Mimetic finite differences for elliptic problems

DOI:10.1051/m2an:2008046 www.esaim-m2an.org

MIMETIC FINITE DIFFERENCES FOR ELLIPTIC PROBLEMS

Franco Brezzi

, Annalisa Buffa

and Konstantin Lipnikov

Abstract. We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependentH¹norm are derived.

Mathematics Subject Classification. 65N06, 65N12, 65N15, 65N30.

Received December 5, 2007. Revised July 21, 2008.

Published online December 5, 2008.

1. Introduction

For numerical solution of partial differential equations, polyhedral meshes can provide several advantages.

For instance, a polyhedral mesh has fewer mesh faces than a tetrahedral one with the same mesh resolution, which increases performance of linear solvers. Moreover in computational fluid dynamics it is often desirable to have element faces perpendicular to the flow. A polyhedral element with many faces increases the probability of having such faces. As mentioned in [11], this results in a smaller numerical diffusion and a more accurate solution.

More generally, polyhedral meshes have enormous flexibility in representing complex geometries. The adap- tive mesh refinement technique, which is used to optimize available computational resources and is an essential part of modern multi-physics codes, results in polyhedral meshes with degenerate elements. Non-matching meshes can also be seen as polyhedral meshes with degenerate elements. Another technique for modeling com- plex porous media structures, such as pinch-outs and faults, is to collapse edges of a hexahedral element to points which results in polyhedral elements with strongly curved faces [12].

In this article, we use the mimetic finite difference (MFD) discretization technique which has been designed to work on general polyhedral meshes without any special treatment of degenerate elements. From each polyhedron, the MFD method requires only boundary data such as areas, barycenters and normals to faces, which simplifies its usage for elements with irregular shapes.

The MFD method produces a compatible discretization where discrete analogs of differential operators retain their important properties, so that conservation laws, solution symmetries, and the fundamental identities of vector and tensor calculus do hold for discrete systems. For example, the discrete divergence and gradient operators are negatively adjoint with respect to inner products in discrete spaces. This property has a number of useful consequences. For diffusion problems, it results in symmetric discretizations and simplifies their

Keywords and phrases. Finite differences, polyhedral meshes, diffusion equation, error estimates.

1 Istituto Universitario di Studi Superiori, Pavia, Italy. brezzi@imati.cnr.it

2 Instituto di Matematica Applicata e Tecnologie Informatiche, Pavia, Italy. annalisa@imati.cnr.it

3 Los Alamos National Laboratory, Theoretical Division, MS B284, Los Alamos, NM, 87545, USA.lipnikov@lanl.gov

Article published by EDP Sciences c EDP Sciences, SMAI 2008

convergence analysis [4]. For compressible flow simulations, it helps to build discretizations that preserve total momentum and energy, see e.g. [13].

In articles [4–7], we developed new analysis of mimetic discretization methods for solving elliptic equations on polyhedral meshes. In the case of polyhedral meshes, these discretizations use one flux unknown per mesh face and one scalar unknown (pressure, temperature, etc.) per mesh element. In the case of generalized polyhedral meshes, three flux unknowns per curved faces are required to build a mimetic method [6,7]. For simplicial meshes, the family of MFD methods contains the mixed finite element method with the lower-order Raviart-Thomas elements [14]. In this article, we develop and analyze new nodal mimetic methods.

As in [5,7], here we build againa familyof discretization methods which is reduced to the standardP₁ finite element method [9] in the case of simplicial meshes. Under weak assumptions on the mesh regularity, each method in the family provides the first-order convergence rate in the mesh dependent energy norm.

There are a few advantages of using of a polyhedral mesh rather than an equivalent tetrahedral mesh with the same nodes. First, building of a conformal tetrahedral partition requires analysis of geometry which comes with additional computational overhead, especially for moving mesh methods. Indeed, for a given partition of polyhedron’s faces into triangles, a tetrahedral partition using only the polyhedron vertices may not exist!

Second, there are two ways to break a quadrilateral element into two triangles. The question of choosing the better partition is transformed to finding a proper member in a family of MFD methods, which provides a new numerical and analytical tool for future research. Third, a symmetric breaking may be required for special problems and it can be hardly done without using additional points. In shock calculations, where the nodal discretization of an elliptic equation is used to add a numerical viscosity to the system [8], a non-symmetric breaking may quickly destroy solution symmetry.

Recently, more general frameworks for mimetic discretizations have been developed using algebraic topology and cochain approximations of differential forms [1,3]. The key concept of [1] is a natural inner product on cochains which induces a combinatorial Hodge theory on the cochain complex. This article provides the constructive method for building one of the inner products.

The article outline is as follows. In Section 2, we define the elliptic problem. In Section 3, a class of admissible polyhedral meshes is described. In Section 4, the discrete operators are introduced. In Section 5, the discrete MFD method is formulated. In Section 6, first-order error estimates in energy norm are proved. The theoretical results are verified with numerical experiments in Section 7.

2. The continuous problem

Let Ω⊂R³ be a bounded Lipschitz polyhedron, g∈L²(Ω) andKbe a regular symmetric positive definite tensorK∈

W^1,∞(Ω)_3×3

. We look for the solutionu∈H₀¹(Ω) of the boundary value problem

−divKgradu=g in Ω. (2.1)

Extension to other boundary conditions is straightforward. This problem admits the variational formulation:

Find u∈H₀¹(Ω) such that

ΩKgradu·gradvdx=

Ωg vdx ∀v∈H₀¹(Ω), (2.2)

and we will discretize the problem in this form. For further use, we set:

u, v :=

ΩKgradu·gradvdx. (2.3)

In what follows, we assume that there exist two constantsκ andκsuch that:

κv^Tv≤v^TK(x)v≤κv^Tv ∀v∈R³, x∈Ω. (2.4) All constants in the estimates proposed in this paper will depend uponκ andκ.

Throughout this paper, we shall use · n, Dand| · |n,Dto denote the norm and semi-norm, respectively, on the Hilbert space Hⁿ(D), where D ⊂Ω. If D = Ω, subscriptD may be omitted. Finally, for further use, we set H¹(Ω) =H₀¹(Ω)∩C⁰(Ω).

3. The decomposition 3.1. Notation

We assume that on Ω we are given a sequence{Th}hof regular polyhedral meshesThin a sense of assumption (HG). This means that for each Th the domain Ω is split into n_P polyhedra P₁, ...,P_nP, with n_V vertices V₁, V₂, ..., V_nV.

For every geometric object Q(edge, face, polyhedron,etc.), we will denote its diameter byh_Q. Moreover, for every decompositionT_h we set

|h|_Th := max

P∈Th

h_P. (3.1)

Most of the times, the subscript will be omitted, and we shall simply write it as|h|. We denote byV(T_h),L(T_h) and F(T_h) the set of vertices, edges and faces of the decomposition T_h. The corresponding sets of internal vertices, edges and faces are denoted byV₀(T_h),L₀(T_h) andF₀(T_h), respectively.

3.2. Assumptions on the decompositions

As we shall see, the properties of the decompositions that are needed in our approach are very weak, meaning that we are allowed a great freedom in the choice of the shape of the polyhedral elements.

However, to make the description simpler, we will make some assumptions that arestronger than necessary.

It will nevertheless be clear in the following discussion that more general situations can be tackled.

We assume that there exist two positive real numbers N_s and ρ_s (the same for all the sequence) such that for every decompositionTh in the sequence we have:

(HG) (Regular polyhedral decompositionTh): There exists acompatiblesub-decompositionShintoshape- regulartetrahedra, such that

• Every polyhedronP ∈T_hadmits a decompositionS_h|P made of less thanN_stetrahedra;

• The shape-regularity of the tetrahedra K ∈ Sh is defined as follows [9]: the ratio between the radiusr_K of the inscribed sphere and the diameterh_K is bounded from below by constantρ_s:

r_K

h_K ≥ρ_s>0. (3.2)

It is important to point out, from the very beginning, that there is no need, in practice, to build the decompositionSh. We are only assuming thatit does exist(or, better, thatit could be built). In practice, we are essentially avoiding sequences of decompositions in which there are polyhedra that are, asymptotically, more and more hourglass-shaped or having thinner and thinner tails (see Fig.1): a choice which is hardly conceivable by any user of our numerical method.

3.3. Consequences of the assumption (HG)

The above requirements have several consequences, that can be easily verified. Among them we underline the following ones which will be used later.

(C1): There exist integer numbers N_f, N_e and N_v (depending only onN_s) such that every polyhedron in every decomposition has less than N_f faces, less thanN_eedges, and less thanN_v vertices.

Figure 1. Hourglass (left) and thin-tailed (right) polyhedra.

(C2): There exists a positive numberσ_s(depending only onN_sandρ_s) such that

h_e≥σ_sh_f ≥σ²_sh_P, (3.3)

whenevereis an edge off and f is a face ofP.

(C3): For every face f, there exists a decomposition of f in a finite number (≤ N_s) of regular shaped triangles, meaning that there exists a positive constant σ_φ depending only on ρ_s such that for every triangleT we have

r_T ≥σ_φh_T, (3.4)

wherer_T is the radius of the inscribed circle and h_T is the diameter ofT.

(C4): There exists a constantγ_a, depending only onN_sandρ_ssuch that for every polyhedronP and for every facef ofP we have theAgmon inequality

f

ϕ²dS≤γ_a

h⁻¹_P ||ϕ||²_L2(P)+h_P||gradϕ||²_L2(P)

. (3.5)

4. The discrete operators 4.1. The discrete unknowns

We consider now the setV(Th) ofverticesinThand the setNofnodal valuesonV(Th), that is the mappings from V(Th) into R. We will also consider the subset N0 of the nodal unknowns that vanish at the vertices V ∈∂Ω, that is

N0:={u∈Nsuch thatu(V) = 0 ∀V ∈V(Th), V ∈∂Ω}· (4.1) In a similar way we can consider the setEofedge unknownsas the mappings from the set of alloriented edges ofTh toR.

4.2. Restrictions of unknowns

When considering the restrictions ofunknowns(or, more generally,mappings) to a given geometrical objectQ we would generally use the subscript_|Q or simply_Q. For instance, bothN_|QandN_Qwill denote the restriction ofNto the nodes belonging toQ.

4.3. The GRAD operator

It will often be convenient to consider theGRAD operator, defined from the set of nodal unknownsNto the set of edge unknownsE as follows: for each elementu∈Nand for each edge ewith vertices (V₁, V₂), oriented

fromV₁ toV₂,

GRADu

|e=u(V₂)−u(V₁). (4.2)

Sometimes, it will be convenient to consider the application of theGRAD operator to a subset ofN. Given a polyhedronP ∈T_h, the operatorGRAD_P (defined exactly as in (4.2)) mapsN_P intoE_P. It is obvious that GRAD_P is a restriction ofGRAD, and it will also be denoted byGRAD when no confusion can occur. Finally we set

E₀={τ ∈E : τ(e) = 0 ∀e∈L(T_h), e∈∂Ω}·

It is easy to see thatGRAD maps alsoN₀→E₀.

If u_h ∈ N₀ is taken as an approximation of the scalar function u that solves (2.2), then GRADu_h is an element ofE₀and the operatorGRAD is related to the operatorgrad.

4.4. The norm in N

In the space of our unknownsN₀, we can now introduce the following norm:

|||v_h|||²:=

P∈Th

|||v_h|||²_P :=

P∈Th

h_P

e∈∂P

GRAD_Pv_h

|e². (4.3)

Note that, essentially from (3.3), the norm in the above (4.3) is equivalent to

|||v_h|||²

P∈Th

h³_P

e∈∂P

GRAD_Pv_h

|e

|h_e|

2

, (4.4)

mimicking theH₀¹(Ω)-norm.

4.5. The interpolation and reconstruction operators

We shall now define the natural interpolation operators Π_Nfrom H¹(Ω) to the discrete spaceN. For eachu in H¹(Ω) we define Π_Nu∈Nby

(Π_Nu)|_V =u(V) ∀V ∈V(Th). (4.5)

Let us consider the problem of finding continuous right inverses of the interpolation operator Π_N. We shall see in Appendix A that, under assumption(HG)on the decompositionTh, there exists a constantγdepending solely onN_s andρ_s, and a linear operatorv_h→R_Nv_h fromNintoH¹(Ω) with the following properties:

• For everyv_h∈N,

Π_NR_Nv_h=v_h. (4.6)

• For everyv_h∈N₀ and for every polyhedronP ∈T_h,

|R_Nv_h|²_1,P ≤γ|||v_h|||²_P. (4.7)

• For everyv_h∈N0, for every polyhedronP ∈Th and for every vertexV ∈P,

||R_Nv_h−v_h(V)||²_0,P ≤γ h²_P|||v_h|||²_P. (4.8) The existence of such a reconstruction is the only reason why we ask for the assumption(HG)to hold. It is clear then that the assumption(HG) is abundant. We have chosen it only to allow asimpleconstruction (see App. A), and in particular one that does not require too much functional analysis.

For further use, we note that from (4.7), (4.8) and (3.5), we immediately have the following result:

• For everyv_h∈N₀, for every polyhedronP ∈T_h, for every facef∈∂P, and for every vertexV ∈f

||R_Nv_h−v_h(V)||²_0,f ≤γ h_P|||v_h|||²_P (4.9) whereγ is a constant independent ofv_h,P andThand depending only onN_s,ρ_s.

Remark 4.1. Actually, the properties of the decomposition that we really need are(C1)–(C4)and (4.6)–(4.8), and we could take them as ourassumptionson the decomposition. Readers with a sufficiently solid background in Partial Differential Equations and Functional Analysis will soon recognize that these assumptions require very little regularity properties for the polyhedra inTh. However, in practice, all this generality is not needed, since the decompositionTh is essentially at the choice of the user.

5. The discrete problem

As it is reasonable to expect, the discrete version of the problem (2.2) will have the following structure Find u_h∈N₀ such that

GRADu_h,GRADv_h

E= g,v_h

N ∀v_h∈N0, (5.1)

where [·,·]_E is a suitable scalar product in E₀ and g,·

N a suitable linear functional on N₀, that need to be properly defined.

We shall use theH₀¹-type inner product u_h,v_h

inN₀defined by analogy with (2.3), u_h,v_h

:=

GRADu_h,GRADv_h

E, (5.2)

and write the discrete problem as follows:

Findu_h∈N0such that u_h,v_h

= g,v_h

N ∀v_h∈N₀. (5.3)

5.1. Numerical integration formulae

In order to define the terms appearing in the previous subsection we need to introduce suitable numerical integration formulae. Towards this aim, we choose a numerical integration formula for each elementP and for each facef. More precisely, for each polyhedronP with V_Pnodes, we assume that we are given V_P non-negative weights

ω_P¹, ..., ω_P^VP (5.4)

such that the corresponding numerical integration formula overP,

P

χdP ^VP

i=1

χ(V_Pⁱ)ωⁱ_P, (5.5)

is exact wheneverχ is a constant. Similarly, for every facef with V_f nodes, we assume that we are given V_f non-negative weights

ω_f¹, ..., ω_f^Vf (5.6)

such that the corresponding numerical integration formula overf,

f

χdS

Vf

i=1

χ(V_fⁱ)ω_fⁱ, (5.7)

is exact wheneverχ is a polynomial of degree≤1.

Remark 5.1. To derive an integration formula for a facef which is exact for linear functions, we could use a linear relation expressing the center of mass off in terms of its vertices. Since the integral of a linear function equals to the function value at the center of mass times|f|, the coefficients in the linear expression scaled by|f| define the weightsωⁱ_f. A similar argument would work for a polyhedron (although it will not be needed here).

5.2. Scalar products

Once we choose our two numerical integration formulae, we can choose the linear functional g,v_h

N:=

P

g¯_|P^VP

i=1

v_h(V_Pⁱ)ω_Pⁱ, (5.8)

where, in each elementP, we take ¯g_|P as the average ofg overP, that is

¯g_|P := 1

|P|

P

gdP. (5.9)

In the definition of the scalar product [·,·]_E, the tensorKenters into play and we need to construct a suitable approximation of it. We denote byK the piecewise constant tensor onThobtained by averaging each component ofKover each element P inTh. Thus,K theL²-projection ofKonto the space of piecewise constant tensors.

It is easy to see that

||K−K|| ∞,P ≤γh_P, (5.10)

where (as we shall assume from now on) γ is a generic constant depending only onK, onN_s and onρ_s. For each facef ofP, we define the outward unit normal vectorn^P_f and the co-normal vectors

ν^P_f :=K_|Pn^P_f and ν˜^P_f :=K_|Pn^P_f. (5.11) When no confusion will occur, we will simply useνf andνf instead ofν^P_f and ˜ν^P_f, respectively. Using (5.10) it is immediate to see that on each facef

||ν^P_f −ν˜^P_f||∞,f ≤γh_P. (5.12)

Now, for every polyhedronP, for every functionχ∈H¹(P), and for every polynomialpof degree≤1, the Gauss-Green formula is

P

K∇χ· ∇pdP =

∂P

χK∇ p·ndS=

f∈∂P

f

χ ∂p

∂νf dS. (5.13)

Inspired by (5.13) we make our final choice. For every polyhedron P we choose a symmetric bilinear form u,v

P onNP×NP verifying the following properties:

• For every polynomialpof degree≤1, settingp^I := Π_Np(as defined in (4.5)), we have v, p^I

P ≡

f∈∂P Vf

i=1

v(V_fⁱ) ∂p

∂νfω_fⁱ ∀v∈NP. (5.14)

• There exist two constantscandCindependent ofP and of hsuch that c|||v|||²_P ≤

v,v

P ≤C|||v|||²_P ∀v∈N_P. (5.15)

Then we set, in a natural way,

GRADu,GRADv

E≡ u,v

:=

P

u,v

P. (5.16)

In Section7, we show that there exist a family of bilinear forms with the above properties. For the moment, only (5.15) and (5.16) are needed for the convergence analysis.

Remark 5.2. Let p and q be polynomials of order ≤ 1. Taking into account (5.13) and the fact that the integration formula (5.7) is exact for polynomials of degree≤1, we have immediately that (5.14) implies

Π_Np,Π_Nq

P =

P

K∇ p· ∇qdP ≡

PK∇p· ∇qdP, (5.17)

so that the assumption of symmetry and (5.14) are compatible.

5.3. Mimetic finite differences

It is easy to put all this in the framework of Mimetic Finite Differences. The gradient operator GRAD is the primary operator, and the divergence operator DIV_K is the derived operator. Operators GRAD and DIV_Kapproximate operatorsgrad·and div(K·), respectively. Let [·,·]_N be a suitable scalar product inN. The divergence operator is formally defined through the discrete Green formula:

DIV_KG_h,v_h

N=−

G_h,GRADv_h

E ∀G_h∈E₀, v_h∈N₀. (5.18) Then, the MFD method is

⎧⎪

⎨

⎪⎩

Findu_h∈N₀ andG_h∈E₀ such that G_h=GRADu_h,

DIV_KG_h=−Π_N0(g).

(5.19) For a more general framework on Cochain approximations of Differential Forms (that however does not include the present discussion), see [3].

6. Error estimates

We point out that our choices of the scalar product

·,·

and of the linear functional g,·

Ndepend on three choices:

• the integration formula (5.5) in each polyhedronP;

• the integration formula (5.7) on each facef;

• the bilinear forms u,v

P for eachP.

All the properties of the numerical scheme, includinga priorierror estimates, will be derived by the properties of the integration formulae and of the bilinear forms defining the scalar product.

Letu_h be the solution of the discrete problem (5.3) andube the solution of the continuous problem (2.2).

We assume that u∈H¹(Ω) and setu^I = Π_Nu. We shall also consider the discontinuous function w which is linear in each polyhedronP ofTh. The restriction ofwtoP, denoted byw_P, is defined as theL²(P)-projection ofuonto the space of polynomials of degree≤1. We shall also denote byw^I_P the element ofNP that assumes the values ofw_P at the nodes of P.

Finally, we set

δ:=u_h−u^I (6.1)

and estimateδin the norm||| · |||.

6.1. Six easy pieces

c|||δ|||² = c

P |||δ|||²_P (use (5.15) and (5.16))

≤

δ, δ

(use (6.1))

=

u_h, δ

− u^I, δ

(use (5.3))

=

g, δ

N− u^I, δ

≡I− u^I, δ

.

(6.2)

On the other hand, starting with (5.16), we get u^I, δ

=

P

u^I, δ

P (add and subtractw^I_P)

=

P

u^I−w^I_P, δ

P +

P

w^I_P, δ

P

≡ II+

P

w^I_P, δ

P (use (5.14))

= II+

P

f∈∂P Vf

i=1

δ(V_fⁱ)∂w_P

∂νf ω_i^f.

(6.3)

Moreover, for everyP ∈Th and for every facef ∈∂P, using that (5.7) is exact on constants, we have

Vf

i=1

δ(V_fⁱ)∂w_P

∂ν_fω^f_i =

Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]∂w_P

∂ν_fω^f_i +

Vf

i=1

δ(V_f¹)∂w_P

∂ν_f ω^f_i

=

Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]∂w_P

∂ν_fω^f_i +

f

δ(V_f¹)∂w_P

∂ν_f dS.

Thus,

P

f∈∂P Vf

i=1

δ(V_fⁱ)∂w_P

∂νfω_i^f =

P

f∈∂P Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]∂w_P

∂νfω^f_i +

P

f∈∂P

fδ(V_f¹)∂w_P

∂νf dS

≡III+

P

f∈∂P

f

δ(V_f¹)∂w_P

∂ν_f dS.

We can now add and subtract a functionR_N(δ)∈H¹(Ω) that, for the moment, is justany functioninH¹(Ω) having the same value asδ at the nodes. Later, we shall require that it satisfies (4.6)–(4.8). We obtain

P

f∈∂P

fδ(V_f¹)∂w_P

∂νf dS (add and subtractR_N(δ))

=

P

f∈∂P

f

[δ(V_f¹)−R_N(δ)]∂w_P

∂νf dS+

P

f∈∂P

f

R_N(δ)∂w_P

∂νf dS

≡ IV +

P

f∈∂P

f

R_N(δ)∂w_P

∂νf dS (use (5.13))

= IV +

P

P

K∇R_N(δ)· ∇w_PdP.

(6.4)

Finally,

P

P

K∇R_N(δ)· ∇w_PdP (add and subtract∇u)

=

P

P

K∇R_N(δ)· ∇(wP−u) dP+

Ω

K∇R_N(δ)· ∇udP

≡ V +

Ω

K∇R_N(δ)· ∇udP (add and subtractK)

= V +

Ω(K−K)∇R_N(δ)· ∇udP+

ΩK∇R_N(δ)· ∇udP (use (2.2))

≡ V +V I+

Ω

gR_N(δ) dP.

(6.5)

Collecting the above equations, we have c|||δ|||²≤

g, δ

N−

Ω

gR_N(δ) dP

−

P

u^I−w^I_P, δ

P

−

P

f∈∂P Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]∂w_P

∂ν_fω^f_i −

P

f∈∂P

f

[δ(V_f¹)−R_N(δ)]∂w_P

∂ν_f dS

−

P

P

K∇R_N(δ)· ∇(w_P−u) dP−

Ω(K −K)∇R_N(δ)· ∇udP. (6.6) In the next section, we shall estimate separately each of the Six Easy Pieces in the above equation.

6.2. Four useful lemmata

We shall need four simple lemmata. Letω_Pⁱ and ωⁱ_f be such that (5.5) and (5.7) are exact for constant and linear functions, respectively.

Lemma 6.1. There exist a constant γ₁, depending only on N_s andρ_s, such that, for every polyhedron P, for every vertex V_P¹ ofP, and for every χin NP:

V_Pⁱ∈P

[χ(V_P¹)−χ(V_Pⁱ)]²ω_Pⁱ ≤γ₁h²_P|||χ|||²_P. (6.7)

Proof. Since all the ωⁱ_P are non-negative and their sum is |P|, then every ωⁱ_P is bounded by h³_P. Then the triangle inequality, and the fact that in every P we have less that N_v vertices (from (C1)), easily imply the

result.

Lemma 6.2. There exists a constantγ₂, depending only onN_s andρ_s, such that for every polyhedron P, for every facef ofP, for every vertexV_f¹ of f, and for everyχ inNP:

V_fⁱ∈f

[χ(V_f¹)−χ(V_fⁱ)]²ω_fⁱ ≤γ₂h_P|||χ|||²_P. (6.8)

Proof. The proof is the same as before; but this time everyωⁱ_f is bounded byh²_P.

Lemma 6.3. Let ϕ ∈H²(Ω)∩H₀¹(Ω), and let ϕ^I be the continuous piecewise linear interpolant of ϕ on the gridSh. Letψ, piecewise linear (and discontinuous) on the partitionTh, be defined as follows: for everyP ∈Th, ψ_P is theL²(P)-projection of ϕover the polynomials of degree≤1. For everyP ∈Th we denote as well (with an abuse of notation) byϕ^I andψ_P the elements inNP that assume the same values ofϕandψ_P (respectively) at the vertices ofP. Then

|||ϕ^I −ψ_P|||²_P ≤ γ₃h²_P|ϕ|²_2,P (6.9) whereγ₃ depends only on N_sandρ_s.

Proof. As bothϕ^I andψ_P are piecewise linear on the regular gridS_h|P, we can use all the classical finite element tools (including inverse inequalities). In particular,

ϕ^I−ψ_P, ϕ^I−ψ_P1/2

P =|||ϕ^I−ψ_P|||_P ≤ γ|ϕ^I −ψ_P|_1,P. (6.10) Moreover,

|ϕ^I−ψ_P|_1,P ≤γh⁻¹_P ||ϕ^I−ψ_P||_0,P ≤γh⁻¹_P

||ϕ^I −ϕ||_0,P+||ϕ−ψ_P||_0,P

≤γh_P|ϕ|_2,P.

The last step above requires additional comments. At a first sight, the estimate of the term||ϕ−ψ_P||0,P may depend upon the shape ofP which is a generic polyhedron. However one can argue as follows. PolyhedronP is the star-shaped domain with respect to a ball of radius ρ^∗_sh_P where the constant ρ^∗_s depends on various constants appeared in(C1)–(C3). Is also satisfies the strong cone condition. Then the result follows from the

revised Bramble-Hilbert lemma for star-shaped domains [2].

Lemma 6.4. In the same assumptions of the previous lemma, for each internal face f (common to the two polyhedra P₁ andP₂), we define

j_f(ψ) :=|∇ψ_P1·ν˜^P_f¹+∇ψ_P2·ν˜^P_f²|. (6.11)

Then,

f∈F0(Th)

||j_f(ψ)||²_0,f≤γ₄

P∈Th

h_P|ϕ|²_2,P, (6.12)

whereγ₄ depends only on N_s,ρ_s, andK.

Proof. By our assumptions,K∇ϕis continuous, so that, recalling the definition (5.11), on each internal face it holds:

∇ϕ·ν^P_f¹+∇ϕ·ν^P_f² = 0. (6.13)

Using (5.12) and the Agmon inequality (3.5), we have

||∇ϕ·ν˜^P_f¹+∇ϕ·ν˜^P_f²||²_0,f ≤γ

h_P1||ϕ||²_2,P1+h_P2||ϕ||²_2,P2

. (6.14)

We have, with obvious notation

j_f(ψ)≤ |∇(ψP1−ϕ)·ν˜^P_f¹|+|∇(ψP2−ϕ)·ν˜^P_f²|+|∇ϕ·ν˜^P_f¹+∇ϕ·ν˜^P_f²|. (6.15) Using again the Agmon inequality (3.5) we have (for i= 1,2):

||∇(ψ_Pi−ϕ)·νf||²_0,f ≤γνf² h⁻¹_P

i|ψ_Pi−ϕ|²_1,Pi+h_Pi|ϕ|²_2,Pi

≤γh_Pi|ϕ|²_2,Pi, (6.16)

and (6.12) follows.

6.3. Estimate of each piece

We begin by observing that for eachP and for each vertexW_P ofP, we easily have g¯_P

i

ω_Pⁱ δ(W_P) = ¯g_P

P

δ(W_P)dP =

P

gδ(W_P)dP. (6.17)

Hence, the First Pieceis bounded by (g, δ)_N−

ΩgR_N(δ)dP (use (5.8))

=

P

¯g_P

i

ωⁱ_Pδ(V_Pⁱ)−

Ω

gR_N(δ)dP (add and subtract

P

gδ(V_P¹)dP)

=

P

¯g_P

i

ωⁱ_P(δ(V_Pⁱ)−δ(V_P¹))−

P

P

g(R_N(δ)−δ(V_P¹))dP (use (6.7))

≤γ||g||_0,Ω

P

h²_P|||δ|||²_P1/2

−

P

P

g(R_N(δ)−δ(V_P¹))dP (use (4.8))

≤γ||g||0,Ω

P

h²_P|||δ|||²_P1/2

≤γ||g||0,Ω|h| |||δ|||.

Using (6.9), we see that theSecond Piece is bounded by

P

u^I−w^I_P, δ

P

≤

P

|||u^I −w^I_P|||_P|||δ|||_P ≤γ

P

h²_P|u|²_{2,P 1/2}

|||δ||| ≤γ|h| |u|_2,Ω|||δ|||.

In order to estimate the third piece, we remark first that δ is equal to zero on each vertex belonging to ∂Ω.

Hence, we can consider only theinternal faces. Taking also into account thatδis single valued, we first rearrange terms to get

P

f∈∂P Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]∂w_P

∂νfω^f_i

≤

f∈F0(Th) Vf

i=2

δ(V_fⁱ)−δ(V_f¹)j_f(w)ω^f_i. (6.18) Then, we use Cauchy-Schwartz, estimate (6.8), the fact that the integration formula is exact on constants, and finally (6.12) to get

f∈F0(Th) Vf

i=2

|δ(V_fⁱ)−δ(V_f¹)|j_f(w)ω^f_i

≤

⎛

⎝

P

f∈∂P Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]²ω_fⁱ

⎞

⎠

1/2⎛

⎝

f∈F0(Th) Vf

i=1

|j_f(w)|²ω_i^f

⎞

⎠

1/2

≤γ

P

h_P|||δ|||²_P

_1/2 ⎛

⎝

f∈F0(Th)

||j_f(w)||²_0,f

⎞

⎠

1/2

≤γ

P

h_P|||δ|||²_{P 1/2}

P

h_P|u|²_{2,P 1/2}

≤ γ|h| |||δ||| |u|2,Ω

that joined with (6.18) gives the estimate of theThird Piece:

P

f∈∂P Vf

i=2

[δ(V_fⁱ)−δ(V_f¹)]∂w_P

∂ν_f ω_i^f

≤ γ|h| |||δ||| |u|2,Ω. (6.19) Following essentially the estimate of the third piece and just using (4.9) instead of (6.8), we can estimatethe Fourth Piece:

P

f∈∂P

f[δ(V_f¹)−R_N(δ)]∂w_P

∂νf dP ≤

f∈F0(Th)

f|δ(V_f¹)−R_N(δ)| |j_f(w)|dP

≤ γ

P

h_P|||δ|||²_P1/2

P

h_P|u|²_2,P1/2

≤ γ|h||||δ||| |u|2,Ω.

(6.20)

We can estimate theFifth Piece using (4.7) and the usual approximation results:

P

P

K∇R_N(δ)· ∇(w_P−u) dP

≤γ|R_N(δ)|_1,Ω

P

h²_P|u|²_2,P1/2

≤γ|h| |||δ||| |u|_2,Ω. Finally, for the Sixth Piece, we use (5.10) and (4.7) to obtain:

Ω(K−K)∇R_N(δ)· ∇udP ≤γ|h| |||δ||| |u|1,Ω. Thus, we proved the following theorem.

Theorem 6.5. LetΩbe a bounded Lipschitz polyhedron andKbe aW^1,∞(Ω)symmetric tensor. Furthermore, let the sequence of decompositions Th satisfy assumption (HG), and the discrete inner product (5.16) satisfy (5.14)and (5.15). Finally, letuandu_h be solutions of(2.2)and(5.3), respectively, andu^I = Π_Nu. Then,

|||u^I−u_h||| ≤γ|h|(||g||0,Ω+|u|1,Ω+|u|2,Ω), whereγ depends only on N_s,ρ_s andK.

7. Numerical results 7.1. Algebraic issues

In this section, we construct explicitly the bilinear form v,u

P onNP×NP verifying (5.14) and (5.15).

Let the polyhedronP haven_v vertices. Our definition of the bilinear form implies that we have to construct a symmetric positive definiten_v×n_v matrixMP such that

v, u

P =v^TMPu,

where v anduare vectors in^nv with entries u(V) and v(V), respectively,V ∈P. In each N_P we construct a new basis as follows. The first four elements of the new basis will be the nodal values of the polynomials of degree≤1:

B₁= Π_N1, B_i+1= Π_N(x_i−x^P_i ) i= 1,2,3, (7.1)

wherex^P is the barycenter ofP. Using (5.17), we have immediately B₁,v

P = 0 ∀v∈N_P (7.2)

and

B_i+1, B_j+1

P =K_i,j|P| i, j = 1,2,3. (7.3)

Moreover, using (5.14), we have that the scalar product v, B_i

P can be computed in a unique way, for every v∈NP andi= 1, ...,4. Hence, the problem offinding linearly independent B_k,k= 5, ..., n_v, such that

B_i, B_k

P = 0 (7.4)

for i= 1, ...,4 and k= 5, ..., n_v makes perfect sense. To simplify the following discussion, we can also assume that theB_k are normalized by

|||B_k|||²_P =|P| k= 5, ..., n_v.

LetBi, i= 1, . . . , n_v, be vectors in^nv with entries B_k(V), for any vertexV ofP. If ˜MP is the matrix that represents our scalar product in the new basis B1, . . . ,Bnv, from (5.17) and (7.4) we already know explicitly the first four lines and the first four columns of ˜MP. Hence, we only have to decide the (n_v−4)×(n_v−4) block at the bottom-right. It is easy to see that every symmetric and positive definite matrix Uthat satisfies (5.15) will do. For instance we can take

U= trace(K) |P|Inv−4

whereInv−4 is the identity matrix inn_v−4 dimensions. Hence, ˜MP will be given by M˜_P =

⎡

⎣ 0 0 0 0 K| P| 0

0 0 U

⎤

⎦. (7.5)

Returning to the original basis, we get

MP =B^−TM˜PB⁻¹, B= [B1, . . . ,Bnv]. (7.6) Remark 7.1. In the case of tetrahedral meshes, the matrix MP coincides with the stiffness matrix in the standardP₁ finite element method.

In practice, we avoid inversion of matrixBusing different representation of a family of admissible matricesM_P. Following essentially [5], we define vectors A_i,i= 2,3,4, as follows:

A_i=M_PB_i.

The vectorsA_i can be calculated directly from the right-hand side of (5.14). Formula (7.3) implies that A^T_i+1Bj+1=Ki,j|P|, i, j= 1,2,3.

LetA1be an_v×3 matrix with columnsAi,i= 2,3,4, andB1be an_v×4 matrix with columnsBi,i= 1, . . . ,4.

Furthermore, let D₁ be an_v×(n_v−4) matrix with columns that span the null space of B^T₁, i.e. B^T₁ D₁ = 0.

Then, the general form of the matrixM_P is (see [5] for more details) MP = 1

|P|A1K⁻¹A^T₁ +D1U D˜ ^T₁, (7.7) where ˜Uis an arbitrary (n_v−4)×(n_v−4) symmetric positive definite matrix.