,p(∂Ω) which can be described precisely (we do not need this description here and we refer to [21] for further developments on this topic).
We also recall that for any p >2 the embedding ofW1−1p,p(∂Ω) in the H¨older space C0,1−2p(∂Ω) holds true and that we have the following sharp estimate.
Lemma 1.4. Let p >2, there exists a constant C depending only on psuch that, for any connected subset σ of ∂Ωand any g∈W1−1p,p(∂Ω)we have
|g(z)−g(z0)| ≤C|z−z0|1−2p Z
σ
Z
σ
g(x)−g(y)
|x−y|1−1p
p dλ(x)dλ(y)
|x−y|
!1p
, ∀z, z0∈σ. (1.7)
Proof. The embedding of W1−1p,p(∂Ω) in the H¨older space C0,1−2p(∂Ω) is given by the Morrey theorem.
In particular, there exists C > 0 such that (1.7) holds for the unit segment σ =]0,1[⊂ R and any g ∈ W1−1p,p(]0,1[). It is now easy, using a linear change of the variables, to see that (1.7) holds with the same constantC for anyσ.
1.4. Finite volume approach. Finite elements approximation of problems like (1.1) are now quite classical (see for instance [5, 24, 7, 19]). Nevertheless, it is also natural to consider finite volume methods for these problems. Indeed, finite volume methods allow more flexibility on the geometry of the meshes and ensure the local consistency of the numerical fluxes inside the domain. Furthermore, this kind of dicretization is well-adapted if one adds a convective term in the problem (1.1).
The nonlinearity and the possible anisotropy of the fluxϕwith respect to∇uemakes it difficult to approximate the problem by standard cell-centered finite volume methods (as presented, e.g., in [16]), since only the normal component of ∇ueon the interface between two adjacent volumes can be easily approximated on conformal meshes.
In the case of linear anisotropic Laplace equation, some studies are available in the literature for instance in [8], [12], [18], with various approaches. To our knowledge, three kinds of gradient reconstruction were proposed for the finite volume approximation of the fully non-linear equation. The one of [4] applies for the p-laplacian on meshes that are dual to triangular ones, moreover, the triangular meshes should be close enough to the structured mesh. The one of [1] applies on rectangular meshes. Finally, for general grids it was recently proposed in [15] to handle fluxes on the edges of a control volume as new unknowns, and reconstruct the discrete gradient, constant per control volume, using these fluxes. In all cases, the crucial feature is that the summation-by-parts procedure permits to reconstitute, starting from the one-dimensional finite differences uK−uL, the whole two-dimensional discrete gradient (see [4, Lemma 8, Proposition 4] and [1, Proposition 2.5, Lemma 2.7]). The coercivity and monotonicity properties of the continuous elliptic operator are then inherited by its discrete finite volume counterparts. For instance, the variational structure of the p-laplacian operator can be inherited by its discrete analogues.
3
We consider in this paper the class of finite volume schemes introduced by Hermeline in [22], by Domelevo, Omn`es in [11] for the Laplace equation and in [9] for other linear equations like the Stokes or the Div-Curl system. More precisely, we show that the method can be successfully extended to the case of the nonlinear diffusion equation (1.1) we are interested in while preserving the main features of the continuous problem. In [9], these schemes are called “Discrete Duality Finite Volume” (DDFV for short) since the discrete gradient and discrete divergence operators are dual one from each other (see Lemma 4.1 below). The equation is approximated simultaneously on two interrelated meshes: the primal and the dual mesh. The number of variables and of equations doubles compared to usual cell-centered FV schemes, but the gradient approximation (the one already used by Coudi`ere and al. [8]) becomes simple and quite efficient. Furthermore, the method is well-suited to almost arbitrary meshes since very few geometrical constraints are imposed to the primal and dual control volumes. Indeed, non convex control volumes, non matching triangulations or locally refined meshes can naturally be handled by this method and also fulfill the assumptions needed for the convergence analysis given in this paper.
1.5. Outline. This paper is organized as follows. The framework of DDFV meshes, the discrete gradient and the finite volume scheme associated to equation (1.1) are described in Section 2. In Section 3, we present the main results of discrete functional analysis necessary for the theoretical study of the finite volume method.
These results include the discrete Poincar´e inequality, the study of the mean-value projection of functions on the mesh and finally a discrete compactness result similar to the Rellich theorem.
Existence and uniqueness of a discrete solution of the scheme as well asa priori estimates are given in section 4. The structure properties of Leray-Lions operators being inherited in the framework of DDFV schemes, the method we use is similar to the one for the continuous problem (1.1).
Section 5 is devoted to the proof of the convergence of the approximate solution in case of general data g ∈ W1−1p,p(∂Ω) and f ∈ W−1,p0(Ω) (see Remark 1.1). Notice that the strong convergence of the discrete gradient of the approximate solution towards the gradient∇ueof the exact solution is obtained in (Lq(Ω))2 for anyq < pin the general case and forq=punder a strong monotonicity assumption on the operator which is satisfied in many situations. In Section 6, we study the stability properties of the approximate solution with respect to the dataf and g and finally in Section 7, we prove error estimates for the discrete gradient in (Lp(Ω))2 in the case where the exact solution lies in W2,p(Ω), which is a usual assumption for the error analysis. The convergence rate obtained is size(T)p−11 forp≥2 and size(T)p−1 forp <2. These rates are the same than the one obtained in [24, 1] for different schemes. As an example, this result implies the first order convergence in the case of the anisotropic Laplace equation which Lipschitz coefficients.
Note that error estimates for general solutions of the p-laplacian equation with source term inLp0(Ω) were obtained in [2], making use of the intrinsic Besov regularity of continuous and discrete solutions, in the case of structured rectangular meshes. It is an open question how to generalize this Besov approach to the unstructured DDFV schemes.
In Section 8, we provide some numerical results which show in particular that, in the truly nonlinear case, the method behaves better than what is expected even for non conformal locally refined meshes. In the concluding Section 9, we discuss the extension of our study to some fully practical variants of the finite volume scheme and to even more general meshes than the ones described in Section 2.1.
2. The finite volume method.
2.1. Definition of the mesh. We call T a triple (M,M∗,D) of meshes on Ω, defined as follows. The meshMis a set of disjoint polygonal control volumesK⊂Ω such that∪K¯= Ω. We note∂Mthe set of edges of the control volumes inMincluded in ∂Ω, which we consider as degenerate control volumes. We associate to (M, ∂M) a family of points P. The set P = Pint∪Pext is composed of one point per control volume
K∈M(calledxK∈Pint) and one point per degenerate control volumeK∈∂M(called xK∈Pext):
Pint={xK,K∈M},Pext={xK,K∈∂M}.
LetP∗denote the set of vertices of the meshM. The setP∗can be decomposed intoP∗=P∗int∪P∗ext where P∗int∩∂Ω = ∅and P∗ext ⊂∂Ω (see Figure 2.1). The sets M∗ and ∂M∗ are two families of “dual” control
4
xL∗
meshM
meshD xK∗
meshM∗
K∗
L∗
K
xL
xK L
xK∈Pext,K∈∂M xK∗∈P∗ext,K∗∈∂M∗
∂M∗
Fig. 2.1.Example of a DDFV mesh
volumes defined as follows. To any point xK∗ ∈P∗int (resp. xK∗ ∈P∗ext) we associate the polygonK∗
∈M∗ (resp. K∗
∈∂M∗) whose vertices are{xK∈P/xK∗ ∈K¯,K∈M∪∂M}(resp. {xK∗} ∪ {xK∈P/xK∗ ∈K¯,K∈ M∪∂M}).
For all adjacent control volumesKandL, denote byσ=K|Lthe edge betweenKandLand byEthe set of such edges. The corresponding notations σ∗ =K∗|L∗ and E∗ refer to the dual meshM∗∪∂M∗. For each couple (σ, σ∗)∈ E × E∗ such that σ = K|L= (xK∗, xL∗) andσ∗ =K∗|L∗ = (xK, xL), we introduce the quadrilateral diamond cellDσ,σ∗whose diagonals areσandσ∗, as shown in Figure 2.2. Ifσ∈ E ∩∂Ω, then the quadrilateral¯ Dσ,σ∗ degenerate in a triangle. The set of diamond cells is denoted byD. Notice that we have
Ω = [
D∈D
D.
Let us point out the fact that the diamond cells can be non convex and that dual control volumes can be non convex and also may overlap in some very particular situations (see [11]).
2.2. Notations. For any control volumeK∈M, we define
• mK, the measure ofK;
• DK={Dσ,σ∗∈D/σ⊂∂K}.
• νK, the outward unit normal vector to∂K.
• dK, the diameter ofK.
5
For any degenerate control volumeK∈∂M,νKstands for the outward unit normal vector to∂Ω. In the same way, for a “dual” control volumeK∗∈M∗∩∂M∗, we set
• mK∗, the measure ofK∗;
• DK∗ ={Dσ,σ∗∈D/σ∗⊂∂K∗}.
• νK∗, the outward unit normal vector to∂K∗.
• dK∗, the diameter ofK∗.
xK
xL
xK
σ∗=K∗|L∗
σ=K|L
αD
τ xK∗
xL∗
xK∗
xL∗
ν τ∗ ν∗
σ∗=K∗|L∗
σ=K|L
xL
Fig. 2.2.Notations in a diamond cellDσ,σ∗
For a diamond cellDσ,σ∗, recall that (xK, xK∗, xL, xL∗) are the vertices ofDσ,σ∗ and note :
• mσ, the length ofσ,mσ∗ the length ofσ∗ andmD the measure of the diamond cell.
• τ,the unit vector parallel toσ, oriented fromxK∗ toxL∗.
• ν, the unit vector normal toσ, oriented fromxK toxL.
• τ∗, the unit vector parallel toσ∗, oriented fromxK toxL.
• ν∗, the unit vector normal toσ∗, oriented fromxK∗ toxL∗.
• αD, the angle between τ andτ∗.
• dD, the diameter ofDσ,σ∗.
2.3. Unknowns and boundary data. The finite volume method associates to all primal control volumes
K∈M, an unknown valueuK and to all dual control volumesK∗
∈M∗, an unknown valueuK∗. We denote the approximate solution on the meshT by
uT = (uK)K∈M,(uK∗)K∗∈M∗
. (2.1)
In this paper we deal with non-homogeneous Dirichlet boundary condition. We describe here the way the boundary data will enter the scheme. Note first that
∂Ω = [
K∈∂M
K= [
K∗∈∂M∗
K∗∩∂Ω .
For any boundary datag∈W1−1p,p(∂Ω) we introduce its discrete counterpart by defining for eachK∈∂M, a valuegK and for eachK∗∈∂M∗, a valuegK∗. The family ((gK),(gK∗)) is denoted bygT and is also identified to a piecewise constant function inLp(∂Ω) as follows
gT ∼ 1 2
X
K∈∂M
1KgK+1 2
X
K∗∈∂M∗
1K∗∩∂ΩgK∗,
where here and in the sequel, we denote by1E the characteristic function of any setE.
6
We will mainly consider the discrete boundary data denoted by PT
mg= ((gK),(gK∗)) defined by gK= 1
mσK
Z
σK
g(s)ds, gK∗ = 1 mσK∗
Z
σK∗
g(s)ds. (2.2)
Here
σK
def=B(xK, ρK)∩∂Ω, and σK∗
def=B(xK∗, ρK∗)∩∂Ω (2.3) andρKandρK∗ are positive numbers associated to the meshT and such that
σK⊂K, σK∗ ⊂∂K∗.
Finally, introduce numbersρK andρK∗ for any K∈M(resp. K∗∈M∗) such that BK
def=B(xK, ρK)⊂Ω, BK∗
def=B(xK∗, ρK∗)⊂Ω. (2.4) These balls are only introduced to prove the convergence of the scheme (but not to prove the error estimates).
In particular, they do not enter the definition of the scheme. Of course, some assumptions are needed on the radiiρKandρK∗ as stated in the next paragraph.
2.4. Regularity of meshes. We note size(T) the maximum of the diameters of the diamond cells inD. The following bounds follow:
mσ≤size(T), ∀σ∈ E; mσ∗ ≤size(T), ∀σ∗∈ E∗; mK≤πsize(T)2, ∀K∈M; mK∗ ≤πsize(T)2, ∀K∗∈M∗; mD ≤1
2size(T)2, ∀D ∈D.
We introduce now a positive number that quantifies the regularity of a given mesh and is useful to perform the convergence analysis of the finite volume schemes. We first define αT to be the unique real number in ]0,π2] such that
sinαT
def= min
D∈D|sinαD|, (2.5)
that is the minimal angle between the diagonals of the diamond cells in the mesh.
Let us introduce the number NT def= sup
x∈Ω
#
K∗ s.t. x∈K∗\∪BK∗ + sup
x∈Ω
#
Ks.t. x∈K\∪BK
+ sup
x∈Ω
#
Ds.t. x∈ \
D ∪BK, D ∈DK + sup
x∈Ω
#
Ds.t. x∈ \
D ∪BK∗, D ∈DK∗ , (2.6) whereEb denotes the convex hull of any set E⊂R2. In the usual case where the primal control volumes, the dual control volumes and the diamond cells are convex and whereBK⊂K,BK∗ ⊂K∗thenNT can be bounded by a function of the maximal number of edges per primal and dual control volumes. Since we do not impose any convexity assumption on the meshes, we need to control this numberNT in the convergence analysis. We can now introduce
reg(T)def= max 1
αT
,NT,max
D∈D
dD
√mD
,max
K∈M
dK
√mK
, max
K∗∈M∗
dK∗
√mK∗
,
K∈maxM∪∂M
dK
ρK
+ρK
dK
, max
K∗ ∈M∗∪∂M∗
xK∗ 6=corner
dK∗
ρK∗
+ρK∗
dK∗
,
K∗ ∈∂M∗max
xK∗=corner
ζp(ρK∗) dK∗
, max
K∈M D∈DK
dK
dD
,K∗∈M∗max
D∈D K∗
dK∗
dD
,
(2.7)
7
where, for anys≥0, we have
ζp(s) =s,ifp <2, ζp(s) =sp
0
2,ifp >2, ζ2(s) =s|logs|.
(2.8)
This special treatment of the corner vertices of the mesh is a purely technical assumption which is used in the convergence rate analysis (see Lemma 7.3).
Let us point out that reg(T) essentially measures:
• how flat the diamond cells are.
• how large is the difference between the size of a primal control volume (resp. a dual control volume) and the size of a diamond cell as soon as they intersect.
Convention : In any estimate given in this paper, the dependence of the constants in reg(T) is implicitly assumed to be non-decreasing.
Remark 2.1. For conformal finite volume meshes (see [16]), it is assumed that forαD =π2 for any diamond D ∈D, so thatαT =π2. In our case, not only this orthogonality condition is relaxed but also Mcan present atypical edges, non convex dual control volumes and non convex diamond cells (see Figure 2.3).
Atypical edge
αD6= π2
Non-convex dual control volume
Fig. 2.3.Non conformal meshes
Remark 2.2. The boundedness of reg(T)imposes only local restriction on the mesh. It is easy to construct a family of locally refined mesh such thatreg(T)is bounded independently on the level of the refinement. Figure 2.4 provides a very simple example of such a construction.
Finally, let us point out the fact that there existsC >0 depending only onpand reg(T) such that kPT
mgkLp(∂Ω)≤CkgkW1−1p,p(∂Ω), ∀g∈W1−1p,p(∂Ω), (2.9) and that, for any sequence of meshesTn such that reg(Tn) is bounded and size(Tn)→0 we have
PTn
m g−−−−→n→∞ g, in Lp(∂Ω), ∀g∈W1−1p,p(∂Ω). (2.10) 2.5. Discrete gradient. We consider the discrete gradient introduced by Coudi`ere and al. in [8] and applied by Hermeline [22] and Domelevo and Omn`es [11] in the framework of DDFV schemes described above.
8
D M
Fig. 2.4.Local refinement allowed by the boundedness ofreg(T)
For a given discrete Dirichlet datagT as defined above, the discrete gradient operator∇TgT can be defined as follows : for anyuT ∈RT,∇TgTuT is the function, constant on each diamond cellDσ,σ∗, given by
∇TgTuT = X
D∈D
∇DgTuT1D, with
∇DgTuT,τ
= uL∗−uK∗
mσ
∇DgTuT,τ∗
= uL−uK
mσ∗
, (2.11)
whereDσ,σ∗is notedDwhen no confusion can arise. The dependence ongT only appears whenDσ,σ∗intersects
∂Ω, in which case we replace the values ofuK or uK∗ bygK or gK∗, for the points xK or xK∗ located on the boundary. Remark that∇DgTuT can be expressed in the (ν,ν∗) basis in the following way :
∇DgTuT = 1 sinαD
uL−uK
mσ∗
ν+uL∗−uK∗
mσ
ν∗
(2.12) thanks to the following lemma.
Lemma 2.1. Using the notations of Section 2.2 and Figure 2.2 (in particular the orientation conventions), for any vector ξ∈R2 we have
(sinαD)ξ = (ξ,τ)ν∗+ (ξ,τ∗)ν.
In the framework of DDFV schemes on general meshes, this lemma is the crucial argument which ensures the coercivity and monotonicity properties of the finite volume approximates of Leray-Lions type operators;
compare to [4, Lemma 8] and [1, Proposition 2.5], which only work due to the particular geometry of meshes.
Remark 2.3. Our notation for the discrete gradient can be easily handled thanks to the following property:
for any discrete boundary data g1T andg2T and for any discrete functionsuT1 anduT2, we have
∇DgT1uT1 +∇Dg2TuT2 =∇DgT1+g2T (uT1 +uT2). (2.13)
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