On a stabilized colocated finite volume scheme for the Stokes problem

Vol. 40, N 3, 2006, pp. 501–527 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006024

ON A STABILIZED COLOCATED FINITE VOLUME SCHEME FOR THE STOKES PROBLEM

Robert Eymard

, Rapha` ele Herbin

and Jean Claude Latch´ e

Abstract. We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem. It has been developed following two main ideas. On one hand, the dis- cretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non- standard interpolation formula for the expression of the pressure on the edges of the control volumes.

On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitk¨aranta technique. We prove that, under usual regularity assumptions for the solution (each component of the velocity in H²(Ω) and pressure in H¹(Ω)), the scheme is first order convergent in the usual finite volume discrete H¹ norm and the L² norm for respectively the velocity and the pressure, provided, in partic- ular, that the approximation of the mass balance flux is of second order. With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles trian- gulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions. Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L² norm.

Mathematics Subject Classification. 65N12, 65N15, 65N30, 76D07, 76M12.

Received: May 4, 2005.

1. Introduction

Finite volumes enjoy many favorable properties for the discretization of conservation equations: to cite only a few examples, (computing time) efficiency, local conservativity and the possibility to hand-build, to some extent, the discrete operators to recover properties of the continuous problem like, for instance, the positivity of the advection operator. In addition, the complexity raised by the design of actually high order schemes for multidimensional problems with finite volumes discretizations, in particular compared to finite elements methods, may often be of little concern in real-life applications, when the regularity which can be expected for the solution is rather poor. These features make finite volume attractive for industrial problems, as encountered for instance in nuclear safety, which is a part of the context of this study.

Keywords and phrases. Finite volumes, colocated discretizations, Stokes problem, incompressible flows, analysis.

1 Universit´e de Marne-la-Vall´ee, France. [email protected]

2 Universit´e de Provence, France. [email protected]

3 Institut de Radioprotection et de Sˆuret´e Nucl´eaire (IRSN) – Direction de la Pr´evention des Accidents Majeurs (DPAM), France. [email protected]

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006024

On the opposite, the difficulty to build stable pairs of discretization for the velocity and pressure in incom- pressible flow problems remains, to our opinion, a severe drawback of the method. In most applications, this stability is obtained by using a staggered arrangement for the velocity and pressure unknowns: the celebrated MAC scheme (see [13] for the pioneering work and [16, 17] for an analysis). Although this discretization has proved its low cost and reliability, it turns to be difficult to handle from a programming point of view: rather in- tricate treatment of particular boundary conditions, as inner corners for instance, difficulty to deal with general computational domains, to implement multilevel local refinement techniques, etc. For this reason, some schemes making use of discretizations where the degrees of freedom for the velocity components and pressure are seen as an approximation of the continuous solution at the same location (or as an average of the continuous solution over the same control volume) have been proposed in the last two decades [11, 18, 19, 21]. These discretizations are referred to as “colocated”.

The purpose of this paper is to present and analyse a novel colocated scheme for the Stokes problem, which has the following essential features. On one hand, the discretization of the pressure gradient term is designed to be the discrete transposed operator of the velocity divergence term, the latter being evaluated using a natural finite volume approximation. On the other hand, the scheme is stabilized using a finite volume analogue of the Brezzi-Pitk¨aranta technique. We choose to restrict ourselves here to the linear case (i.e. the Stokes problem), for the sake of readability, and to specific meshes for which an optimal order of convergence can be proven. An extension of these results to the Navier-Stokes equations and to general meshes can be found in [8].

The outline of the paper is as follows: we first present the scheme under consideration, then the next section is devoted to establish consistency error estimates which will be used to prove the convergence result (Sect. 4);

finally, we show some numerical experiments which substantiate the theoretical analysis.

2. The continuous problem and the discrete scheme 2.1. The Stokes problem

For the sake of simplicity, we restrict the presentation to homogeneous Dirichlet boundary conditions and regular right-hand sides. In the so-called strong or differential form, the problem under consideration reads:

⎧⎪

⎨

⎪⎩

−∆u+∇p=f in Ω,

∇ ·u= 0 in Ω,

u= 0 on∂Ω

(1)

where Ω is a polygonal open bounded connected subset of R^d, d= 2 or 3, ∂Ω stands for its boundary, f is a function of L²(Ω)^d, uand pare respectively a vector valued (i.e. taking values inR^d) and a scalar (i.e. taking values inR) function defined over Ω.

The weak solution of (1) (seee.g. [12]) (u, p) is the unique solution in H¹₀(Ω)^d×L²(Ω) of:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

Ω

∇u:∇v−

Ω

p∇ · v=

Ω

f ·v ∀v∈H¹0(Ω)^d

Ω

q ∇ ·u= 0 ∀q∈L²(Ω)

Ω

p= 0.

(2)

2.2. Discretization and discrete functional spaces

The finite volume discretizations of the polygonal domain Ω which are considered here consist in a finite familyMof disjoint non-empty convex subdomainsKof Ω (the “control volumes”) such that:

– ifd= 2, each control volume is either a rectangle or a triangle with internal angles strictly lower than π/2;

– ifd= 3, each control volume is a rectangular parallelepiped;

– the discretization is conforming in the sense that two neighbouring control volumes share a complete (d−1)-dimensional side, which will be called hereafter an edge of the meshing.

To each control volumeK, we associate the following point, denoted by x_K: if d= 2, the intersection of the perpendicular bisectors of each edge, ifd= 3 the intersection of the lines issued from the barycenter of the edge and orthogonal to the edge. Note that, whenK is a simplex ofR², the fact that the interior angles are lower thanπ/2 impose thatx_K is always located inside the triangleK.

The set of edges is denoted byE. It can be split into the set of internal edges,i.e. separating two control volumes, denoted byEint, and the set of external ones denoted byEext. The setE(K) stands for the set of the edges of the control volumeK.

An internal edge separating two control volumes K and L is denoted by K|L. The segment [xK, x_L] is orthogonal toK|L and crossesK|Lat x_K|L. We denote byd_K|Lthe distance betweenx_K and x_L andd_K,K|L the distance betweenx_K andx_K|L.

Remark 2.1. The class of admissible meshings is far smaller here than usual for elliptic problems (see [7]).

Additional constraints come from the need to suppose in the analysis thatx_K|L is located at the barycenter of K|L, to obtain a second order approximation for the mass fluxes throughK|L.

We will adopt hereafter the following conventions. The expression:

σ∈Eint(σ=K|L)

F(σ, K|L, K, L) or max

σ∈Eint(σ=K|L)F(σ, K|L, K, L)

will stand for a summation or a maximum taken over the internal edges, the control volumes sharing the edge σbeing denoted byKandL. Similarly, the expression:

σ∈Eext (σ∈E(K))

F(σ, K) or max

σ∈Eext (σ∈E(K))F(σ, K)

will stand for a summation or a maximum taken over the external edges, the unique control volume to which the edgeσbelongs being denoted byK. Finally:

σ=K|L

F(σ, K|L, K, L)

will stand, in a relation related to the control volume K, for a summation over the internal edgesσ ofK, the second control volume to whichσbelongs being denoted byL.

We denote byh_K the diameter of each control volume andhthe maximum of the values ofh_K, forK∈ M.

Here and throughout the paper,m(K) andm(σ) will stand for, respectively, thed-measure of the control volume K and the (d−1)-measure of the edge σ. The regularity of the meshing is quantified by the following set of real numbers:

regul(M) =

σ∈Eintmax(σ=K|L)

m(σ)

d_σ (h_K+h_L)^d−2, max

σ∈Eext (σ∈E(K))

m(σ) d_K,σh^d−_K ²,

K∈M, L∈Nmax (K)

h_K

h_L, max

K∈M

h_K ρ_K

(3)

whereρ_K is the diameter of the greatest ball included inKandN(K) stands for the set of neighbouring control volumes ofK.

Throughout the paper, the following notation:

F1 ≤

reg

F2

means that there exists a positive number c, (possibly) depending on Ω and on the meshing only through regul(M) and being a non-decreasing function of all the four elements of regul(M), such that:

F1≤c F2. 2.2.2. Discrete functional spaces

Definition 2.2. LetMbe a discretization as described in the preceding section. We denote by H_D(Ω)⊂L²(Ω) the space of functions which are piecewise constant over each control volumeK∈ M. For allu∈H_D(Ω) and for allK∈ M, we denote byu_K the constant value ofuinK. The space H_D(Ω) is equipped with the following Euclidean structure. For (u, v)∈(H_D(Ω))², we define the following inner product (corresponding to Dirichlet boundary conditions):

[u, v]_D=

σ∈Eint(σ=K|L)

m(σ)

d_σ (u_L−u_K)(v_L−v_K) +

σ∈Eext(σ∈E(K))

m(σ)

d_K,σ u_K v_K. (4) Thanks to the discrete Poincar´e inequality (7) given below, this scalar product defines a norm on H_D(Ω):

u1,D= [u, u]¹_D^/2. (5)

Definition 2.3. We also define the following different inner product (corresponding to Neumann boundary conditions), together with its associated seminorm:

u, v_D =

σ∈Eint(σ=K|L)

m(σ)

d_σ (u_L−u_K)(v_L−v_K) |u|1,D=u, u¹_D^/2. (6) These definitions naturally extend to vector valued functions as follows. Foru= (u⁽ⁱ⁾)_i=1,...,d ∈H_D(Ω)^d and v= (v⁽ⁱ⁾)_i=1,...,d∈H_D(Ω)^d, we define:

[u, v]_D = d i=1

[u⁽ⁱ⁾, v⁽ⁱ⁾]_D u1,D= d i=1

[u⁽ⁱ⁾, u⁽ⁱ⁾]_D ¹/2

.

Proposition 2.4. The following discrete Poincar´e inequalities hold (see[7]):

uL²(Ω)≤diam(Ω)u1,D ∀u∈H_D(Ω)

u_L²(Ω)≤C(Ω) |u|1,D ∀u∈H_D(Ω) such that

Ω

u= 0, (7)

whereC(Ω) only depends on the computational domainΩ.

In addition, we define a mesh dependent seminorm| · |h over H_D(Ω) by:

∀u∈H_D(Ω), |u|²_h=

σ∈Eint(σ=K|L)

(h²_K+h²_L) m(σ)

d_σ (u_L−u_K)².

The following inverse inequality holds:

Proposition 2.5. Let ube a function of H_D(Ω). Then:

|u|h ≤

regu_L²(Ω) (8)

Proof. We have:

σ∈Eint(σ=K|L)

(h²_K+h²_L) m(σ)

d_σ (u_L−u_K)² ≤2

σ∈Eint(σ=K|L)

(h²_K+h²_L) m(σ)

d_σ (u²_L+u²_K)

= 2

K∈M

u²_K

σ=K|L

(h²_K+h²_L) m(σ) d_σ · Owing to the fact that card(E(K)) is bounded and h_L

h_K, m(σ)

d_σ (h_K+h_L)^d−2 and h_K

ρ_K are controlled by elements of regul(M), we get:

2

K∈M

u²_K

σ=K|L

(h²_K+h²_L) m(σ) d_σ ≤

reg

K∈M

ρ^d_Ku²_K

which concludes the proof.

2.3. The finite volume scheme

The finite volume scheme under consideration reads:

Find u∈H_D(Ω)^d andp∈H_D(Ω) such that for each control volumeK of the mesh:

σ=K|L

−m(σ)

d_σ (u_L−u_K) +

σ∈E(K)∩Eext

−m(σ)

d_K,σ (−uK) +

σ=K|L

m(σ)d_L,σ

d_σ (p_L−p_K)n_σ=

K

f

σ=K|L

m(σ) d_L,σ

d_σ u_K+d_K,σ d_σ u_L

·n_σ−λ

σ=K|L

(h²_K+h²_L) m(σ)

d_σ (p_L−p_K) = 0

(9)

where λis a positive parameter and n_K|L stands for the normal to the edge K|Loriented fromK to L. It is easily seen that this system of equations is singular, the vector of unknowns corresponding to a zero velocity and a constant pressure belonging to the kernel of the associated discrete operator. Consequently, we impose to the solution to fulfill the following constraint:

K∈M

m(K)p_K = 0.

As the functions of H_D(Ω) are constant over each control volume, this relation is an exact reformulation of the

third relation of (2):

Ω

p= 0. (10)

On the discrete gradient expression

While the expression of the velocity divergence term is built with a natural interpolation for the velocity on the side σ, the expression of the pressure gradient is not. In fact, the latter is specially constructed to ensure

that the discrete gradient is the transposed operator of the discrete divergence,i.e.:

∀u∈(H_D(Ω))^d, ∀p∈H_D(Ω),

K∈M

p_K

σ=K|L

m(σ) d_L,σ

d_σ u_K+d_K,σ d_σ u_L

·n_σ=−

K∈M

u_K·

σ=K|L

m(σ)d_L,σ

d_σ (p_L−p_K) n_σ. This property seems to be of crucial importance in the analysis of the stability of the scheme.

Another equivalent expression can be derived for the discrete gradient term. Indeed, using the fact that, for each elementK,

σ∈E(K)

m(σ)n_σ= 0, we get:

σ=K|L

m(σ)d_L,σ

d_σ (p_L−p_K)n_σ =

σ=K|L

m(σ)d_L,σ

d_σ (p_L−p_K)n_σ +p_K

⎡

⎣

σ=K|L

m(σ)n_σ+

σ∈E(K)∩Eext

m(σ)n_σ

⎤

⎦

=

σ=K|L

m(σ) (d_L,σ

d_σ p_L+ (1−d_L,σ

d_σ )p_K)n_σ+

σ∈E(K)∩Eext

m(σ)p_K n_σ

=

σ=K|L

m(σ) (d_L,σ

d_σ p_L+d_K,σ

d_σ p_K)n_σ+

σ∈E(K)∩Eext

m(σ)p_K n_σ.

(11)

This last expression can be seen as a rather natural discretization of the integral over the edgeσof the quantity p n_σ, although with a less natural interpolation formula to estimate the pressure on each edge. Note also that this latter formulation is conservative. Both formulations of the discrete gradient are used hereafter.

On the necessity of adding a stabilization term

Let us consider the particular case of a regular meshing of a square two-dimensional domain by square control volumes. It is then easy to see that, without stabilization term, the considered scheme becomes the same as the usual colocated finite volume scheme, which is known to be hardly usable. In particular, the pressure gradient term reads, with the usual notations for the computational molecule:

1 2 h^d−1

pW−pE

pN−pS

.

This relation enlights the risk of odd-even decoupling of the pressure (even if checkerboard pressure modes may be filtered out by boundary conditions); this phenomena has indeed been evidenced in numerical experiments.

On the stabilization term

In the particular case of a uniform meshing, the stabilization term is the discrete counterpart of −2λh²∆p, which explains why we consider that this stabilization falls into the Brezzi-Pitk¨aranta family [4]. However, it can also be seen as a sum of jump terms across each internal edge, as classically used for stabilizing finite element discretizations using a discontinuous approximation space for the pressure [14, 22].

Other stabilizations for colocated finite volumes are possible. In particular, Brezzi and Fortin proposed a few years ago the so-called “minimal stabilization procedure” [3] which consists in adding, as in the present work, a stabilization term in the continuity equation, which reads in the framework of a Galerkin numerical scheme:

c(p, q) =

Ω

p−ΠQ¯hp)(q−ΠQ¯hq

wherep∈Q_his the pressure unknown,q∈Q_his a test function, ΠQ¯h is a projection operator fromQ_h onto ¯Q_h and ¯Q_his a subset ofQ_hsuch that the pair of discrete spaces obtained by associating the velocity discretization space to ¯Q_h is inf-supstable. Later on, Dohrmann and Bochev [6] introduced a stabilized scheme for equal degree (let sayk) finite element discretizations of the Stokes problem based of the same relation, choosing for Q¯_hthe discontinuous space of piecewisek−1 degree polynomials. A finite volume analogue to these procedures is to use for ¯Q_h the space of pressures keeping a constant value over clusters of control volumes; this is the line which we follow in ongoing works [9, 10].

3. A quasi-interpolation operator and consistency error bounds

This section is devoted to state and prove the consistency error estimates useful for the analysis. It is written so as to give, as much as possible, a self-consistent presentation of the matter at hand. We thus give a proof of the whole set of relevant estimates, even if some of them can be found in already published literature (in particular [7]). These proofs are new, and rely on the Cl´ement quasi-interpolation technique [5], which presents two advantages: first, they allow some straightforward generalizations (for instance, dealing with cases when the full regularity of the solution is not achieved); second, they make the presentation closer to the literature of some related topics, as for instancea posteriorierror estimates.

3.1. Two technical preliminary lemma

The two following lemmas will be useful in the rest of the paper. The first one is a trace lemma, the proof of which can be found in the case of a simplex in ([23], Sect. 3). Applying a similar technique to parallelepipeds leads to the same relation, with the space dimensiondreplaced by the (lower) constant (1 +√

5)/2.

Lemma 3.1. LetK be an admissible control volume as defined in Section 2.2.1, h_K its diameter andσone of its edges. Let ube a function of H¹(K). Then:

uL²(σ)≤

d m(σ) m(K)

1/2

uL²(K)+h_K|u|H¹(K)

.

For vector functions of H¹(K)^d, applying this lemma to each component yields the following estimate:

u_L²(σ)≤

2d m(σ) m(K)

1/2

u_L²(K)+h_K|u|_H¹(K)

. (12)

Similarly, ifu∈H²(K), we have:

|u|_H¹(σ)≤

2d m(σ) m(K)

1/2

|u|_H¹(K)+h_K|u|_H²(K)

. (13)

The second lemma allows to bound the L^∞(K) norm of a degree one polynomial by its L²(K) norm:

Lemma 3.2. Let P1be the space of linear polynomials andφbe an element ofP1. Then there exists a constant c_∞,2 such that:

φL^∞(K)≤c_∞,2

1

m(K)^1/2 φL²(K).

Proof. To each type of control volume under consideration, we can associate a reference control volume by an affine mapping: for instance, for two-dimensional simplices, we can choose the triangle of vertices (0,0), (1,0) and (0,1). The vector space of degree one polynomials on the reference control volume is a finite dimensional space, on which the L^∞(K) and L²(K) norm are equivalent. The result then follows by a change of coordinates

in the integral.

In addition, we will repeatedly use hereafter the following inequality, which is an easy consequence of the Cauchy-Schwarz inequality:

∀u∈L²(ω)

ω

u≤m(ω)^1/2u_L²(ω) (14) whereω is (for the cases under consideration here) a polygonal (necessarily bounded) domain ofR^d orR^d−1.

3.2. Definition and properties of a quasi-interpolation operator

Definition 3.3. Let ube a function in L²(Ω). For each control volumeK ∈ M, we define ω_K as the convex hull ofK∪(∪_L∈N(K)L). LetP1 be the space of linear polynomials andφ_K ∈P1 be defined by:

ωK

(u−φ_K)ψ= 0 ∀ψ∈P1. (15)

Then we define Π_Du∈H_D(Ω) by (Π_Du)_K =φ_K(x_K), ∀K∈ M.

By extension, we will keep the same notation for vector valued functions.

The following estimates hold:

Lemma 3.4. Let ¯h_K be the diameter of ω_K. Ifuis a function from ω_K toRandφ_K is defined by (15), then:

if u∈H¹(ω_K) : u−φ_KL²(ωK) ≤c^app_0,1 ¯h_K |u|H¹(ωK)

|u−φ_K|_H¹(ωK) ≤c^app_1,1 |u|_H¹(ωK)

if u∈H²(ω_K) : u−φ_KL²(ωK) ≤c^app_0,2 ¯h²_K |u|H²(ωK)

|u−φ_K|_H¹(ωK) ≤c^app_1,2 ¯h_K |u|_H²(ωK)

wherec^app_0,1,c^app_1,1,c^app_0,2 andc^app_1,2 only depend on d.

The existence of these constants is due to the independence of the shape of a convex domain of the constants in Jackson’s type inequalities, as stated in [24].

Remark 3.5. LetMbe a meshing as described in Section 2.2.1. Each control volume K is intersected by a finite number of domainsω_L, L ∈ M. This number is known to be bounded by a constantN_ω which can be expressed as a non-decreasing function of the parameters in regul(M) (for simplicial discretizations, the number of simplices sharing a vertex is limited by max

K∈M(h_K/ρ_K), see ([2], Sect. 2)).

In addition, it is easy to see that:

∀K∈ M, ¯h_K ≤

regh_K.

The continuity of the projection operator Π_D from H¹(Ω) to H_D(Ω) is addressed in the following proposition:

Proposition 3.6. Let ube a function of H¹(Ω). Then the following estimate holds:

|Π_Du|1,D ≤

reg|u|_H¹(Ω). If in addition ubelongs toH¹₀(Ω), then:

Π_Du1,D ≤

reg|u|_H¹(Ω).

The following result gives some insight into the way the projection ofuapproximates the functionuitself.

Proposition 3.7. Let ube a function of H¹(Ω). Then the following estimate holds:

u−Π_Du_L²(Ω) ≤

reg h|u|_H¹(Ω). (16)

Let us now suppose that u belongs to H²(Ω)∩H¹₀(Ω). As a consequence, u is continuous and we can define

¯

u∈H_D(Ω)by u¯_K =u(x_K),∀K∈ M. Then we have:

ΠDu−u¯ 1,D ≤

reg h|u|H²(Ω). (17)

The proofs of both preceding propositions are given in appendix.

3.3. Consistency error bounds

In this section, we successively provide estimates for the consistency residuals associated to the diffusive term (Lem. 3.8), the pressure gradient term (Lem. 3.9) and, finally, the velocity divergence term (Lem. 3.10).

Lemma 3.8. Let ube a function of H²(Ω)∩H¹0(Ω). For any sideσ of E, we define:

ifσ∈ Eint, σ=K|L: R∆,K|L(u) =m(K|L)

d_K|L ((Π_Du)_L−(Π_Du)_K)−

K|L∇u·n_K|L ifσ∈ Eext, σ∈ E(K) : R∆,σ(u) = m(σ)

d_K,σ (−(Π_Du)_K)−

σ

∇u·n_σ

wheren_σ is the normal vector to the edge σoriented outward to K. Then:

ifσ∈ Eint, σ=K|L:

|R∆,K|L(u)| ≤f(c_∞,2, c^app_0,2, c^app_1,2) m(K|L)

d_K|L min [m(K), m(L)]^1/2 (¯h²_K |u|_H²(ωK)+ ¯h²_L |u|_H²(ωL));

ifσ∈ Eext, σ∈ E(K) :

|R∆,σ(u)| ≤f(c^app_0,2, c^app_1,2) m(σ) d_K,σ m(K)^1/2

¯h²_K |u|H²(ωK).

Proof. We begin with the case of an internal side. By definition of the projection operator Π_D, the quantity R∆,K|L reads:

R∆,K|L(u) = m(K|L)

d_K|L (φ_L(x_L)−φ_K(x_K))−

K|L∇u·n_K|L

= m(K|L)

d_K|L (φ_K(x_L)−φ_K(x_K))−

K|L∇u·n_K|L

T1,K|L

+m(K|L)

d_K|L (φ_L(x_L)−φ_K(x_L))

T2,K|L

. (18)

Using the fact thatφ_K is a linear polynomial, the first term of the right hand side of the preceding relation can be expressed as:

T1,K|L=m(K|L)∇φK·−−−→x_Kx_L d_K|L −

K|L∇u·n_K|L=−

K|L∇(u−φ_K)·n_K|L.

Making use of inequality (14), withω=K|L, applying inequality (13) tou−φ_K and finally using lemma 3.4, we obtain the following estimate:

|T1,K|L| ≤m(K|L)^1/2 |u−φ_K|_H¹(K|L)

≤m(K|L)^1/2

2d m(K|L) m(K)

1/2

|u−φ_K|_H¹(K)+h_K|u−φ_K|_H²(K)

≤(2d)^1/2 m(K|L) m(K)^1/2

|u−φ_K|H¹(ωK)+h_K|u−φ_K|H²(ωK)

≤(1 +c^app_1,2) (2d)^1/2 m(K|L) m(K)^1/2

¯h_K |u|_H²(ωK).

On the other hand, using lemma 3.2, the triangular inequality, the fact thatLis included in bothω_L andω_K, then finally lemma 3.4, the second term at the right hand side of equation (18) can be estimated as follows:

|T2,K|L| ≤m(K|L)

d_K|L φK−φ_LL^∞(L)

≤c_∞,2

m(K|L)

d_K|L m(L)^1/2 φ_K−φ_LL²(L)

≤c_∞,2 m(K|L)

d_K|L m(L)^1/2 (φK−u_L²(L)+φL−u_L²(L))

≤c_∞,2

m(K|L)

d_K|L m(L)^1/2 (φ_K−u_L²(ωK)+φ_L−u_L²(ωL))

≤c_∞,2c^app_0,2 m(K|L)

d_K|Lm(L)^1/2 (¯h²_K |u|_H²(ωK)+ ¯h²_L |u|_H²(ωL)).

The proof is then easily completed by collecting the bounds ofT1,K|LandT2,K|L and using the fact thatd_K|L is smaller than ¯h_K.

On an external side σassociated to the control volumeK, we have:

R∆,σ(u) = m(σ)

d_K,σ (−φK(x_K))−

σ

∇u·n_σ

= m(σ)

d_K,σ (φ_K(x_σ)−φ_K(x_K))−

σ

∇u·n_σ

T1,σ

+m(σ)

d_K,σ (−φK(x_σ))

T2,σ

.

The first termT1,σcan be estimated strictly as the termT1 of the relation (18). As the functionuvanishes on the boundary of the domain, the second one reads:

T2,σ=− 1 d_K,σ

σ

φ_K =− 1 d_K,σ

σ

(φ_K−u).

Making use of inequality (14), lemma 3.1 and lemma 3.4, we get:

|T2,σ| ≤m(σ)^1/2

d_K,σ φK−uL²(σ)

≤d^1/2 m(σ) d_K,σm(K)^1/2

φ_K−uL²(K)+h_K |φ_K−u|H¹(K)

≤(c^app_0,2 +c^app_1,2)d^1/2 m(σ)

d_K,σ m(K)^1/2 ¯h²_K|u|_H²(ωK).

Once again, collecting the bounds forT1,σ andT2,σ yields the result.

Lemma 3.9. Let ube a function ofH¹(Ω). For any sideσofE, we define the following vector valued quantity:

ifσ∈ Eint, σ=K|L: Rgrad,K|L(u) =m(K|L)

d_K,K|L

d_K|L (Π_Du)_K+d_L,K|L

d_K|L (Π_Du)_L

n_K|L−

K|Lu n_K|L, ifσ∈ Eext, σ∈ E(K) : Rgrad,σ(u) =m(σ) (Π_Du)_K n_σ−

σ

u n_σ,

wheren_σ is the normal vector to the edge σoriented outward to K. Then:

ifσ∈ Eint, σ=K|L:

|Rgrad,K|L(u)| ≤f(c_∞,2, c^app_0,1, c^app_1,1) m(K|L) min(m(K), m(L))^1/2

¯h_K |u|_H¹(ωK)+ ¯h_L |u|_H¹(ωL)

,

ifσ∈ Eext, σ∈ E(K) :

|Rgrad,σ(u)| ≤f(c_∞,2, c^app_0,1, c^app_1,1) m(σ) m(K)^1/2

¯h_K |u|_H¹(ωK),

where| · | stands in the preceding relation for the Euclidean norm inR^d.

Proof. First of all, we remark that we can recast the case of an external edge into the formulation associated to an internal one by defining a fictitious external control volumeL, setting d_L,K|L= 0 and giving to (Π_Du)_L any finite value. So the major part of the proof addresses both cases.

Using the linearity ofφ_K, the quantityRgrad,K|L can be decomposed as follows:

Rgrad,K|L(u) =m(K|L) (d_K,K|L

d_K|L φ_K(x_K) +d_L,K|L

d_K|L φ_L(x_L))n_K|L−

K|Lu n_K|L

=m(K|L)φ_K(d_K,K|L

d_K|L x_K+d_L,K|L

d_K|L x_L)n_K|L−

K|Lu n_K|L +m(K|L)d_L,K|L

d_K|L (φ_L(x_L)−φ_K(x_L))n_K|L.

Letx_G be the point defined byx_G= d_K,K|L

d_K|L x_K+d_L,K|L

d_K|L x_L. We recall thatx_K|L is defined as the midpoint of the edgeK|L. We then have, as the midpoint integration rule is exact for the linear polynomialφ_K:

Rgrad,K|L(u) =m(K|L)(φK(x_G)−φ_K(x_K|L))n_K|L

T1,K|L

+

K|L(φ_K−u)n_K|L

T2,K|L

+m(K|L)d_L,K|L

d_K|L (φ_L(x_L)−φ_K(x_L))n_K|L

T3,K|L

.

Next step is to bound successively the three termsT1,K|L,T2,K|L andT3,K|L. First, we have:

T1,K|L=m(K|L) (φK(x_G)−φ_K(x_K|L))n_K|L=m(K|L) (∇φK· −−−−−→x_K|L x_G)n_K|L.

As the distance betweenx_K|Landx_G is smaller than ¯h_K, by Lemma 3.2 then 3.4, the following estimate holds:

|T1,K|L| ≤m(K|L) ¯h_K|∇φK| ≤c_∞,2 m(K|L) m(K)^1/2

¯h_K |φK|_H¹(ωK)≤c_∞,2(1 +c^app_1,1) m(K|L) m(K)^1/2

¯h_K |u|_H¹(ωK).

The termT2,K|L is estimated as follows, using successively inequality (14), Lemmas 3.1 and 3.4:

|T2,K|L| ≤m(K|L)^1/2φ_K−uL²(K|L)

≤m(K|L)^1/2

d m(K|L) m(K)

1/2

φ_K−uL²(K)+h_K |φ_K−u|H¹(K)

≤(c^app_0,1 +c^app_1,1)d^1/2 m(K|L) m(K)^1/2

¯h_K|u|_H¹(ωK).

If the edge under consideration is an external one, the termT3,K|L is zero (since d_L,K|L = 0). Otherwise, by Lemma 3.2 then 3.4,T3,K|Lis bounded by:

|T3,K|L| ≤m(K|L)φL−φ_KL^∞(L)

≤c_∞,2

m(K|L)

m(L)^1/2 φ_L−φ_KL²(L)

≤c_∞,2 m(K|L) m(L)^1/2

φL−u_L²(L)+φK−u_L²(L)

≤c_∞,2 m(K|L) m(L)^1/2

φL−uL²(ωL)+φK−uL²(ωK)

≤c_∞,2 c^app_0,1 m(K|L) m(L)^1/2

¯h_L|u|_H¹(ωL)+ ¯h_K |u|_H¹(ωK)

.

Collecting the bounds ofT1,K|L,T2,K|L andT3,K|L, the proof is over.