A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
Jerome Droniou, Robert Eymard, Thierry Gallouët, Raphaele Herbin
aa r X i v : . [ m a t h . NA ] D ec A unified approach to Mimetic Finite Difference, Hybrid FiniteVolume and Mixed Finite Volume methods J´erˆome Droniou , Robert Eymard , Thierry Gallou¨et , Rapha`ele Herbin .October 26, 2018 Abstract
We investigate the connections between several recent methods for the discretization of aniso-tropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Differencescheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical upto some slight generalizations. As a consequence, some of the mathematical results obtained for each ofthe method (such as convergence properties or error estimates) may be extended to the unified commonframework. We then focus on the relationships between this unified method and nonconforming FiniteElement schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operatorclose to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also showthat for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unifiedmethod boils down to the well-known efficient two-point flux Finite Volume scheme.
A benchmark was organized at the last FVCA 5 conference [19] in June 2008 to test the recently developedmethods for the numerical solution of heterogeneous anisotropic problems. In this benchmark and in thispaper, we consider the Poisson equation with homogeneous boundary condition − div(Λ ∇ p ) = f in Ω , (1.1a) p = 0 on ∂ Ω , (1.1b)where Ω is a bounded open subset of R d ( d ≥ → M d ( R ) is bounded measurable symmetric anduniformly elliptic (i.e. there exists ζ > x ∈ Ω and all ξ ∈ R d , Λ( x ) ξ · ξ ≥ ζ | ξ | ) and f ∈ L (Ω).The results of this benchmark, in particular those of [6, 15, 21, 22] seem to demonstrate that the behaviorof three of the submitted methods, namely the Hybrid Finite Volume method [14, 17, 16], the MimeticFinite Difference method [2, 4], and the Mixed Finite Volume method [10, 11] are quite similar in anumber of cases (to keep notations light while retaining the legibility, in the following we call “Hybrid”,“Mimetic” and “Mixed” these respective methods).A straightforward common point between these methods is that they are written using a general partitionof Ω into polygonal open subsets and that they introduce unknowns which approximate the solution p and the fluxes of its gradient on the edges of the partition. However, a comparison of the methods is stilllacking, probably because their mathematical analysis relies on different tool boxes. The mathematicalanalysis of the Mimetic method [2] is based on an error estimate technique (in the spirit of the mixedfinite element methods). For the Hybrid and the Mixed methods [16, 10], the convergence proofs relyon discrete functional analysis tools. The aim of this paper is to point out the common points of thesethree methods. To this purpose, we first gather, in Table 1 of Section 2, some definitions and notationsassociated with each method, and we present the three methods as they are introduced in the literature;we also present a generalized or modified form of each of the methods. The three resulting methods are This work was supported by GDR MOMAS CNRS/PACEN Universit´e Montpellier 2, Institut de Math´ematiques et de Mod´elisation de Montpellier, CC 051, Place Eug`ene Bataillon,34095 Montpellier cedex 5, France. email: [email protected] Universit´e Paris-Est, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UMR 8050, 5 boulevard Descartes,Champs-sur-Marne, 77454 Marne-la-Vall´ee Cedex 2, France. email:
[email protected] L.A.T.P., UMR 6632, Universit´e de Provence, [email protected] L.A.T.P., UMR 6632, Universit´e de Provence, [email protected] p -Laplacian type [9, 12], non-linear coupled problems [7], etc.However, the main ideas to apply these methods in such more complex situations stem from the studyof their properties on the linear diffusion equation. The unifying framework which is proposed here hasa larger field of application than (1.1); it facilitates the transfer of the ideas and techniques used for onemethod to another and can also give some new leads for each method. We first provide the definitions and notations associated with each method (in order for the readers whoare familiar with one or the other theory to easily follow the subsequent analysis, we shall freely use oneor the other notation (this of course yields some redundant notations).
Mimetic notation Finite Volume notation
Partition of Ω in polygonal sets Ω h M Elements of the partition (grid cells)
E K (“control volume”)Set of edges/faces of a grid cell ∂E , or numberedfrom 1 to k E E K Edges/faces of grid cell e σ
Space of discrete p unknowns (piecewise approxi-mations of p on the partition) Q h H M Approximation of the solution p on the grid cell E = K p E p K Discrete flux: approximation of | e | R e − Λ ∇ p · n eE (with e = σ an edge of E = K and n eE = n K,σ theunit normal to e outward E ) F eE F K,σ
Space of discrete fluxes (approximations of | e | R e/σ − Λ ∇ p · n ) X h F Table 1: Usual notations and definitions in Mimetic and Finite Volume frameworks.
Remark 2.1
In the usual Finite Volume literature, the quantity F K,σ is usually rather an approximationof R σ − Λ ∇ p · n K,σ than | σ | R σ − Λ ∇ p · n K,σ ; we choose here the latter normalization in order to simplifythe comparison.
In all three methods, a natural condition is imposed on the gradient fluxes (called conservativity in FiniteVolume methods and continuity condition in Mimetic methods): for any interior edge σ (or e ) betweentwo polygons K and L (or E and E ), F K,σ + F L,σ = 0 , (or F eE = − F eE ) . (2.1)This condition is included in the definition of the discrete flux space F (or X h ).2he Mimetic, Hybrid and Mixed methods for (1.1) all consist in seeking p ∈ H M (or Q h ) and F ∈ F (or X h ), which approximate respectively the solution p and its gradient fluxes, writing equations on theseunknowns which discretize the continuous equation (1.1). Each method is in fact a family of schemesrather than a unique scheme: indeed, there exists some freedom on the choice of some of the parametersof the scheme (for instance in the stabilization terms which ensure the coercivity of the methods). The standard Mimetic method first consists in defining, from the Stokes formula, a discrete divergenceoperator on the space of the discrete fluxes: for G ∈ X h , DIV h G ∈ Q h is defined by( DIV h G ) E = 1 | E | k E X i =1 | e i | G e i E . (2.2)The space Q h of piecewise constant functions is endowed with the usual L inner product[ p, q ] Q h = X E ∈ Ω h | E | p E q E and a local inner product is defined on the space of fluxes unknowns of each element E :[ F E , G E ] E = F tE M E G E = k E X s,r =1 M E,s,r F e s E G e r E , (2.3)where M E is a symmetric definite positive matrix of order k E . Each local inner product is assumed tosatisfy the following discrete Stokes formula (called Condition ( S2 ) in [2, 4]): ∀ E ∈ Ω h , ∀ q affine function, ∀ G ∈ X h : [(Λ ∇ q ) I , G ] E + Z E q ( DIV h G ) E dV = k E X i =1 G e i E Z e i q d Σ (2.4)where ((Λ ∇ q ) I ) eE = | e | R e Λ E ∇ q · n eE d Σ and Λ E is the value, assumed to be constant, of Λ on E (if Λ isnot constant on E , one can take Λ E equal to the mean value of Λ on E ). Remark 2.2
Note that Condition ( S1 ) of [2, 4] is needed in the convergence study of the method, butnot in its definition; therefore it is only recalled in Section 4, see (4.49) . The local inner products (2.3) allow us to construct a complete inner product [
F, G ] X h = P E ∈ Ω h [ F, G ] E ,which in turn defines a discrete flux operator G h : Q h → X h as the adjoint operator of DIV h : for all F ∈ X h and p ∈ Q h , [ F, G h p ] X h = [ p, DIV h F ] Q h (notice that this definition of G h p takes into account the homogeneous boundary condition (1.1b)). Usingthese definitions and notations, the standard Mimetic method then reads: find ( p, F ) ∈ Q h × X h suchthat DIV h F = f h , F = G h p (2.5)where f h is the L projection of f on Q h , or in the equivalent weak form: find ( p, F ) ∈ Q h × X h suchthat ∀ G ∈ X h : [ F, G ] X h − [ p, DIV h G ] Q h = 0 , (2.6) ∀ q ∈ Q h : [ DIV h F, q ] Q h = [ f h , q ] Q h . (2.7)3he precise definition of the Mimetic method requires to choose the local matrices M E defining the localinner products [ · , · ] E . It can be shown [4] (see also Lemma 6.2 in the appendix) that this matrix definesan inner product satisfying (2.4) if and only if it can be written M E N E = ¯ R E or equivalently M E = 1 | E | ¯ R E Λ − E ¯ R TE + C E U E C TE (2.8)where ¯ R E is the k E × d matrix with rows ( | e i | (¯ x e i − ¯ x E ) T ) i =1 ,k E , with ¯ x e the center of gravity of the edge e and ¯ x E the center of gravity of the cell E, (2.9) C E is a k E × ( k E − d ) matrix such that Im( C E ) = (Im( N E )) ⊥ ,with N E the k E × d matrix with columns( N E ) j = (Λ E ) j · n e E ...(Λ E ) j · n e kE E for j = 1 , . . . , d, (Λ E ) j being the j -th column of Λ E (2.10) U E is a ( k E − d ) × ( k E − d ) symmetric definite positive matrix . (2.11)Here, we consider a slightly more general version of the Mimetic method, replacing ¯ x E by a point x E which may be chosen different from the center of gravity of E . We therefore take M E = 1 | E | R E Λ − E R TE + C E U E C TE (2.12)where R E is the k E × d matrix with rows ( | e i | (¯ x e i − x E ) T ) i =1 ,k E ,with ¯ x e the center of gravity of the edge e and x E any point in the cell E . (2.13)The other matrices C E and U E remain given by (2.10) and (2.11). The choice [(2.12),(2.13)] of M E nolonger gives, in general, an inner product [ · , · ] E which satisfies (2.4), but it yields a generalization of thisassumption; indeed, choosing a weight function w E : E → R such that Z E w E ( x ) d x = | E | and Z E xw E ( x ) d x = | E | x E , (2.14)we prove in the appendix (Section 6.1) that the matrix M E can be written (2.12) with R E defined by(2.13) if and only if the corresponding inner product [ · , · ] E satisfies ∀ E ∈ Ω h , ∀ q affine function, ∀ G ∈ X h : [(Λ ∇ q ) I , G ] E + Z E q ( DIV h G ) E w E dV = k E X i =1 G e i E Z e i q d Σ . (2.15) Definition 2.3 (Generalized Mimetic method)
The Generalized Mimetic scheme for (1.1) reads:Find ( p, F ) ∈ Q h × X h which satisfies [(2.2),(2.3),(2.6),(2.7),(2.12),(2.13)] .Its parameters are the family of points ( x E ) E ∈ Ω h (which are freely chosen inside each grid cell) and thefamily of stabilization matrices ( C E , U E ) E ∈ Ω h satisfying (2.10) and (2.11) . .2 The Hybrid method The standard Hybrid method is best defined using additional unknowns p σ playing the role of approx-imations of p on the edges of the discretization of Ω; if E is the set of such edges, we let e H M be theextension of H M consisting of vectors e p = (( p K ) K ∈M , ( p σ ) σ ∈E ). It will also be useful to consider thespace e H K of the restrictions e p K = ( p K , ( p σ ) σ ∈E K ) to a control volume K and its edge of the elements e p ∈ e H M . A discrete gradient inside K is defined for e p K ∈ e H K by ∇ K e p K = 1 | K | X σ ∈E K | σ | ( p σ − p K ) n K,σ . (2.16)The definition of this discrete gradient stems from the fact that, for any vector e , any control volume K and any x K ∈ R d , we have | K | e = X σ ∈E K | σ | e · (¯ x σ − x K ) n K,σ (2.17)where ¯ x σ is the center of gravity of σ and x K is any point of K . Hence the discrete gradient is consistentin the sense that if p σ = ψ ( x σ ) and p K = ψ ( x K ) for an affine function ψ on K , then ∇ K e p K = ∇ ψ on K (note that this consistency is not sufficient to ensure strong convergence, and in fact, only weakconvergence of the discrete gradient will be obtained). A stabilization needs to be added to the discretegradient (2.16) in order to build a discrete coercive bilinear form expected to approximate the bilinearform ( u, v ) R Ω Λ ∇ u · ∇ v d x occurring in the weak formulation of (1.1). Choosing a point x K for eachcontrol volume K (for instance the center of gravity, but this is not mandatory), and keeping in mind that p K is expected to be an approximation of the solution p of (1.1) at this point, a second-order consistencyerror term S K ( e p K ) = ( S K,σ ( e p K )) σ ∈E K is defined by S K,σ ( e p K ) = p σ − p K − ∇ K e p K · (¯ x σ − x K ) . (2.18)Note that thanks to (2.17), X σ ∈E K | σ | S K,σ ( e p K ) n K,σ = 0 , (2.19)and that S K,σ ( e p K ) = 0 if p σ = ψ ( x σ ) and p K = ψ ( x K ) for an affine function ψ on K .The fluxes ( F K,σ ) σ ∈E K on the boundary of a control volume K associated with some e p ∈ e H M are thendefined by imposing that ∀ K ∈ M , ∀ e q K = ( q K , ( q σ ) σ ∈E K ) ∈ e H K : X σ ∈E K | σ | F K,σ ( q K − q σ ) = | K | Λ K ∇ K e p K · ∇ K e q K + X σ ∈E K α K,σ | σ | d K,σ S K,σ ( e p K ) S K,σ ( e q K ) (2.20)where Λ K is the mean value of Λ on K , d K,σ is the distance between x K and the hyperplane containing σ and α K,σ >
0. Note that F K,σ is uniquely defined by (2.20), since this equation is equivalent to | σ | F K,σ = | K | Λ K ∇ K e p K · ∇ K e q K + X σ ∈E K α K,σ | σ | d K,σ S K,σ ( e p K ) S K,σ ( e q K ) (2.21)where e q K satisfies q K − q σ = 1 and q K − q σ ′ = 0 for σ ′ = σ. To take into account the boundary condition(1.1b), we denote by e H M , = { e p ∈ e H M such that p σ = 0 if σ ⊂ ∂ Ω } and the Hybrid method can then bewritten: find e p ∈ e H M , such that, with F K,σ defined by (2.20), ∀ e q ∈ e H M , : X K ∈M X σ ∈E K | σ | F K,σ ( q K − q σ ) = X K ∈M q K Z K f. (2.22)5n particular, taking q K = 0 for all K , and q σ = 1 for one σ and 0 for the others in (2.22) yields that F satisfies (2.1); once this conservativity property is imposed by requiring that F ∈ F , we may eliminatethe q σ from (2.22) and reduce the Hybrid method to: find ( e p, F ) ∈ e H M , × F satisfying (2.20) and ∀ K ∈ M : X σ ∈E K | σ | F K,σ = Z K f. (2.23)This last equation is the flux balance, one of the key ingredients of the finite volume methods.Let us now introduce a generalization of the Hybrid method for the comparison with the other methods.As previously mentioned, the stabilization term S K is added in (2.20) in order to obtain a coercive scheme(the sole discrete gradient ( ∇ K e p K ) K ∈M does not allow to control p itself); the important characteristicof S K is that it yields a coercive bilinear form, so that we may in fact replace (2.20) by the more generalequation ∀ K ∈ M , ∀ e q K ∈ e H K : X σ ∈E K | σ | F K,σ ( q K − q σ ) = | K | Λ K ∇ K e p K · ∇ K e q K + X σ,σ ′ ∈E K B HK,σ,σ ′ S K,σ ( e p K ) S K,σ ′ ( e q K )= | K | Λ K ∇ K e p K · ∇ K e q K + S K ( e q K ) T B HK S K ( e p K ) , (2.24)where B HK,σ,σ ′ are the entries of a symmetric definite positive matrix B HK (the superscript H refers to theHybrid method). Definition 2.4 (Generalized Hybrid method)
A Generalized Hybrid scheme for (1.1) reads:Find ( e p, F ) ∈ e H M , × F which satisfies [(2.16),(2.18),(2.23),(2.24)] .Its parameters are the family of points ( x K ) K ∈M (which are freely chosen inside each grid cell) and thefamily ( B HK ) K ∈M of symmetric definite positive matrices. Remark 2.5
Another presentation of the Generalized Hybrid method is possible from (2.22) and (2.24) by eliminating the fluxes: Find e p ∈ e H M , such that ∀ e q ∈ e H M , : X K ∈M | K | Λ K ∇ K e p K · ∇ K e q K + X K ∈M S K ( e q K ) T B HK S K ( e p K ) = X K ∈M q K Z K f. (2.25) As in the Hybrid method, we use the unknowns e p in e H M for the Mixed method (that is to say approximatevalues of the solution to the equation inside the control volumes and on the edges), and fluxes unknownsin F . However, contrary to the Hybrid method, the discrete gradient is no longer defined from p , butrather from the fluxes, using the dual version of (2.17), that is: | K | e = X σ ∈E K | σ | e · n K,σ (¯ x σ − x K ) . (2.26)For F ∈ F , we denote F K = ( F K,σ ) σ ∈E K its restriction to the edges of the control volume K and F K isthe set of such restrictions; if F K ∈ F K , then v K ( F K ) is defined by | K | Λ K v K ( F K ) = − X σ ∈E K | σ | F K,σ (¯ x σ − x K ) (2.27)6here, again, Λ K is the mean value of Λ on K , ¯ x σ is the center of gravity of σ and x K a point choseninside K . Recalling that F K,σ is an approximation of | σ | R σ − Λ ∇ p · n K,σ , Formula (2.26) then shows that v K ( F K ) is expected to play the role of an approximation of ∇ p in K .The Mixed method then consists in seeking ( e p, F ) ∈ e H M , × F (recall that F ∈ F satisfies (2.1), andwe impose p σ = 0 if σ ⊂ ∂ Ω) which satisfies the following natural discrete relation between p and thisdiscrete gradient, with a stabilization term involving the fluxes and a positive parameter ν > ∀ K ∈ M , ∀ σ ∈ E K : p σ − p K = v K ( F K ) · (¯ x σ − x K ) − ν diam( K ) F K,σ (2.28)and the flux balance: ∀ K ∈ M : X σ ∈E K | σ | F K,σ = Z K f. (2.29)Multiplying, for any G K ∈ F K , each equation of (2.28) by | σ | G K,σ and summing on σ ∈ E K , we obtain ∀ K ∈ M , ∀ G K ∈ F K : X σ ∈E K ( p K − p σ ) | σ | G K,σ = | K | v K ( F K ) · Λ K v K ( G K ) + X σ ∈E K ν diam( K ) | σ | F K,σ G K,σ . The term P σ ∈E K ν diam( K ) | σ | F K,σ G K,σ in this equation can be considered as a strong stabilization term,in the sense that it vanishes (for all G K ) only if F K vanishes. We modify here the Mixed method byreplacing this strong stabilization by a weaker stabilization which, as in the Hybrid method, is expectedto vanish on “appropriate fluxes”. To achieve this, we use the quantity T K,σ ( F K ) = F K,σ + Λ K v K ( F K ) · n K,σ , (2.30)which vanishes if ( F K,σ ) σ ∈E K are the genuine fluxes of a given vector e . Then, taking B MK to be asymmetric positive definite matrix, we endow F K with the inner product h F K , G K i K = | K | v K ( F K ) · Λ K v K ( G K ) + X σ,σ ′ ∈E K B MK,σ,σ ′ T K,σ ( F K ) T K,σ ′ ( G K )= | K | v K ( F K ) · Λ K v K ( G K ) + T K ( G K ) T B MK T K ( F K ) (2.31)and the stabilized formula (2.28) linking p and F is replaced by ∀ K ∈ M , ∀ G K ∈ F K : h F K , G K i K = X σ ∈E K ( p K − p σ ) | σ | G K,σ . (2.32)We can get back a formulation more along the lines of (2.28) if we fix σ ∈ E K and take G K ( σ ) ∈ F K defined by G K ( σ ) σ = 1 and G K ( σ ) σ ′ = 0 if σ ′ = σ : (2.32) then gives p σ − p K = − | σ | | K | v K ( F K ) · Λ K v K ( G K ( σ )) − | σ | T K ( G K ( σ )) T B MK T K ( F K ) . But | K | Λ K v K ( G K ( σ )) = −| σ | (¯ x σ − x K ) and thus p σ − p K = v K ( F K ) · (¯ x σ − x K ) − | σ | T K ( G K ( σ )) T B MK T K ( F K ) . This equation is precisely the natural discrete relation (2.28) between p and its gradient, in which thestrong stabilization involving F K,σ has been replaced by a “weak” stabilization using T K ( F K ). Definition 2.6 (Modified Mixed method)
A Modified Mixed scheme for (1.1) reads:Find ( e p, F ) ∈ e H M , × F which satisfies [(2.27),(2.29),(2.30),(2.31),(2.32)] .Its parameters are the family of points ( x K ) K ∈M (which are freely chosen inside each grid cell) and thefamily ( B MK ) K ∈M of symmetric definite positive matrices. Algebraic correspondence between the three methods
We now focus on the main result of this paper, which is the following.
Theorem 3.1 (Equivalence of the methods)
The Generalized Mimetic, Generalized Hybrid and Mo-dified Mixed methods (see Definitions 2.3, 2.4 and 2.6) are identical, in the sense that for any choice ofparameters for one of these methods, there exists a choice of parameters for the other two methods whichleads to the same scheme.
The proof of this result is given in the remainder of this section, and decomposed as follows: for comparisonpurposes, the Generalized Mimetic method is first written under a hybridized form in Section 3.1; then,the correspondence between the Generalized Mimetic and the Modified Mixed methods is studied inSection 3.2; finally, the correspondence between the Generalized Mimetic and the Generalized Hybridmethods is carried out in Section 3.3.
Although the edge unknowns introduced to define the Generalized Hybrid and Modified Mixed methodsmay be eliminated, they are in fact essential to these methods; indeed, both methods can be hybridizedand reduced to a system with unknowns ( p σ ) σ ∈E only. In order to compare the methods, we thereforealso introduce such edge unknowns in the Generalized Mimetic method; this is the aim of this section.Let E be a grid cell and e ∈ ∂E be an edge. If e is an interior edge, we denote by e E the cell on theother side of e and define G ( e, E ) ∈ X h by: G ( e, E ) eE = 1, G ( e, E ) e e E = − G ( e, E ) e ′ E ′ = 0 if e ′ = e .We notice that G ( e, E ) = − G ( e, e E ) and, using G ( e, E ) in (2.6), the definitions of DIV h and of the innerproducts on X h and Q h give | e | ( p E p e E ) = [ F, G ( e, E )] E + [ F, G ( e, E )] e E = [ F, G ( e, E )] E − [ F, G ( e, e E )] e E .It is therefore natural to define p e (only depending on e and not on the grid cell E such that e ⊂ ∂E ) by ∀ E ∈ Ω h , ∀ e ∈ ∂E : | e | ( p E − p e ) = [ F, G ( e, E )] E . (3.33)This definition can also be applied for boundary edges e ⊂ ∂ Ω, in which case it gives p e = 0 (thanks to(2.6)).We thus extend p ∈ Q h into an element e p ∈ e H M , having values inside the cells and on the edges of themesh. Denoting, as in Section 2.3, F E the space of restrictions G E to the edges of E of elements G ∈ X h and writing any G E ∈ F E as a linear combination of ( G ( E, e )) e ∈ ∂E , it is easy to see from (3.33) that theGeneralized Mimetic method [(2.6),(2.7)] is equivalent to: find ( e p, F ) ∈ e H M , × X h such that ∀ E ∈ Ω h , ∀ G E ∈ F E : [ F E , G E ] E = X e ∈ ∂E | e | ( p E − p e ) G eE (3.34)and ∀ E ∈ Ω h : X e ∈ ∂E | e | F eE = Z E f. (3.35) ↔ Modified Mixed
The simplest comparison is probably to be found between the Modified Mixed and Generalized Mimeticmethods. Indeed, we see from [(2.29),(2.32)] and [(3.34),(3.35)] that both methods are identical providedthat, for any grid cell K = E , the local inner products defined by (2.31) and (2.12) are equal: h· , ·i K =[ · , · ] E ,. We therefore have to study these local inner products and understand whether they can beidentical (recall that there is some latitude in the choice of both inner products).Let us start with the term | K | v K ( F K ) · Λ K v K ( G K ) in the definition of h· , ·i K . Thanks to (2.27), | K | v K ( F K ) · Λ K v K ( G K ) = 1 | K | X σ ∈E K Λ − K (cid:0) | σ | (¯ x σ − x K ) (cid:1) F K,σ ! · X σ ∈E K | σ | (¯ x σ − x K ) G K,σ ! . R E , P σ ∈E K | σ | (¯ x σ − x K ) F K,σ = R TE F K and thus | K | v K ( F K ) · Λ K v K ( G K ) = 1 | K | (cid:0) Λ − K R TE F K (cid:1) · (cid:0) R TE G K (cid:1) = G TK (cid:18) | K | R E Λ − K R TE (cid:19) F K . Hence, the term | K | v K ( · ) · Λ K v K ( · ) in the definition of h· , ·i K is identical to the first term | E | R E Λ − E R TE in the definition of the matrix M E of [ · , · ] E (see (2.12)). Therefore, in order to prove that h· , ·i K = [ · , · ] E , itonly remains to prove that, for appropriate choices of C E , U E and B MK , we have for any ( F K , G K ) ∈ F K : T K ( G K ) T B MK T K ( F K ) = G TK C E U E C TE F K (3.36)(see (2.31) and (2.12)); in fact, this is the consequence of Lemma 6.3 in the appendix and of the followinglemma. Lemma 3.2
The mappings T K : R k E → R k E and C TE : R k E → R k E − d have the same kernel. Proof of Lemma 3.2
We first prove that:i) Im( N E ) ⊂ Ker( T K ) i.e. T K,σ (( N E ) j ) = 0 for all σ ∈ E K and all j = 1 , . . . , d (which also amountsto the fact that the lines of T K are orthogonal to the vectors ( N E ) j ).ii) dim(Im( T K )) = k E − d , and therefore dim(Ker( T K )) = d .Item i) follows from (2.27) and (2.26): we have Λ K v K (( N E ) j ) = − | K | P σ ∈E K | σ | (Λ K ) j · n K,σ (¯ x σ − x K ) = − (Λ K ) j and thus T K,σ (( N E ) j ) = (Λ K ) j · n K,σ − (Λ K ) j · n K,σ = 0.In order to obtain Item ii), we first remark that k E − d is an upper bound of the rank of T K since thelines of T K are in the orthogonal space of the independent vectors (( N E ) j ) j =1 ,...,d ( ). Moreover, (2.27)shows that the rank of v K : R k E → R k E is the rank of the family (¯ x σ − x K ) σ ∈E K , that is to say d , andthe kernel of v K therefore has dimension k E − d ; since T K = Id on this kernel, we conclude that the rankof T K is at least k E − d , which proves ii).These properties show that Ker( T K ) = Im( N E ) = (Im( C E )) ⊥ = ker( C TE ), and the proof is complete.The comparison between T K ( G K ) T B MK T K ( F K ) and G TK C E U E C TE F K is now straightforward. Indeed,let ( C E , U E ) be any pair chosen to construct the Generalized Mimetic method; applying Lemma 6.3,thanks to Lemma 3.2, with A = C TE , B = T K and {· , ·} R kE − d the inner product on R k E − d corre-sponding to U E , we obtain an inner product {· , ·} R kE on R k E such that { T K ( F K ) , T K ( G K ) } R kE = { C TE F K , C TE G K } R kE − d = G TK C E U E C TE F K ; this exactly means that, if we define B MK as the matrix of {· , ·} R kE , (3.36) holds. Similarly, inverting the role of C TE and T K when applying Lemma 6.3, for any B MK used in the Modified Mixed method we can find U E satisfying (3.36) and the proof of the correspondencebetween the Generalized Mimetic and Modified Mixed methods is concluded. ↔ Generalized Hybrid
To compare the Generalized Mimetic and Generalized Hybrid methods, we use a result of [4, 23] whichstates that the inverse of the matrix M E in (2.12) can be written M − E = W E = 1 | E | N E Λ − E N TE + D E e U E D TE (3.37)where D E is a k E × ( k E − d ) matrix such thatIm( D E ) = (Im( R E )) ⊥ (3.38) Let us notice that the independence of (( N E ) j ) j =1 ,...,d can be deduced from the independence of the columns of Λ K :thanks to (2.26), any non-trivial linear combination of the ( N E ) j gives a non-trivial combination of the columns of Λ K . e U E is a symmetric definite positive ( k E − d ) × ( k E − d ) matrix (note that the proof in [4, 23] assumes x E to be the center of gravity of E , i.e. M E to satisfy (2.8), but that it remains valid for any choiceof x E , i.e. for any matrix M E satisfying (2.12)). This result is to be understood in the following sense:for any ( C E , U E ) used to construct M E by (2.12), there exists ( D E , e U E ) such that W E defined by (3.37)satisfies W E = M − E , and vice-versa .For e p E = ( p E , ( p e ) e ∈ ∂E ) ∈ e H E , we denote by P E the vector in R k E with components ( | e | ( p E − p e )) e ∈ ∂E .The relation (3.34) can be re-written M E F E = P E , that is to say F E = W E P E , which is also equivalent,taking the inner product in R k E with an arbitrary Q E (built from a e q E ∈ e H E ), to ∀ E ∈ Ω h , ∀ e q E ∈ e H E : X e ∈ ∂E | e | ( q E − q e ) F eE = Q TE W E P E (3.39)The Generalized Mimetic method [(3.34),(3.35)] is therefore identical to the Generalized Hybrid method[(2.23),(2.24)] provided that, for all E = K ∈ Ω h , ∀ ( e p E , e q E ) ∈ e H E : Q TE W E P E = | K | Λ K ∇ K e p K · ∇ K e q K + S K ( e q K ) T B HK S K ( e p K ) . (3.40)As in the comparison between the Generalized Mimetic and Modified Mixed methods, the proof of (3.40)is obtained from the separate study of each term of (3.37).First, by the definition (2.10) of the matrix N E and the definition (2.16) of ∇ K e p K , we have ( N TE P E ) j = P e ∈ ∂E | e | (Λ E ) j · n eE ( p E − p e ) = −| K | (Λ K ) j · ∇ K e p K for all j = 1 , . . . , d , that is to say, by symmetry ofΛ, N TE P E = −| K | Λ K ∇ K e p K . Hence, Q TE (cid:18) | E | N E Λ − K N TE (cid:19) P E = | K | (Λ K ∇ K e q K ) T Λ − K (Λ K ∇ K e p K ) = | K | Λ K ∇ K e p K · ∇ K e q K . The first part of the right-hand side in (3.40) thus corresponds to the first part of W E in (3.37),and it remains to prove that (with appropriate choices of D E , e U E and B HK ), for all ( e p E , e q E ) ∈ e H E , Q TE D E e U E D TE P E = S K ( e q K ) T B HK S K ( e p K ). Let us notice that, defining L K : R k E → R k E by L K ( V ) = (cid:18) | σ | V σ − D K V · (¯ x σ − x K ) (cid:19) Tσ ∈E K with D K V = 1 | K | X σ ∈E K V σ n K,σ , (3.41)this boils down (letting V = P E and V ′ = Q E ) to proving that ∀ ( V, V ′ ) ∈ R k E : ( V ′ ) T D E e U E D TE V = L K ( V ′ ) T B HK L K ( V ) . (3.42)As previously, this will be a consequence of Lemma 6.3 and of the following result. Lemma 3.3
The mappings L K : R k E → R k E and D TE : R k E → R k E − d have the same kernel. Proof of Lemma 3.3
From (3.38), we get that Ker( D TE ) = Im( R E ). Hence it remains to prove thatIm( R E ) = Ker( L K )Let us first prove that Im( R E ) ⊂ Ker( L K ). The j -th column of R E is ( R E ) j = ( | σ | (¯ x jσ − x jK )) Tσ ∈E K (thesuperscript j denotes the j -th coordinate of points in R d ). Thus, for any e ∈ R d , by (2.26), D K ( R E ) j · e = 1 | K | X σ ∈E K | σ | e · n K,σ (¯ x jσ − x jK ) = e j , which means that D K ( R E ) j is the j -th vector of the canonical basis of R d . We then have D K ( R E ) j · (¯ x σ − x K ) = ¯ x jσ − x jK and thus( L K (( R E ) j )) σ = ¯ x jσ − x jK − D K ( R E ) j · (¯ x σ − x K ) = 0;10his proves that the columns of R E are in the kernel of L K , that is Im( R E ) ⊂ Ker( L K ).Next we notice that the rank of the mapping D K : V ∈ R k E D K V ∈ R d (i.e. the rank of the fam-ily ( n K,σ ) σ ∈E K ) is d and that the mapping L K is one-to-one on Ker( D K ). Hence dim(Im( L K )) ≥ dim(Ker( D K )) = k E − d , and therefore dim(Ker( L K )) ≤ d . But since Im( R E ) ⊂ Ker( L K ) anddim(Im( R E ) = d (the rank of the rows of R E ), we thus get that Im( R E ) = Ker( L K ).From Lemmas 6.3 and 3.3, we deduce as in Section 3.2 that for any choice of ( D E , e U E ) there correspondsa choice of B HK such that (3.42) holds, and vice-versa , which concludes the proof that the GeneralizedMimetic method is identical to the Generalized Hybrid method. Remark 3.4
These proofs and Remark 6.4 show that one can explicitly compute the parameters of onemethod which gives back the parameters of another method. Notice also that the algebraic computationsrequired in Remark 6.4 to obtain these parameters are made in spaces with small dimensions; the cost oftheir practical computations is therefore very low. However, to implement the methods, it is not necessaryto understand which ( C E , U E ) or ( D E , e U E ) corresponds to which B MK or B HK , since the only useful quan-tities are C E U E C TE , D E e U E D TE , T K ( · ) T B MK T K ( · ) and L K ( · ) T B HK L K ( · ) , and the relations between thesequantities are trivial (see (3.36) and (3.42) ). We showed in Section 3 that the three families of schemes which we called Generalized Mimetic, Gener-alized Hybrid FV and Modified Mixed FV are in fact one family of schemes, which we call the HMMF(Hybrid Mimetic Mixed Family) for short in the remainder of the paper. We now give convergence anderror estimate results for the HMMF method.
In this section, we are interested in convergence results which hold without any other regularity assump-tion on the data than those stated in Section 1. We therefore consider that Λ is only bounded anduniformly elliptic (not necessarily Lipschitz continuous or even piecewise continuous), that f ∈ L (Ω)and that the solution to (1.1) only belongs to H (Ω) (and not necessarily to H (Ω)).We study how existing results, previously established for the Hybrid scheme, can be extended to theHMMF. In [16], we proved the L convergence of the “standard” Hybrid method for a family of partitionsof Ω such that any cell K is star-shaped with respect to a point x K and such that there exists θ > max σ ∈E int K,L ∈M σ d K,σ d L,σ , max K ∈M σ ∈E K diam( K ) d K,σ ≤ θ, (4.43)where E int denotes the set of internal edges of the mesh and M σ the set of cells to which σ is an edge. Wenow show how the convergence of the HMMF may be deduced from an easy extension of [16, Theorem4.1, Lemma 4.4] provided that: • in the Generalized Mimetic formulation [(2.2),(2.3),(2.6),(2.7),(2.12),(2.13)],there exist s ∗ > S ∗ > , independent of the mesh, such that , for all cell K and all V = ( V σ ) σ ∈E K ,s ∗ X σ ∈E K | σ | d K,σ ( V σ ) ≤ V T M K V ≤ S ∗ X σ ∈E K | σ | d K,σ ( V σ ) , (4.44)11 in the Generalized Hybrid formulation [(2.16),(2.18),(2.23),(2.24)], using the notation (3.41),there exist ¯ s ∗ > S ∗ > , independent of the mesh, such that , for all cell K and all V = ( V σ ) σ ∈E K , ¯ s ∗ P σ ∈E K | σ | d K,σ ( L K ( V )) σ ≤ L K ( V ) T B HK L K ( V ) ≤ ¯ S ∗ P σ ∈E K | σ | d K,σ ( L K ( V )) σ , (4.45) • in the Modified Mixed formulation [(2.27),(2.29),(2.30),(2.31),(2.32)],there exist ˜ s ∗ > S ∗ > , independent of the mesh, such that , for all cell K and all V = ( V σ ) σ ∈E K , ˜ s ∗ P σ ∈E K | σ | d K,σ ( T K,σ ( V )) ≤ T K ( V ) T B MK T K ( V ) ≤ ˜ S ∗ P σ ∈E K | σ | d K,σ ( T K,σ ( V )) . (4.46)The three conditions (4.44), (4.45) and (4.46) are in fact equivalent (this is stated in the next theorem),and one only has to check the condition corresponding to the chosen framework. Theorem 4.1 (Convergence of the HMMF method)
Assume that
Λ : Ω → M d ( R ) is boundedmeasurable symmetric and uniformly elliptic, that f ∈ L (Ω) and that the solution to (1.1) belongsto H (Ω) . Let θ > be given. Consider a family of polygonal meshes of Ω such that any cell K isstar-shaped with respect to a point x K , and satisfying (4.43) . Then the three conditions (4.44) , (4.45) and (4.46) are equivalent. Moreover, if for any mesh of the family we choose a HMMF scheme suchthat one of the conditions (4.44) , (4.45) or (4.46) holds, then, the family of corresponding approximatesolutions converges in L (Ω) to the unique solution of (1.1) as the mesh size tends to 0. Proof of Theorem 4.1.
Let us first prove the equivalence between (4.44) and (4.45), assuming (4.44) to begin with. Using (3.42),we get that, for all V , V T D K e U K D TK V = L K ( V ) T B HK L K ( V ) . Let us apply the above relation to e L K ( V ) defined by e L K ( V ) σ = | σ | L K ( V ) σ . From (2.26) (see also(2.17)) and recalling the operator D K defined in (3.41), we easily get that D K e L K ( V ) = 0, which provides L K ( e L K ( V )) = L K ( V ). Hence we get( e L K ( V )) T D K e U K D TK e L K ( V ) = L K ( V ) T B HK L K ( V ) . We then remark that N TK e L K ( V ) = 0 (once again from (2.26)) and therefore, using (3.37), L K ( V ) T B HK L K ( V ) = ( e L K ( V )) T W K e L K ( V ) . (4.47)Let I K be the diagonal matrix I K = diag(( p | σ | d K,σ ) σ ∈E K )). Condition (4.44) applied to G K = I K F K gives, for all G K ∈ F K , s ∗ G TK G K ≤ G TK I − K M K I − K G K ≤ S ∗ G TK G K . This shows that the eigenvalues of I − K M K I − K are in [ s ∗ , S ∗ ], and thus that the eigenvalues of I K M − K I K = I K W K I K belong to [ S ∗ , s ∗ ]. Translating this into bounds on ( I − K e L K ( V )) T I K W K I K ( I − K e L K ( V )), wededuce 1 S ∗ X σ ∈E K | σ | d K,σ ( e L K ( V ) σ ) ≤ ( e L K ( V )) T W K e L K ( V ) ≤ s ∗ X σ ∈E K | σ | d K,σ ( e L K ( V ) σ ) , and (4.45) follows from (4.47) (with ¯ s ∗ = S ∗ and ¯ S ∗ = s ∗ ). Reciprocally, from (4.45), following the proofof [16, Lemma 4.4] and setting V σ = | σ | ( p K − p σ ) in that proof, we get the existence of c > c > c X σ ∈E K | σ | d K,σ V σ ≤ V T W K V ≤ c X σ ∈E K | σ | d K,σ V σ . (4.48)12sing I K as before, we then get (4.44) with s ∗ = c and S ∗ = c .Let us turn to the proof of the equivalence between (4.44) and (4.46), beginning by assuming that (4.44)holds. Using (2.26), one has T K ((Λ K v K ( F K ) · n K,σ ) σ ∈E K ) = 0, and thus T K ( T K ( F K )) = T K ( F K ); hence,noting that R TK T K ( F K ) = 0 (once again thanks to (2.26)) and remembering (2.12), (3.36), we see that(4.44) applied to V = T K ( F K ) directly gives (4.46). The reciprocal property follows, in a similar way as(4.48), from a simple adaptation of classical Mixed FV manipulations (used for example in the proof of a priori estimates on the approximate solution), see [10].Note also that obtaining (4.44) from (4.45) or (4.46) is very similar to [4, Theorem 3.6].We can now conclude the proof of the theorem, that is to say establish the convergence of the approximatesolutions: using (4.45), this convergence is a direct consequence of [16, Theorem 4.1] with a straightfor-ward adaptation of the proof of [16, Lemma 4.4]. The convergence could also be obtained, using (4.46),by an easy adaptation of the techniques of proof used in the standard Mixed setting (see [10]).Mimetic schemes are usually studied under a condition on the local inner products usually called ( S1 )and which reads [2]: there exist s ∗ > S ∗ > ∀ E ∈ Ω h , ∀ G ∈ X h : s ∗ k E X i =1 | E | ( G e i E ) ≤ [ G, G ] E ≤ S ∗ k E X i =1 | E | ( G e i E ) . (4.49)The mesh regularity assumptions in [2, 4] also entail the existence of C , independent of the mesh, suchthat C K ) d − ≤ | σ | , ∀ σ ∈ E K ,C K ) ≤ d K,σ , ∀ σ ∈ E K . (4.50)Under such mesh assumptions and since | σ | ≤ d − diam( L ) whenever σ ∈ E L , it is easy to see that thereexists θ only depending on C | σ | d K,σ are of the same order as | K | and thus that (4.49) implies (4.44) (with possibly different s ∗ and S ∗ ). Wecan therefore apply Theorem 4.1 to deduce the following convergence result under the usual assumptionsin the mimetic literature (except for the regularity assumptions on the data). Corollary 4.2 (Convergence under the usual mimetic assumptions)
We assume that
Λ : Ω → M d ( R ) is bounded measurable symmetric and uniformly elliptic, that f ∈ L (Ω) and that the solutionto (1.1) is in H (Ω) . Consider a family of polygonal meshes of Ω such that any cell E is star-shapedwith respect to a point x E , and such that (4.50) holds (with C ( S1 ) of [2] (i.e. (4.49) with s ∗ , S ∗ not depending onthe mesh) and (2.15) . Then, the family of approximate solutions given by the corresponding GeneralizedMimetic schemes converges in L (Ω) to the solution of (1.1) as the mesh size tends to 0.In particular, taking x E as the center of gravity ¯ x E of E , (2.15) reduces to Condition ( S2 ) of [2] (thatis to say (2.4) ), and the family of approximate solutions given by the corresponding “standard” Mimeticschemes converges in L (Ω) to the solution of (1.1) as the mesh size tends to 0. Remark 4.3 (Compactness and convergence of a gradient)
The proofs of the above convergenceresults rely on the use of the Kolmogorov compactness theorem on the family of approximate solutions, see[10, Lemma 3.3] or [16, Section 5.2]. In fact, this latter study shows that for p ∈ [1 , + ∞ ) , any family ofpiecewise constant functions that is bounded in an adequate discrete mesh-dependent W ,p norm convergesin L p (Ω) (up to a subsequence) to a function of W ,p (Ω) . The regularity of the limit is shown thanks tothe weak convergence of a discrete gradient to the gradient of the limit [16, Proof of Lemma 5.7, Step 1].Note that this compactness result and the construction of this weakly converging gradient are completelyindependent of the numerical scheme, which is only used to obtain the discrete W ,p estimate and thefact that the limit is indeed the solution of (1.1) .Moreover, it is possible, from the approximate solutions given by the HMMF schemes, to reconstructa gradient (in a similar way as [16, (22)-(26)]) which strongly converges in L (Ω) to the gradient of he solution of (1.1) , see [16, Theorem 4.1]. One can also directly prove that the gradient v K , definedfrom the fluxes by (2.27) , is already a strongly convergent gradient in L (Ω) , see [10]; in fact, the strongconvergence of this gradient is valid in a more general framework than the one of the methods presentedhere, see [1]. Let us first consider the original Mimetic Finite Difference method (2.2)–(2.11). This method is consistentin the sense that it satisfies the condition ( S2 ) of [2]. Under the assumptions that: • Condition ( M1 ) of [2] on the domain Ω (namely Ω is polyhedral and its boundary is Lipschitzcontinuous) and Conditions ( M2 )-( M6 ) of [2] (corresponding to ( M1 )-( M4 ) in [4]) hold; • The stability condition ( S1 ) of [2, 4] holds; this condition concerns the eigenvalues of the matrix U E in (2.8), and are connected with the discretization, • Λ ∈ W , ∞ (Ω) d × d and p ∈ H (Ω),then Theorem 5.2 in [2] gives an order 1 error estimate on the fluxes, for an adequate discrete norm;moreover, if Ω is convex and the right-hand side belongs to H (Ω), an order 1 error estimate on p in the L norm is also established.By the equivalence theorem (Theorem 3.1), this result yields similar error estimates for the GeneralizedHybrid and Modified Mixed methods in the case where the point x K is taken as the center of gravity ¯ x K of K .In the case of the original Hybrid method (2.22), an order 1 error estimate is proved in [16] only under theassumption of an homogeneous isotropic medium, that is Λ = Id, and in the case where the solution to(1.1) also belongs to C (Ω) (but with no convexity assumption on the domain Ω). As stated in Remark4.2 of [16], the proof is readily extended if the solution is piecewise H (in fact, in [16] the situation ismore complex because the Hybrid scheme is considered in the more general setting of the SUSHI scheme,which involves the elimination of some or all edge unknowns; see Section 5.4 below). Under an additional assumption of existence of a specific lifting operator, compatible with the consideredMimetic Finite Difference method, a theoretical L -error estimate of order 2 on p is proved in [2]. Acondition on the matrix M E which ensures the existence of such a lifting operator is given in [3]: inparticular, if the smallest eigenvalue of U E is large enough (with respect to the inverse of the smallesteigenvalue of C TE C E and to the largest eigenvalue of a local inner product involving a generic liftingoperator), then the existence of a lifting operator compatible with M E , and thus the super-convergenceof the associated Mimetic scheme, can be proved.Regarding the consequence on the HMMF method, this means that, if x K is the center of gravity of K and if the symmetric positive definite matrices ( U E ) E ∈ Ω h or ( B HK ) K ∈M or ( B MK ) K ∈M are “large enough”,then the approximate solution p given by the corresponding HMMF method converges in L with order2. It does not seem easy to give a practical lower bound on these stabilization terms which ensure thatthey are indeed “large enough” for the theoretical proof; however, several numerical tests (both with x K the center of gravity of K , or x K elsewhere inside K ) suggest that the HMMF methods enjoy asuperconvergence property for a wider range of parameters than those satisfying the above theoreticalassumptions. We now show that, under one of its three forms and according to the choice of its parameters, a schemefrom the HMMF (Hybrid Mimetic Mixed Family) may be interpreted as a nonconforming Finite Elementscheme, as a mixed Finite Element scheme, or as the classical two-point flux finite volume method.14 .1 A nonconforming Finite Element method
In this section, we aim at identifying a hybrid FV scheme of the HMMF with a nonconforming FiniteElement method. Hence we use the notations of Definition 2.4 and Remark 2.5. For any K ∈ M and σ ∈ E K , we denote by △ K,σ the cone with vertex x K and basis σ . For any given e p ∈ e H M we define thepiecewise linear function b p : Ω → R by: ∀ K ∈ M , ∀ σ ∈ E K , for a.e. x ∈ △ K,σ : b p ( x ) = p K + (cid:18) ∇ K e p K + β K d K,σ S K,σ ( e p K ) n K,σ (cid:19) · ( x − x K )where β K > n K,σ · (¯ x σ − x K ) = d K,σ , in the particular case where β K = 1 we have b p (¯ x σ ) = p σ ). We let b H M = { b p , e p ∈ e H M } and we define the “broken gradient” of b p ∈ b H M by ∀ K ∈ M , ∀ σ ∈ E K , for a.e. x ∈ △ K,σ : b ∇ b p ( x ) = ∇ b p ( x ) = ∇ K e p K + β K d K,σ S K,σ ( e p K ) n K,σ . We consider the following nonconforming finite element problem: find b p ∈ b H M such that ∀ b q ∈ b H M : Z Ω b Λ( x ) b ∇ b p ( x ) · b ∇ b q ( x ) d x = Z Ω f ( x ) b q ( x ) d x, (5.51)where b Λ( x ) is equal to Λ K for a.e. x ∈ K . Then the above method leads to a matrix which belongs to thefamily of matrices corresponding to the Generalized Hybrid method (and thus also to the GeneralizedMimetic and Modified Mixed methods). Therefore, since | △ K,σ | = | σ | d K,σ d , we get from (2.19): Z Ω b Λ( x ) b ∇ b p ( x ) · b ∇ b q ( x ) d x = X K ∈M | K | Λ K ∇ K e p K · ∇ K e q K + X K ∈M S K ( e q K ) T B HK S K ( e p K ) , with B HK,σ,σ = | σ | ( β K ) d d K,σ Λ K n K,σ · n K,σ , and, for σ = σ ′ , B HK,σ,σ ′ = 0 . Note that this definition for B HK fulfills (4.45) under the regularity hypothesis (4.43), hence ensuringconvergence properties which can easily be extended to this nonconforming Finite Element method (5.51)even though it is not completely identical to a scheme of the HMMF, since the right-hand sides do notcoincide: in general, Z Ω f ( x ) b q ( x ) d x = X K ∈M q K Z K f. Indeed, this does not prevent the study of convergence since the difference between the two right-hand-sides is of order h . In this section, we aim at identifying a particular scheme of the HMMF, under its mixed version with aMixed Finite Element method. Let us first recall the remark provided in [2, Section 5.1] and [5, Example1]: on a simplicial mesh (triangular if d = 2, tetrahedral if d = 3), the Raviart-Thomas RT0 Mixed FiniteElement method fulfills properties (4.49) and (2.4), and it is therefore possible to include this MixedFinite Element scheme in the framework of the HMMF. Our purpose is to show that our framework alsoprovides a Mixed Finite Element method on a general mesh (note that, even in the case of simplices, thismethod differs from the Raviart-Thomas RT0 method). We use here the notations provided by Definition2.6. Still denoting △ K,σ the cone with vertex x K and basis σ , we define, for F ∈ F , ∀ K ∈ M , ∀ σ ∈ E K , ∀ x ∈ △ K,σ : b F K,σ ( x ) := − Λ K v K ( F K ) + T K,σ ( F K ) x − x K d K,σ . (5.52)15f x belongs to the interface ∂ △ K,σ ∩ ∂ △ K,σ ′ between two cones of a same control volume, we have( x − x K ) · n ∂ △ K,σ ∩ ∂ △ K,σ ′ = 0 and thus the normal fluxes of b F K,σ are conservative through such interfaces;moreover, for all σ ∈ E K and all x ∈ σ , we have ( x − x K ) · n K,σ = d K,σ and thus, by (2.30), b F K,σ ( x ) · n K,σ = F K,σ ; since the elements of F satisfy (2.1), these observations show that the function b F , defined by ∀ K ∈ M , ∀ σ ∈ E K , for a.e. x ∈ △ K,σ : b F ( x ) = b F K,σ ( x ) , (5.53)satisfies b F ∈ H div (Ω). Noting that X σ ∈E K T K,σ ( F K ) Z △ K,σ x − x K d K,σ d x = X σ ∈E K T K,σ ( F K ) | σ | (¯ x σ − x K ) d + 1 = 0 (5.54)(thanks to (2.27), (2.30) and (2.26)), we have, with b Λ again denoting the piecewise-constant functionequal to Λ K in K , Z K b Λ( x ) − b F ( x ) · b G ( x ) d x = | K | v K ( F K ) · Λ K v K ( G K ) + X σ ∈E K γ K,σ T K,σ ( F K ) T K,σ ( G K ) (5.55)with γ K,σ = Z △ K,σ b Λ( x ) − x − x K d K,σ · x − x K d K,σ d x > . The right-hand side of (5.55) defines an inner product h· , ·i K which enters the framework defined by(2.31), setting B MK,σ,σ = γ K,σ , and B MK,σ,σ ′ = 0 for σ = σ ′ , (5.56)which fulfills (4.46) under the regularity hypothesis (4.43) (hence ensuring convergence properties). There-fore the form [(2.6),(2.7)] of the resulting HMMF scheme resumes to the following Mixed Finite Elementformulation: find ( p, b F ) ∈ H M × b F such that ∀ b G ∈ b F : Z Ω b Λ( x ) − b F ( x ) · b G ( x ) d x − Z Ω p ( x )div b G ( x ) d x = 0 , (5.57) ∀ q ∈ H M : Z Ω q ( x )div b F ( x ) d x = Z Ω q ( x ) f ( x ) d x , (5.58)where b F = { b F , F ∈ F} . Indeed, since | △
K,σ | = | σ | d K,σ d and P σ ∈E K | σ | n K,σ = 0, we have, for all F ∈ b F , Z K div b F ( x ) d x = X σ ∈E K | σ | T K,σ ( F K ) = X σ ∈E K | σ | F K,σ (5.59)and (5.58) written on the canonical basis of H M is exactly (2.29). Summing (2.32) on K , the termsinvolving p σ vanish thanks to the conservativity of G and we get (5.57). Reciprocally, to pass from (5.57)to (2.32), one has to get back the edge values p σ , which can be done exactly as for the hybridizationof the Generalized Mimetic method in Section 3.1 (using G = G ( K, σ ) such that G ( K, σ ) K,σ = 1, G ( K, σ ) L,σ = − L is the control volume on the other side of σ and G ( K, σ ) Z,θ = 0 for other controlvolumes Z and/or edges θ ).An important element of study of the standard Mimetic method seems to be the existence of a suitablelifting operator, re-constructing a flux unknown inside each grid cell from the fluxes unknowns on theboundary of the grid cell (see [2] and Section 4.3). We claim that, for the Generalized Mimetic methodcorresponding to the choice (5.56), the flux b F given by [(5.52),(5.53)] provides a (nearly) suitable liftingoperator : it does not completely satisfy the assumptions demanded in [2, Theorem 5.1], but enough sothat the conclusion of this theorem still holds. 16 roposition 5.1 For F K ∈ F K , let b F K be the restriction to K of b F defined by (5.53) . Then the operator F K ∈ F K b F K ∈ L ( K ) satisfies the following properties: ∀ F K ∈ F K , ∀ σ ∈ E K , ∀ x ∈ σ : b F K ( x ) · n K,σ = F K,σ , (5.60) ∀ F K ∈ F K , ∀ q affine function : Z K q ( x )div( b F K )( x ) d x = Z E q ( x ) DIV h ( F K ) w E ( x ) d x , (5.61) ∀ F ∈ R d , defining F K = ( F · n K,σ ) σ ∈E K : b F K = F , (5.62) for any w E satisfying (2.14) . Remark 5.2
Properties (5.60) and (5.62) are the same as in [2, Theorem 5.1], but Property (5.61) isreplaced in this reference by the stronger form “ div( b F K ) = DIV h ( F K ) on K ” ( x K is also taken as thecenter of gravity, which corresponds to w E = 1 in (5.61) ). Proof of Proposition 5.1
We already noticed (5.60) (consequence of (2.30) and the fact that ( x − x K ) · n K,σ = d K,σ for all x ∈ σ ).If F ∈ R d , and F K = ( F · n K,σ ) σ ∈E K , then (2.26) and (2.27) show that Λ K v K ( F K ) = − F and thus that T K ( F K ) = 0, in which case b F K = − Λ K v K ( F K ) = F and (5.62) holds.Let us now turn to (5.61). For q ≡
1, this relation is simply (5.59). If q ( x ) = x , then Z K q ( x )div( b F K )( x ) d x = X σ ∈E K T K,σ ( F K ) d Z △ K,σ xd K,σ d x and (5.54) then gives Z K q ( x )div( b F K )( x ) d x = X σ ∈E K T K,σ ( F K ) d x K d K,σ | △
K,σ | = X σ ∈E K | σ | T K,σ ( F K ) x K = X σ ∈E K | σ | F K,σ ! x K = Z K q ( x ) DIV h ( F K ) w E ( x ) d x by assumption on w E . We now consider isotropic diffusion tensors: Λ = λ ( x )Id , (5.63)with λ ( x ) ∈ R and exhibit cases in which the HMMF provides two-point fluxes, in the sense that thefluxes satisfy F eE = τ E,E ′ ( p E − p E ′ ), in the case where e is the common edge of two neighboring cells E and E ′ , with τ E,E ′ ≥ x K the intersection of the orthogonal bisectors [18, 13].Therefore, whenever possible, one should strive to recover this scheme when using admissible meshes (in17he sense of [13] previously mentioned). In the HMMF framework, two-point fluxes are obtained, usingthe notations provided by Definition 2.1, if the matrix of the bilinear form [ F, G ] X h in (2.6) is diagonal.This implies, in the case of meshes such that two neighboring grid cells only have one edge in common,that the matrices M E defining the local inner product (2.3) are diagonal. If the matrix M E is diagonal,we get from the property M E N E = R E and from (5.63) that there exists µ eE ∈ R such that µ eE n eE = ¯ x e − x E . (5.64)This implies that x E is, for any face e of E , a point of the orthogonal line to e passing through ¯ x e ( x E is then necessarily unique) and that µ eE is the orthogonal distance between x E and e . In the case ofa triangle, x E is thus the intersection of the orthogonal bisectors of the sides of the triangle, which isthe center of gravity only if the triangle is equilateral; hence , except in this restricted case, the originalMimetic method cannot yield a two-point flux method. Note also that there are meshes such that theorthogonal bisectors of the faces do not intersect, but for which nevertheless some “centers” in the cellsexist and are such that the line joining the centers of two neighboring cells is orthogonal to their commonface: these meshes are referred to “admissible” meshes in [13, Definition 9.1 p. 762]; a classical exampleof such admissible meshes are the general Voronoi meshes. On such admissible meshes, the HMMF doesnot provide a two-point flux scheme for isotropic diffusion operators, although a two-point flux FiniteVolume scheme can be defined, with the desired convergence properties (see [13]).In this section, we shall call “super-admissible discretizations” the discretizations which fulfill the property(5.64) for some choice of ( x E ) E ∈ Ω h . We wish to show that for all super-admissible discretizations and inthe isotropic case (5.63), the HMMF provides a two-point flux scheme. Using the notations of Definitions2.4 and 2.6, (5.64) is written n K,σ = (¯ x σ − x K ) /d K,σ and defines our choice of parameters ( x K ) K ∈M .Let us take α K,σ = λ K in the Hybrid presentation (2.20), denoting by λ K the mean value of the function λ ( x ) in K . Using Definition (2.16) for ∇ K e p K and thanks to (2.17), a simple calculation shows that X σ ∈E K | σ | λ K d K,σ S K,σ ( e p K ) S K,σ ( e q K ) = X σ ∈E K | σ | λ K d K,σ ( p K − p σ )( q K − q σ ) − | K | λ K ∇ K e p K · ∇ K e q K . Hence we deduce from (2.20) that F K,σ = λ K d K,σ ( p K − p σ ) and the conservativity (2.1) leads, for an internaledge between the control volumes K and L , to p σ = d K,σ λ L p L + d L,σ λ K p K d K,σ λ L + d L,σ λ K . The resulting expression for theflux becomes F K,σ = λ K λ L d K,σ λ L + d L,σ λ K ( p K − p L ) , which is the expression of the flux for the standard 2-points Finite Volume scheme with harmonic averagingof the diffusion coefficient.It is also easy to find back this expression from the Mixed presentation (2.31), taking B MK,σ,σ = | σ | d K,σ λ K , and, for σ = σ ′ , B MK,σ,σ ′ = 0 . The property X σ ∈E K | σ | d K,σ λ K T K,σ ( F K ) T K,σ ( G K ) = X σ ∈E K | σ | d K,σ λ K F K,σ G K,σ − | K | v K ( F K ) · λ K v K ( G K ) , which results from (2.27), (2.30), (2.17) and (5.64) (under the form d K,σ n K,σ = ¯ x σ − x K ), shows that h F K , G K i K = X σ ∈E K | σ | d K,σ λ K F K,σ G K,σ . Thanks to (2.32), this gives F K,σ = λ K d K,σ ( p K − p σ ) and we conclude as above.18 .4 Elimination of some edge unknowns In the study of the hybrid version (2.25) of the HMMF [15, 16], it was suggested to replace the space e H M , by the space e H BM , defined by e H BM , = { e p ∈ e H M , such that p σ = P K ∈M σ β Kσ p K if σ ∈ B} , where:i) M σ is a subset of M , including a few cells (in general, less than d + 1, where we recall that d isthe dimension of the space) “close” from σ ,ii) the coefficients β Kσ are barycentric weights of the point ¯ x σ with respect to the points x K , whichmeans that P K ∈M σ β Kσ = 1 and P K ∈M σ β Kσ x K = ¯ x σ ,iii) B is any subset of the set of all internal edges (the cases of the empty set or of the full set itselfbeing not excluded).Then the scheme is defined by: find e p ∈ e H BM , such that ∀ e q ∈ e H BM , : X K ∈M | K | Λ K ∇ K e p K · ∇ K e q K + X K ∈M S K ( e q K ) T B HK S K ( e p K ) = X K ∈M q K Z K f. (5.65)In the case where B = ∅ , then the method belongs to the HMMF and in the case where B is the fullset of the internal edges, then there is no more edge unknowns, and we get back a cell-centered scheme.In the intermediate cases, we get schemes where the unknowns are all the cell unknowns, and the edgeunknowns p σ with σ / ∈ B .This technique has been shown in [15] and [16] to fulfill the convergence and error estimates requirementsin the case of diagonal matrices B HK . It can be applied, with the same convergence properties, to theHMMF with symmetric positive definite matrices B HK as in Section 4. We prove here two results linked with the definition of the Generalized Mimetic method: the existenceof a weight function satisfying (2.14) and the equivalence between (2.15) and (2.12).
Lemma 6.1 If E is a bounded non-empty open subset of R d and x E ∈ R d , then there exists an affinefunction w E : R d → R satisfying (2.14) . Proof of Lemma 6.1
We look for ξ ∈ R d such that w E ( x ) = 1 + ξ · ( x − ¯ x E ) = 1 + ( x − ¯ x E ) T ξ satisfies the properties (where ¯ x E is the center of gravity of E ). The first property of (2.14) is straightforward since R E ( x − x E ) d x = 0, andthe second property is equivalent to | E | ¯ x E + R E x ( x − ¯ x E ) T ξ d x = | E | x E ; since R E ¯ x E ( x − ¯ x E ) T d x = 0,this boils down to (cid:18)Z E ( x − ¯ x E )( x − ¯ x E ) T d x (cid:19) ξ = | E | ( x E − ¯ x E ) . (6.66)Let J E be the d × d matrix R E ( x − ¯ x E )( x − ¯ x E ) T d x : we have, for all η ∈ R d \{ } , J E η · η = R E (( x − ¯ x E ) · η ) d x and the function x → ( x − ¯ x E ) · η vanishes only on an hyperplane of R d ; this proves that J E η · η > η = 0. Hence, J E is invertible and there exists (a unique) ξ satisfying (6.66), whichconcludes the proof. Lemma 6.2
Let [ · , · ] E be a local inner product on the space of the fluxes unknowns of a grid cell E , andlet M E be its matrix. Then [ · , · ] E satisfies (2.15) (with w E satisfying (2.14) ) if and only if M E satisfies (2.12) (with R E defined by (2.13) and ( C E , U E ) defined by (2.10) and (2.11) ). roof of Lemma 6.2 It is known [4] that, for the standard Mimetic method, (2.4) is equivalent to M E N E = ¯ R E with ¯ R E defined by (2.9) and N E defined in (2.10); similarly, it is quite easy to see that (2.15) (with w E satisfying(2.14)) is equivalent to M E N E = R E (6.67)with R E defined by (2.13): indeed, (2.15) with q = 1 is simply the definition (2.2) of the discretedivergence operator (because R E w E ( x ) d x = | E | ) and, with q ( x ) = x j , since R E x j w E ( x ) d x = | E | ( x E ) j ,(2.15) boils down to G TE M E ( N E ) j + k E X i =1 G e i E | e i | ( x E ) j = k E X i =1 G e i E | e i | (¯ x e i ) j , which is precisely G TE M E ( N E ) j = G TE ( R E ) j with ( R E ) j the j -th column of R E . We therefore only needto compare (2.12) with (6.67).Let us first assume that M E satisfies (2.12). The generic formula (2.26) implies R TE ( N E ) j = | E | (Λ E ) j (6.68)and thus 1 | E | R E Λ − E R TE N E = R E . (6.69)Since C TE N E = 0 by definition of C E , this shows that M E satisfies (6.67).Let us now assume that M E satisfies (6.67) and let us consider the symmetric matrix e M E = M E − | E | R E Λ − E R TE . (6.70)By (6.67) and (6.69), we have e M E N E = 0, and the columns of N E are therefore in the kernel of e M E .The definition of C E shows that the columns of N E span the kernel of C TE : we therefore have ker( C TE ) ⊂ ker( e M E ) and we deduce the existence of a k E × ( k E − d ) matrix A such that e M E = AC TE . (6.71)By symmetry of e M E we have AC TE = C E A T and thus AC TE C E = C E A T C E ; but C TE C E is an invertible( k E − d ) × ( k E − d ) matrix (since C E is of rank k E − d ) and therefore A = C E A T C E ( C TE C E ) − = C E U E forsome ( k E − d ) × ( k E − d ) matrix U E . Gathering this result with (6.70) and (6.71), we have proved that M E satisfies (2.12) for some U E , and it remains to prove that this last matrix is symmetric definite positiveto conclude. By (2.12) and the symmetry of M E and | E | R E Λ − E R TE we have C E U E C TE = C E U TE C TE and,since C TE is onto and C E is one-to-one, we deduce U E = U TE . To prove that U E is definite positive, weuse (2.12) and the fact that M E is definite positive to write ∀ ξ ∈ ker( R TE ) , ξ = 0 : ( U E C TE ξ ) · ( C TE ξ ) > . (6.72)This shows in particular that ker( R TE ) ∩ ker( C TE ) = { } and thus, since ker( R TE ) has dimension k E − d ,that the image by C TE of ker( R TE ) is R k E − d . Equation (6.72) then proves that U E is definite positive onthe whole of R k E − d and the proof is complete. Lemma 6.3
Let X , Y and Z be finite dimension vector spaces and A : X → Y , B : X → Z be two linearmappings with identical kernel. Then, for all inner product {· , ·} Y on Y , there exists an inner product {· , ·} Z on Z such that, for all ( x, x ′ ) ∈ X , { Bx, Bx ′ } Z = { Ax, Ax ′ } Y . roof of Lemma 6.3 Let N = ker( A ) = ker( B ). The mappings A and B define one-to-one mappings ¯ A : X/N → Y and¯ B : X/N → Z such that, if ¯ x is the class of x , Ax = ¯ A ¯ x and Bx = ¯ B ¯ x . We can therefore work with ¯ A and ¯ B on X/N rather than with A and B on X , and assume in fact that A and B are one-to-one.Then A : X → Im( A ) and B : X → Im( B ) are isomorphisms and, if {· , ·} Y is an inner product on Y , wecan define the inner product {· , ·} Im( B ) on Im( B ) the following way: for all z, z ′ ∈ Im( B ), { z, z ′ } Im( B ) = { AB − z, AB − z ′ } Y (this means that { Bx, Bx ′ } Im( B ) = { Ax, Ax ′ } Y for all x, x ′ ∈ X ). This innerproduct is only defined on Im( B ), but we extend it to Z by choosing W such that Im( B ) ⊕ W = Z ,by taking any inner product {· , ·} W on W and by letting { z, z ′ } Z = { z B , z ′ B } Im( B ) + { z W , z ′ W } W for all z = z B + z ′ W ∈ Z = Im( B ) ⊕ W and z ′ = z ′ B + z ′ W ∈ Z . This extension of {· , ·} Im( B ) preserves theproperty { Bx, Bx ′ } Z = { Ax, Ax ′ } Y . Remark 6.4
The proof gives a way to explicitly compute {· , ·} Z from {· , ·} Y , A and B : find a supplemen-tal space G of ker( A ) = ker( B ) in X , compute an inverse of B between G and Im( B ) , deduce {· , ·} Im( B ) and extend it to Z by finding a supplemental space of Im( B ) . References [1] L. Agelas, D. Di Pietro, and J. Droniou. The G method for heterogeneous anisotropic diffusion ongeneral meshes. submitted, november 2008.[2] F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method fordiffusion problems on polyhedral meshes.
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