Discrete Poincaré inequalities for arbitrary meshes in the discrete duality finite volume context

HAL Id: cea-00726543

https://hal-cea.archives-ouvertes.fr/cea-00726543

Submitted on 30 Aug 2012

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

the discrete duality finite volume context

Anh Ha Le, Pascal Omnes

To cite this version:

Anh Ha Le, Pascal Omnes. Discrete Poincaré inequalities for arbitrary meshes in the discrete duality

finite volume context. Electronic Transactions on Numerical Analysis, Kent State University Library,

2013, 40, pp. 94–119. �cea-00726543�

DISCRETE DUALITY FINITE VOLUME CONTEXT

ANH HA LE^†‡ANDPASCAL OMNES^†‡

Abstract. We establish discrete Poincar´e type inequalities on a two-dimensional polygonal domain covered by arbitrary, possibly nonconforming meshes. On such meshes, discrete scalar fields are defined by their values both at the cell centers and vertices, while discrete gradients are associated with the edges of the mesh, like in the discrete duality finite volume scheme. We prove that the constants that appear in these inequalities depend only on the domain and on the angles in the diagonals of the diamond cells constructed by joining the two vertices of each mesh edge and the centers of the cells that share that edge.

Key words. Poincar´e inequalities, finite volumes, discrete duality, arbitrary meshes AMS subject classifications. 65N08, 46E35

1. Introduction. LetΩbe a two-dimensional polygonal domain. Let us introduce the following two Poincar´e inequalities which will be mentioned throughout this article: The Friedrichs (also called Poincar´e) inequality

Z

Ω

u²(x)dx≤cF

Z

Ω

|∇u(x)|²dx , ∀u∈H₀¹(Ω) (1.1)

and the Poincar´e (also called mean Poincar´e) inequality Z

Ω

u²(x)dx≤cP

Z

Ω

|∇u(x)|²dx , ∀u∈H¹(Ω)such that Z

Ω

u(x)dx= 0, (1.2)

wherecFandcPare constants depending only onΩ. These two inequalities play an important role in the theory of partial differential equations. Here,H¹(Ω)is the Sobolev space ofL²(Ω) functions with generalized derivatives in(L²(Ω))², andH₀¹(Ω) is the subspace ofH¹(Ω) with zero boundary values in the sense of traces on∂Ω. More details on the Sobolev spaces H¹(Ω),H₀¹(Ω)may be found, e.g., in [1].

This article considers discrete versions of Poincar´e inequalities for the so-called discrete duality finite volume (DDFV) method of discretization on arbitrary meshes, as presented, e.g., in [11]. Originally developed for the discretization of (possibly heterogeneous, anisotropic, nonlinear) diffusion equations on arbitrary meshes [3,6,11,15,16,20], this technique has found applications in other fields, like electromagnetics [17], div-curl problems [9] and Stokes flows [8,18,19], drift diffusion and energy transport models [4].

The originality of these schemes is that they work well on all kind of meshes, including very distorted, degenerating, or highly nonconforming meshes (see the numerical tests in [11]). The name DDFV comes from the fact that these schemes are based on the definition of discrete gradient and divergence operators which verify a discrete Green formula.

Details about this method are recalled in section2. In this introduction, let us only mention that in the DDFV discretization, scalar functions are discretized by their values both at the centers and at the vertices of a given mesh, and their gradients are evaluated on the so-called “diamond-cells” associated to the edges of the mesh. Each internal diamond-cell is a quadrilateral; its vertices are the two nodes of a given internal edge and the centers of the

†CEA, DEN, DM2S-SFME, F-91191 Gif-sur-Yvette Cedex, France. (pascal.omnes@cea.fr anh-ha.le@cea.fr)

‡Universit´e Paris 13, LAGA, CNRS UMR 7539, Institut Galil´ee, 99 Avenue J.-B. Cl´ement, F-93430 Villeta- neuse, France.

1

two cells which share this edge. Each boundary diamond cell is a degenerated quadrangle (i.e. a triangle); its vertices are the two nodes of a given boundary edge and the center of the corresponding cell and that of the boundary edge.

Then, the discrete version of theL²norm on the left-hand side of (1.1) and (1.2) is the half-sum of theL²norms of two piecewise constant functions, one defined with the discrete values given at the centers of the original (”primal” in what follows) cells, and the other defined with the discrete values given at the vertices of the primal mesh, to which we associate cells of a dual mesh. Moreover, the discrete version of the gradientL²norm on the right-hand side of (1.1) and (1.2) is theL²norm of the piecewise constant gradient vector field defined with it discrete values on the diamond-cells.

In the finite volume context, discrete Poincar´e-Friedrichs inequalities have previously been proved in [12, Lemma 9.1, Lemma 10.2] and [14], respectively for so-called ”admissi- ble” meshes (roughly speaking, meshes such that each edge is orthogonal to the segment join- ing the centers of the two cells sharing that edge, see the precise definition in [12, Definition 9.1]) and for Voronoi meshes. Similar results on duals of general simplicial triangulations are proved in [21]. In the DDFV context, a discrete version of (1.1) is given for arbitrary meshes in [3]. However, the discrete constantcF which appears in that paper depends on the mesh regularity in a rather intricate way, see [3, Formula (2.6) and Lemma 3.3].

The main result of our contribution is the proof of discrete versions of both (1.1) and (1.2) in the DDFV context, with constantscF andcP depending only on the domain and on the minimum angle in the diagonals of the diamond cells of the mesh.

Our proof of the discrete version of (1.1) is very similar to those given in [12] or [21].

We also prove a discrete version of (1.1) in a slightly more general situation when the domain is not simply connected and the discrete values of the function vanish only on the exterior boundary of the domain and are constant on each of the internal boundaries (this will have a subsequent application in the last section of the present work).

However, the task is more difficult for the mean-Poincar´e inequality. Like in [12], it is divided into three steps. The first is the proof of this inequality on a convex subdomain; in the second, our proof differs from that in [12] because we actually do not need to prove a bound on theL²norm of the difference of discrete functions and their discrete mean value on the boundary of a convex subset, but rather an easier bound on theL¹norm of this difference.

The final step consists in dividing a general polygonal domain into several convex polygonal subdomains and in combining the first two steps to obtain the result.

As a consequence of these results, we derive a discrete equivalent of the following result (which is a particular case of a result given in [13]): Let us consider open, bounded, simply connected, convex polygonal domains(Ωq)_q∈[0,Q]ofR²such thatΩq ⊂Ω0for allq∈[1, Q]

andΩ¯q1 ∩Ω¯q2 = ∅ for all (q1, q2) ∈ [1, Q]² withq1 6= q2. LetΩ be defined byΩ = Ω0\(∪^Q_q=1Ωq). Let us denote byΓ = ∂Ω = ∪^Q_q=0Γq, withΓq = ∂Ωq for allq ∈ [0, Q].

Then, there exists a constant C, depending only on Ω, such that for all vector field v in H(div,Ω)∩H(rot,Ω)withv·n= 0onΓand(v·τ,1)Γq = 0for allq ∈ [1, Q], there holds

||v||L²(Ω)≤C(||∇ ·v||L²(Ω)+||∇ ×v||L²(Ω)).

The discrete equivalent has applications in the derivation of a priori error estimates for the DDFV method applied to the Stokes equations ([10]).

Let us mention that, although 3D extensions of the DDFV scheme have been published [2,5,6], the extension of our results to 3D is beyond the scope of this article.

This article is organized as follows. Section2sets some notations and definitions related to the meshes, to discrete differential operators and to discrete functions. In section3, discrete

Ti Sk Pk

Gi

D j

FIG. 2.1. A nonconforming primal mesh and its associated dual mesh (left) and diamond-mesh (right).

Poincar´e inequalities are presented. First, we prove a discrete Poincar´e inequality for discrete functions vanishing on the boundary of the polygonal domain and then extend this result to the slightly more general case mentioned above. Then, we prove the discrete mean Poincar´e inequality with the 3 steps described above. Finally, we present in section4an application of the previous results to the derivation of another discrete inequality, relating the norm of dis- crete vector fields defined on the diamond cells and verifying special boundary conditions, to that of their divergence and curls defined on the primal and dual meshes. In the AppendixA, we present the details of the proof of a Lemma involved in our main results.

2. Notations and Definitions. The following notations are summarized in Fig.2.1and Fig.2.2. LetΩbe defined as above and be covered by a primal mesh with polygonal cells denoted byTi,i ∈ [1, I]. With eachTi, we associate a pointGi located in the interior of Ti. let us denote bySk, withk∈[1, K]the nodes of the cells. With anySk, we associate a dual cellPk by joining the pointsGi associated with the primal cells surroundingSk to the midpoints of the edges of whichSkis a node.

With any primal edgeAj withj ∈ [1, J], we associate a so-called diamond-cell Dj

obtained by joining the verticesS_k1(j)andS_k2(j)ofAj to the pointsG_i1(j)andG_i2(j)as- sociated with the primal cells that shareAj as a part of their boundaries. WhenAj is a boundary edge (there areJ^Γ such edges), the associated diamond-cell is a flat quadrilateral (i.e. a triangle) and we denote byGi2(j)the midpoint ofAj (thus, there areJ^Γsuch addi- tional pointsGi). The unit normal vector toAj isn_j and points fromGi1(j)toGi2(j). We denote byA^′_j1(resp.A^′_j2) the segment joiningGi1(j)(resp.Gi2(j)) and the midpoint ofAj. Its associated unit normal vector, pointing fromSk1(j) toSk2(j), is denoted byn^′

j1 (resp.

n^′

j2). We also define vectorsτ_j,τ^′_j1andτ^′_j2such that(n_j,τ_j),(n^′

j1,τ^′_j1)and(n^′

j2,τ^′_j2) are orthonormal, positively oriented basis ofR². In the case of a boundary diamond-cell,A^′_j2 reduces to{Gi2(j)}and does not play any role. Finally, for any diamond-cellDj, we shall denote byMiαkβ the midpoint of[Giα(j)Sk_β(j)], with(α, β) ∈ {1; 2}²,Mj the midpoint ofSk1(j)Sk2(j)andθj1 (respθj2) is defined by the angle, lower thanπ/2, between segment Sk1(j)Sk2(j)and segmentGi1(j)Mj(respGi2(j)Mj).

We shall use the following definition

DEFINITION2.1. We denote byθ^∗>0the greatest angle in the mesh such that θj1 ≥θ^∗andθj2 ≥θ^∗ for allj∈[1, J].

Now we shall associate discrete scalar values to the pointsGiandSk and discrete two- dimensional vector fields to the diamond-cells. This leads us to the following definitions.

i₁k₂

M

i₁k₁

M

k₂

S

k₁

i₂ k₁

M

i₂ i₂ k₂

M

A’j 2

Gi 1

S_k

1 n_j

i₂

G = M

S_k

2

A_j n_j

j₂

n’

Mj j₁

n’

A’j 1

j₁

Θ A_j

j₂

Θ

i₁

G

j₁

Θ A’j

1 n’j₁

S

G

j

FIG. 2.2. Notations for the inner diamond-cell (left) and a boundary diamond mesh (right).

DEFINITION2.2. Letφ= (φ^T_i , φ^P_k), andψ= (ψ_i^T, ψ^P_k)be inR^I×R^K. Letv= (v_j) andw= (w_j)be in R^J2

. We define the following scalar products and associated norms

(φ, ψ)T,P := 1 2



 X

i∈[1,I]

|Ti|φ^T_iψ_i^T+ X

k∈[1,K]

|Pk|φ^P_kψ_k^P



,

kφk²_T,P := (φ, φ)T,P, (w,v)D:= X

j∈[1,J]

|Dj|w_j·v_j , kvk²_D:= (v,v)D.

DEFINITION 2.3. Letφ = (φ^T_i , φ^P_k)be inR^I+JΓ ×R^K. We define the traceφ˜ofφ on the boundary edgesAj ⊂Γwithφ˜j := ¹₄

φ^P_k

1(j)+ 2φ^T_i

2(j)+φ^P_k

2(j)

. We also define a discrete scalar product for the traces ofv·nandφ˜on the boundariesΓq

(v·n,φ)˜Γq,h:= X

j∈Γq

|Aj|(v_j·n_j) ˜φj

and onΓ

(2.1) (v·n,φ)˜ Γ,h:= X

q∈[0,Q]

(v·n,φ)˜Γq,h.

In the proof of discrete Poincar´e inequalities, we often use the piecewise constant functions based on the discrete functions defined at the centers of each mesh; we set the following definitions

DEFINITION2.4. Letφ∈R^I+JΓ×R^K. The piecewise constant functionsφ^T(x)and φ^P(x)are defined following, respectively,

φ^T(x) =φ^T_i , ∀x∈Tiandi∈[1, I];

φ^P(x) =φ^P_k , ∀x∈Pkandk∈[1, K].

We recall here the discrete gradient [7,11] and (vector) curl operators [9] which have been constructed on the diamond cells.

DEFINITION2.5. Letφ= (φ^T_i , φ^P_k)be inR^I+JΓ, its discrete gradient∇^D_hφand discrete curl∇^D_h ×φare defined by their values in the cellsDjthrough

(∇^D_hφ)j := 1 2|Dj|

[φ^P_k2−φ^P_k1](|A^′_j1|n^′_j1+|A^′_j2|n^′_j2) + [φ^T_i2−φ^T_i1]|Aj|n_j ,

(∇^D_h ×φ)j :=− 1 2|Dj|

[φ^P_k2−φ^P_k1](|A^′_j1|τ^′_j1+|A^′_j2|τ^′_j2) + [φ^T_i2−φ^T_i1]|Aj|τ_j . In the proof of our results, we shall use the following theorem which is exactly [9, Theorem 4.7]

THEOREM2.6 (Discrete Hodge Decomposition). Let(v_j)j∈[1,J]be a discrete vector field defined by its values on the diamond-cellsDj.

There exist uniqueφ = (φ^T_i, φ^P_k)i∈[1,I+J^Γ],k∈[1,K], ψ = (ψ_i^T, ψ_k^P)i∈[1,I+J^Γ],k∈[1,K] and (c^T_q, c^P_q)q∈[1,Q]such that:

(2.2) v_j = (∇^D_hφ)j+ (∇^D_h ×ψ)j, ∀j∈[1, J],

X

i∈[1,I]

|Ti|φ^T_i = X

k∈[1,K]

|Pk|φ^P_k = 0,

(2.3) ψ^T_i = 0, ∀i∈Γ0 , ψ_k^P = 0, ∀k∈Γ0, and

(2.4) ∀q∈[1, Q], ψ^T_i =c^T_q , ∀i∈Γq , ψ^P_k =c^P_q , ∀k∈Γq.

Moreover, decomposition (2.2) is orthogonal. We shall also need the following construction of discrete divergence and (scalar) curl operators on both primal and dual cells:

DEFINITION2.7. Letv= (v_j)be defined in(R²)^Jby its values on the diamond-cells.

We define

∇^T_h ·v

i:= 1

|Ti| X

j∈∂Ti

|Aj|v_j·n_ji,

∇^P_h ·v

k:= 1

|Pk| X

j∈∂Pk

|A^′_j1|v_j·n^′

j1k+|A^′_j2|v_j·n^′

j2k

+ X

j∈∂Pk∩Γ

|Aj| 2 v_j·n_j

! ,

∇^T_h ×v

i:= 1

|Ti| X

j∈∂Ti

|Aj|v_j·τ_ji,

∇^P_h ×v

k:= 1

|Pk| X

j∈∂Pk

|A^′_j1|v_j·τ^′

j1k+|A^′_j2|v_j·τ^′

j2k

+ X

j∈∂Pk∩Γ

|Aj| 2 v_j·τ_j

! .

The following result [9, Proposition 4.1], which consists in discrete Green formulas, has motivated the name ”discrete duality”:

THEOREM 2.8 (Discrete Green Formulas). Forv ∈ (R²)^J andφ = (φ^T, φ^P) ∈ R^I+JΓ×R^K, it holds that

(v,∇^D_hφ)D=−(∇^T,P_h ·v, φ)T,P + (v·n,φ)˜ Γ,h, (2.5)

(v,∇^D_h ×φ)D= (∇^T,P_h ×v, φ)T,P −(v·τ,φ)˜ Γ,h. (2.6)

3.

Discrete Poincar´e Inequalities.

Our result is a special case of that proved in [3, Lemma 3.3], but our expression of the discrete constantcF is more precise and simple, in that its dependence on the geometry of the cells occurs only through the angles in the diagonals of the diamond-cells. This is an important result in the DDFV context, since a priori error estimations of the discrete solution of the Laplace equation obtained with this method also only depend on the cell geometries through angles in the diamond-cells (see [11]).

THEOREM3.1 (Discrete Poincar´e-Friedrichs Inequality). LetΩbe an open bounded polygonal domain; let us consideru= (u^T_i, u^P_k)∈R^I+JΓ×R^Ksuch that

u^P_k = 0, ∀k∈Γ and u^T_i = 0, ∀i∈Γ.

Letθ^∗be defined by Definition2.1. Then, there exists a constant C only depending onΩand θ^∗such that

kukT,P ≤Ck∇^D_hukD. (3.1)

Proof. Letu^T(·)andu^P(·)be the piecewise constant functions defined in Definition2.4.

Then obviouslykuk²_T,P = (ku^Tk²_L2(Ω)+ku^Pk²_L2(Ω))/2, so that, in order to prove (3.1), it suffices to prove

ku^TkL²(Ω)≤Ck∇^D

hukD, (3.2)

ku^PkL²(Ω)≤Ck∇^D

hukD. (3.3)

We shall first prove (3.2). Letd₁= (0,1)^tandd₂= (1,0)^t; forx∈Ω, letD_x¹andD²_xbe the straight lines going throughxand parallel to the vectorsd₁andd₂. For any edgej ∈[1, J]

and anyx∈Ω, let us defineχ^T,1_j (x)andχ^T,2_j (x)by (3.4) χ^T,ℓ_j (x) =

(1 ifAj∩ D^ℓ_x6=∅

0 ifAj∩ D^ℓ_x=∅ forℓ= 1,2.

REMARK3.2. For anyx= (x1, x2)∈Ω, we note thatχ¹_j(x)only depends onx1and χ²_j(x)only depends onx2. From the first formula of definition2.5and simple geometry, it is easy to see that

(3.5) (∇^D_hu)j·−−−−−−−−→

Gi1(j)Gi2(j)=u^T_i2(j)−u^T_i1(j), ∀j∈[1, J].

Then, for anyi∈[1, I]and a.e.x∈Ti, let us follow the straight lineD^ℓ_xuntil it intersects the boundaryΓ, and let us denote byv1(i) :=i, v2(i),· · · , vn−1(i)the indices of the primal cells

u v (i)

D

u

v (i)

u

v (i)

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 1111

Γ

u x

=0

FIG. 3.1. Straight lineD2

xintersecting primal cells from pointxto the boundary.

that it intersects (in the order they are intersected) and byvn(i)the index in[I+ 1, I+J^Γ] corresponding to the first boundary segment intersected byD^ℓ_x(see Fig. 3.1). Then, since u^T_vn(i)= 0because of the boundary conditions, we may write

u^T_i =u^T_v1(i)= (u^T_v1(i)−u^T_v2(i)) + (u^T_v2(i)−u^T_v3(i)) +· · ·+ (u^T_vn−1(i)−u^T_vn(i))

=

n−1

X

m=1

(u^T_vm(i)−u^T_vm+1(i)),

so that, since any couple(u^T_vm(i), u^T_vm+1(i))is a pair of neighboring values through an edge Ajintersected byD^ℓ_x, there holds, thanks to (3.5)

|u^T(x)|=|u^T_i| ≤

J

X

j=1

(∇^D_j u)j·−−−−−−−−→

Gi1(j)Gi2(j)

χ^T,ℓ_j (x)

forℓ= 1,2. Then, settingvj :=

(∇^D_ju)j·−−−−−−−−→

Gi1(j)Gi2(j)

, one has

(u^T(x))²≤





J

X

j=1

vjχ^T,1_j (x)









J

X

j=1

vj χ^T,2_j (x)



.

Integrating the above inequality overTiand summing overi∈[1, I]yields

(3.6) ku^Tk²_L2(Ω)≤ Z

Ω









J

X

j=1

vj χ^T,1_j (x)









J

X

j=1

vjχ^T,2_j (x)







dx.

Letα = inf{x1; (x1, x2) ∈ Ω}andβ = sup{x1; (x1, x2) ∈ Ω}. For eachx1 ∈(α, β), we denote byH(x1)the set ofx2 such thatx= (x1, x2)∈ Ω. From Remark3.2and the fact thatR

H(x1)χ^T,2_j (x2)dx2≤ |Aj|andRβ

αχ^T,1_j (x1)dx1≤ |Aj|, we infer that (3.6) may be

written in the following way:

ku^Tk²_L2(Ω)≤ Z β

α

dx1

Z

H(x1)

dx2





J

X

j=1

vjχ^T,1_j (x1)

J

X

j=1

vj χ^T,2_j (x2)





≤ Z β

α J

X

j=1

vjχ^T,1_j (x1)



 Z

H(x1) J

X

j=1

vj χ^T,2_j (x2)dx2



dx1

≤ Z β

α J

X

j=1

vjχ^T,1_j (x1)





J

X

j=1

vj

Z

H(x1)

χ^T,2_j (x2)dx2



dx1

≤ Z β

α J

X

j=1

vjχ^T,1_j (x1)





J

X

j=1

vj|Aj|



dx1

≤





J

X

j=1

vj|Aj|





J

X

j=1

vj

Z β α

χ^T,1_j (x1)dx1≤





J

X

j=1

vj|Aj|









J

X

j=1

vj|Aj|



.

We thus obtain

(3.7) ku^Tk²_L2(Ω)≤





J

X

j=1

|(∇^D_hu)j.−−−−−−−−→

Gi1(j)Gi2(j)||Aj|





2

.

Finally, Using the Cauchy-Schwarz inequality, we have ku^Tk²_L²_(Ω)≤





J

X

j=1

|(∇^D_hu)j|²|Gi1(j)Gi2(j)||Aj|









J

X

j=1

|Gi1(j)Gi2(j)||Aj|



.

Since|Dj| = ¹₂(|Aj||Gi1Mj|sinθj1+|Aj||Gi2Mj|sinθj2), we have that|Aj||Gi1Gi2| ≤

2|Dj|

sinθ^∗ by Definition2.1and the triangle inequality. Moreover, sincePJ

j=1|Dj|=|Ω|, there holds

ku^Tk²_L²_(Ω)≤ 4 sin²θ^∗|Ω|

J

X

j=1

|(∇^D_hu)j|²|Dj|.

We have completed inequality (3.2) withC= _sinθ²∗|Ω|^1/2.We now turn to inequality (3.3).

We shall use a very similar process to that employed in the proof of (3.2). A slight difference comes from the fact that dual cells may be non-convex, and that the straight linesD^ℓ_xmay thus intersect twice the boundaryA^′_j1∪A^′_j2between two adjacent dual cells (see Fig. 3.2), in which case it is not useful to introduce the differenceu^P_k2(j)−u^P_k1(j)in the calculation. We thus defineχ^P,1_j (x)andχ^P,2_j (x)by

χ^P,ℓ_j (x) =

(1 if eitherA^′_j1∩ D^ℓ_x6=∅orA^′_j2∩ D_x^ℓ 6=∅ 0 if A^′_j1∪A^′_j2

∩ D^ℓ_x=∅ forℓ= 1,2.

In the above definition, it is meant that the “either’ - or” is exclusive: ifD^ℓ_xintersects both A^′_j1andA^′_j2, thenχ^P,ℓ_j (x) = 0. From the first formula of definition2.5, it is easy to see that

(∇^D_j u)j·−−−−−−−−→

Sk1(j)Sk2(j)=u^P_k2(j)−u^P_k1(j), ∀j∈[1, J].

u

k₁(j)

u

k₂(j)

A

j1

A

j2

M

G

1(j)

G

2(j)

FIG. 3.2. The straight lineD2

xintersects twice the boundaryA^′_j1∪A^′_j2of a non convex dual.

Thus, for anyk∈[1, K]and a.e.x∈Pk, one has

|u^P_k| ≤

J

X

j=1

|(∇^D_hu)j·−−−−−−−−→

S_k1(j)S_k2(j)|χ^P,ℓ_j (x) , ℓ= 1,2.

Using a similar process as in the proof of (3.2) and taking into account that Z β

α

χ^P,1_j (x1)dx1≤ |A^′_j1|+|A^′_j2|and Z

H(x1)

χ^P,2_j (x2)dx2≤ |A^′_j1|+|A^′_j2|, we obtain

ku^Pk²_L²_(Ω)≤





J

X

j=1

|(∇^D_hu)j||Aj|(|A^′_j1|+|A^′_j2|)





2

which allows to obtain, similarly as above ku^Pk²_L2(Ω)≤ 4

sin²θ^∗|Ω|

J

X

j=1

|(∇^D_hu)j|²|Dj|,

which concludes the proof of inequality (3.3) withC= _sinθ²∗|Ω|^1/2.

We now turn to a generalization of Theorem3.1which will be useful in the last section of this work.

THEOREM3.3 (Discrete Poincar´e-Friedrichs Inequality). Let us consider open, boun- ded, simply connected, convex polygonal domains(Ωq)_q∈[0,Q] ofR²such thatΩq ⊂Ω0for allq∈[1, Q]andΩ¯q1 ∩Ω¯q2 =∅for all(q1, q2)∈[1, Q]²withq16=q2. LetΩbe defined by Ω = Ω0\(∪^Q_q=1Ωq). Let us denote byΓ =∂Ω =∪^Q_q=0Γq, withΓq=∂Ωqfor allq∈[0, Q].

Letu= (u^T, u^P)∈R^I+JΓ×R^Kbe such that

u^P_k = 0, ∀k∈Γ0 and u^T_i = 0, ∀i∈Γ0, u^P_k =c^P_q, ∀k∈Γq, and u^T_i =c^T_q, ∀i∈Γq,∀q∈[1, Q].

(3.8)

00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000

11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111

u_i

v (i)

u

2

D_x²

Γq Γ0

v (i)n

u

v (i)n

u u_{v (i)}

n+1q

u_{v (i)n−1}

q

uv (i)n+2 q

x

=

q

=0

FIG. 3.3. Straight lineD2

xintersecting primal cells from pointxto the boundary through internal boundaryΓq.

Withθ^∗given by Definition2.1, there exists a constantCdepending only onΩandθ^∗such that (3.1) holds.

Proof. Like in Theorem3.1, it suffices to prove both (3.2) and (3.3). We shall only prove (3.2), since the proof of (3.3) follows exactly the same lines.

The only difference in the proof of (3.2) in Theorem3.3with respect to Theorem3.1 is that the straight lineD^ℓ_xmay now intersect one or several internal boundary(ies)Γq, with q ∈ [1, Q], before intersecting the external boundary Γ0 (see Fig. 3.3). For the sake of simplicity, we shall consider only one intersection with an internal boundaryΓq (since the alternative may be treated exactly in the same way), and we denote byvnq(i)andvnq+1(i) the indices in[I+ 1, I+J^Γ]corresponding to those intersected boundary edges ofΓq. We may still write

u^T_i =

n−1

X

m=1

(u^T_vm(i)−u^T_vm+1(i)), but, now, the couple(u^T_v

nq(i), u^T_v

nq+1(i))is not a pair of neighboring values through an edge Ajintersected byD^ℓ_x. However, these two values are equal because of (3.8), so that

u^T_i = X m∈[1, n−1]

m6=nq

(u^T_vm(i)−u^T_vm+1(i)).

Now, any couple(u^T_vm(i), u^T_vm+1(i))in the above sum is a pair of neighboring values through an edgeAjof the mesh, intersected byD^ℓ_x, so that there holds, thanks to (3.5)

|u^T_i| ≤

J

X

j=1

(∇^D_ju)j·−−−−−−−−→

Gi1(j)Gi2(j)

χ^T,ℓ_j (x) forℓ= 1,2and we finish the proof just like in the proof of (3.2).

Let us now turn to a discrete version of (1.2). As announced in the Introduction, the proof will be divided in three steps. The first step is to prove it in the case of a convex polygonal domain (Theorem3.4), then we shall prove an inequality related to the mean value on the boundary of a convex polygonal domain (Theorem3.7) and we shall conclude by the general case of a possibly non-convex polygonal domain (Theorem3.9).

THEOREM3.4 (Discrete mean Poincar´e Inequality for a convex polygonal domain).

LetΩbe an open bounded polygonal connected domain, andωbe an open convex polygonal

x_BD yAC

B A

D

x_AC y_BD x

C

y ω

O x

x₁

2

FIG. 3.4. Notation for pointsA,B,C,Dand pointsxAC,xBD,yAC,yBD.

subset ofΩ, withω6=∅. Letu= (u^T_i , u^P_k)∈R^I+JΓ×R^K; the associated piecewise constant functionsu^T,u^Pare defined through Definition2.4. Letθ^∗be defined through Definition2.1.

Let us define the following mean-values:

m^T_ω(u) := 1

|ω|

Z

ω

u^T(x)dx , m^P_ω(u) := 1

|ω|

Z

ω

u^P(x)dx.

Then, there exists a constant C only depending onΩandθ^∗such that (3.9) ku^T −m^T_ω(u)kL²(ω)≤Ck∇^D_hukD, and

(3.10) ku^P−m^P_ω(u)k_L²_(ω)≤Ck∇^D

hukD. (Choosingω= Ωproves the discrete equivalent of (1.2) ifΩis convex.)

Proof. We only prove inequality (3.9). The proof of (3.10) may be adapted just like in the proof of Theorem3.1. We first note that

Z

ω

|u^T(x)−m^T_ω(u)|²dx= Z

ω

u^T(x)− 1

|ω|

Z

ω

u^T(y)dy

2

dx

≤ 1

|ω|

Z

ω

Z

ω

|u^T(x)−u^T(y)|²dydx.

(3.11)

We define pointsA,B,C,Dbelonging toωin the following way

xA= inf{x1; (x1, x2)∈ω}, xC= sup{x1; (x1, x2)∈ω}, yB = inf{y2; (y1, y2)∈ω}, yD= sup{y2; (y1, y2)∈ω}.

REMARK3.5. Up to a rotation ofω, we may always suppose that those four points are different one from the other, except ifωis triangular; in that case, up to a rotation ofω, we may setA=Band the proof is exactly the same as that below.

For anyx= (x1, x2)∈ω, we definexAC ∈[AC]such that(xAC)1 =x1andxBD ∈ [BD]such that(xBD)2 =x2. The notations are summarized in Fig. 3.4. These points are

used because, sincexACdoes not depend onx2, norxBDonx1, they will help us simplify the quadruple integral in the right-hand side of (3.11) into double integrals only. Moreover, since these points are all located on the two fixed straight lines[AC]and[BD], the evaluation of the remaining integrals may be treated in a systematic way, as will be shown below.

Applying the triangle inequality leads to

|u^T(x)−u^T(y)| ≤ |u^T(x)−u^T(xBD)|+|u^T(xBD)−u^T(yAC)|

+|u^T(yAC)−u^T(y)|

(3.12) and also to

|u^T(x)−u^T(y)| ≤ |u^T(x)−u^T(xAC)|+|u^T(xAC)−u^T(yBD)|

+|u^T(yBD)−u^T(y)|.

(3.13)

From (3.12) and (3.13), we have Z

ω

Z

ω

|u^T(x)−u^T(y)|²dxdy≤

9

X

i=1

Ii, (3.14)

whereI1–I9are defined and estimated in what follows:

Treatment ofI1

I1= Z

ω

Z

ω

|u^T(x)−u^T(xBD)| |u^T(x)−u^T(xAC)|dxdy.

(3.15)

Using again (3.4) and (3.5), we may write

|u^T(x)−u^T(xAC)| ≤

J

X

j=1

χ^T,1_j (x)

(∇^D_hu)j·−−−−−−−−→

Gi1(j)Gi2(j)

(3.16)

and

|u^T(x)−u^T(xBD)| ≤

J

X

j=1

χ^T,2_j (x)

(∇^D_hu)j·−−−−−−−−→

G_i1(j)G_i2(j) . (3.17)

Henceforth, we set for conveniencevj =

(∇^D_hu)j·−−−−−−−−→

Gi1(j)Gi2(j)

. Recalling thatχ^T,1_j (x) only depends onx1andχ^T,2_j (x)only depends onx2, and noting that the integrand in (3.15) does not depend ony, there holds

I1≤ |ω|



 Z xA

xC

J

X

j=1

χ^T,1_j (x)vjdx1







 Z yD

yB

J

X

j=1

χ^T,2_j (x)vjdx2





≤ |ω|





J

X

j=1

vj

Z xA

xC

χ^T,1_j (x)dx1









J

X

j=1

vj

Z yD

yB

χ^T,2_j (x)dx2



.

We use that

Z xC

xA

χ^T,1_j (x)dx1≤ |Aj|

and (3.18)

Z yD

yB

χ^T,2_j (x)dx2≤ |Aj| and obtain

I1≤ |ω|





J

X

j=1

|Aj|vj





2

(3.19) .

Treatment ofI2

I2= Z

ω

Z

ω

|u^T(x)−u^T(xBD)| |u^T(xAC)−u^T(yBD)|dxdy.

Using inequality (3.17), we have I2≤

Z

ω

Z

ω





J

X

j=1

χ²_j(x)vj



|u^T(xAC)−u^T(yBD)|dxdy.

By definition,χ²_j(x)only depends onx2(which is in[yB, yD]), whilexAConly depends on x1(which is in[xA, xC]); of course,yBDdoes not depend onx, so that

I2≤





J

X

j=1

vj

Z yD

yB

χ^T,2_j (x)dx2



 Z

ω

Z xC

xA

|u^T(xAC)−u^T(yBD)|dx1dy.

Thanks to (3.18), we thus have I2≤





J

X

j=1

|Aj|vj



 Z

ω

Z xC

xA

|u^T(xAC)−u^T(yBD)|dx1dy.

SinceyBDonly depends ony2andxACdoes not depend ony, the integration with respect to y1(which is in[xA, xC]) is straightforward and yields

I2≤(xC−xA)





J

X

j=1

|Aj|vj



 Z yD

yB

Z xC

xA

|u^T(xAC)−u^T(yBD)|dx1dy2. (3.20)

Treatment ofI3

I3= Z

ω

Z

ω

|u^T(x)−u^T(xBD)| |u^T(yBD)−u^T(y)|dxdy.

This integral clearly decouples into two independent integrals I3=

Z

ω

|u^T(x)−u^T(xBD)|dx Z

ω

|u^T(yBD)−u^T(y)|dy

which may be treated like in the estimation ofI1 thanks to (3.17), (3.18) and the fact that χ^T,2depends only onx2. We obtain

I3= (xC−xA)²





J

X

j=1

|Aj|vj





2

. (3.21)