On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality.

Transcription

1 On 3D DDFV dscretzaton of gradent and dvergence operators. I. Meshng, operators and dscrete dualty. Bors Andreanov, Mostafa Bendahmane, Florence Hubert, Stella Krell To cte ths verson: Bors Andreanov, Mostafa Bendahmane, Florence Hubert, Stella Krell. On 3D DDFV dscretzaton of gradent and dvergence operators. I. Meshng, operators and dscrete dualty.. IMA J. Numer. Analyss, 202, 32 (4), pp <0.093/manum/drr046>. <hal v3> HAL Id: hal Submtted on Mar 20 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés. Dstrbuted under a Creatve Commons Attrbuton - NonCommercal 4.0 Internatonal Lcense

2 ON 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I. MESHING, OPERATORS AND DISCRETE DUALITY. B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL Abstract. Ths work s ntended to provde a convenent tool for the mathematcal analyss of a partcular knd of fnte volume approxmaton whch can be used, for nstance, n the context of nonlnear and/or ansotropc dffuson operators n 3D. Followng the so-called DDFV (Dsctere Dualty Fnte Volume) approach developed by F. Hermelne and by K. Domelevo and P. Omnès n 2D, we consder a double coverng T of a three-dmensonal doman by a rather general prmal mesh and by a well-chosen dual mesh. The assocated dscrete dvergence operator dv T s obtaned by the standard fnte volume approach. A smple and consstent dscrete gradent operator T s defned by local affne nterpolaton that takes nto account the geometry of the double mesh. Under mld geometrcal constrants on the choce of the dual volumes, we show that dv T, T are lnked by the dscrete dualty property, whch s an analogue of the ntegraton-by-parts formula. The prmal mesh need not be conformal, and ts nterfaces can be general polygons. We gve several numercal examples for ansotropc lnear dffuson problems; good convergence propertes are observed. The sequel [3] of ths paper wll summarze some key dscrete functonal analyss tools for DDFV schemes and gve applcatons to provng convergence of DDFV schemes for several nonlnear degenerate parabolc PDEs. Contents. Introducton 2.. The context of DDFV and related schemes 2.2. The goal of the paper 4.3. Bref descrpton of the operators 4.4. Outlne of the paper 5 2. Double meshes and the assocated gradent and dvergence operators Constructon of double meshes Spaces of dscrete functons and felds, dscrete dvergence and gradent operators Cartesan DDFV meshes n 3D 9 3. The dscrete dualty property 0 4. On the choce of the face centers L and other generalzatons 3 5. Numercal experments for lnear dffuson problems 6 Appendx: A reconstructon property n the plane 20 References 22 Date: March, Mathematcs Subject Classfcaton. 65N2,65M2. Key words and phrases. Fnte volume approxmaton, Gradent reconstructon, Dscrete gradent, Dscrete dualty, 3D CeVe-DDFV, Consstency, Ansotropc ellptc problems, General mesh, Non-conformal mesh. A large part of ths paper was completed durng the vst of the frst author to the Unversty of Concepcón, Chle, supported by the FONDECYT program No The authors thank Franços Hermelne and Charles Perre for dscussons and remarks.

3 2 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL. Introducton.. The context of DDFV and related schemes. Dscrete dualty fnte volume (DDFV) dscretzaton of lnear and nonlnear dffuson operators, ntroduced for the Laplace problem n 2D by Hermelne n [44] and by Domelevo and Omnès n [30] (some key deas appear already n the works of Ncolaïdes [56] and Ncolaïdes and Hu [48, 56]), s one of possble dscretzaton strateges whch apples to very general meshes and to a large varety of PDEs ncludng the Stokes problem, the Maxwell equatons, nonlnear and lnear ansotropc dffuson and convectondffuson problems (see e.g. [5, 9, 3, 5, 22, 25, 28, 29, 30, 43, 44, 45, 46, 48, 49, 50, 56, 57] and references theren). The name DDFV stresses one mportant aspect of ths 2D scheme, namely the dualty between the dscrete gradent and the dscrete dvergence operators n use. Yet ths property s shared by varous numercal schemes (e.g. the mmetc ones, for whch the dscrete dualty property underles the defnton of the scheme); actually, the name DDFV refers also to the strategy of usng double meshes n 2D wth both cell and vertex unknowns. Among dfferent dscretzaton approaches ntended to resolve the dffcultes comng ether from ansotropy/nonlnearty of the PDE under consderaton or from the need of workng on non-orthogonal, non-conformal, locally refned meshes, let us menton those of [, 2, 7, 6, 7, 27, 3, 32, 33, 34, 35, 36, 37, 38, 39, 40, 52, 53, 59]; ths lst s by no means exhaustve. Several of the above works present dmenson-ndependent constructons, others are specfc to 2D and ther extenson to 3D requres new deas (ths s the case of the DDFV schemes). A 2D Benchmark that reflects the behavour of some of the above methods on lnear dffuson problems was presented by Herbn and Hubert n [4]. Among the aforementoned works, the closest to the DDFV strategy s the complementary volumes strategy (Walkngton [59]) as descrbed by Handlovčová, Mkula and Sgallar n [40], that was ntroduced n 2D n [54] (see also [55, 39]). The same dea was used n 2D, n slghtly dfferent ways, n [2] and n [0] (see also [4, Sec.2.]). A 3D verson was formulated n [8] and analyzed n [39]. From a dfferent vewpont, the DDFV methodology of [30] s nspred by the gradent reconstructon strategy per damond ntroduced by Coudère, Vla and Vlledeu [27]. In contrast to the above complementary volumes methods, a 2D DDFV mesh s a double mesh. It conssts of a prmal mesh, whch n general can be non-orthogonal and non-conformal (e.g., locally refned), and of an approprately chosen dual mesh. A dscrete functon s then a superposton of two constant per volume functons, one on the prmal mesh and the other one on the dual mesh. From the prmal and the dual mesh, a damond mesh s generated (a more general pont of vew s suggested n [9, IX.B]); the gradent of a dscrete functon s reconstructed as a constant per damond vector feld. Convergence analyss technques for these 2D schemes are well developed by now (see e.g. [9, 3, 30] for detals). In 3D, several types of methods nspred by the 2D DDFV methodology were already proposed. They dffer by the number and the nterpretaton of ther unknowns and/or by the constructon of the addtonal control volumes. We have recently learned of another nterpretaton, developed by Perre [58], n whch the DDFV methods are seen as standard fnte volume methods assocated to only one mesh. (A) The constructon due to Perre (see [57] and [24, 26, 23]) nvolves, lke n the 2D case, unknowns at both centers and vertces of the ntal (prmal) mesh. The faces of the prmal mesh are ether trangles or quadrangles. The dual mesh used by Perre et al. s rather unusual: t recovers the doman twce. The dscrete dualty property does hold for ths constructon (see [57, 26]), makng t a DDFV method. From the practcal pont of vew, the method of [57] was successfully appled to dscretzaton of the ellptc-parabolc bdoman cardac system n [24, 25, 26]. (B) Two constructons generalzng the 2D case are due to Hermelne. The one of [46] provdes a very wde famly of fnte volume schemes usng varous knds of 3D double meshes. In ths approach, addtonal unknowns are added at the faces of the prmal mesh; they are lnked to unknowns at the vertces through nterpolaton n order to overcome the

4 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 3 dffcultes of gradent reconstructon. No study of dualty features was conducted for ths method and, except n some partcular stuatons, the dscrete dualty fals. From the practcal pont of vew, ths method showed good convergence propertes on several tests ncludng mld heterogenety and ansotropy, see [46]; but t was supplanted by the more constraned method [47] of the same author where the face unknown s eventually elmnated. Numercally, the method of [47] mproves over the one of [46], e.g., on strongly dstorted meshes. The mproved method of Hermelne does enjoy the dscrete dualty property, see [47, Th.]; and t s presented wth specfc adaptaton to the more delcate case of dscontnuous dffuson coeffcents (cf. [3] for the 2D case). In [47], the faces of the ntal mesh are ether trangles or quadrangles, and the face centers are barycenters. These restrctons can be bypassed. Indeed, the method of [47] actually enters the framework (C) below when the dffuson tensor s contnuous. (C) The constructon of Karlsen and the frst two authors [4, 5] dffers from the one of (A) by the form of the dual cells, whch s more conventonal (see Sectons 2,4 for detals). Yet (A) and (C) lead to the same matrces on lnear dffuson problems (see [23]); only the projecton operators dffer and ths nduces slghtly dfferent dscretzaton of the source term. The mproved constructon of Hermelne [47] that took over the method (B) actually amounts to the same method (C). The focus n [5] was on the orthogonal(delaunay-voronoï, accordng to [46, 47]) meshes wth trangular faces, whle the gradent reconstructon announced n [4] s applcable for the generalcase. In the present paper, we descrbe ths approachfor a wde class ofmeshes (no restrcton on number of face vertces appears; nether orthogonalty nor conformty s needed) and, most mportantly, we assess the essental dscrete dualty feature of the DDFV method, for ths wde class of meshes. From the practcal pont of vew, the method (C) was used n [6] for a further numercal study of the bdoman cardac model. (D) The last scheme developed by Coudère and the thrd author n [9, 20] modfed the dea of [46] and added unknowns not only at faces, but also at the edges of the prmal mesh. These unknowns used for gradent reconstructon are not nterpolated, contrarly to [46]. Both face and edge ponts are seen as the cell centers for a new mesh whch does not appear n the prevous constructons. Consequently, n the fnte volume approxmaton of [9, 20] three meshes are nvolved, makng t a trple mesh. The dscrete dualty property for ths constructon was shown n [20]. The approach of [20] recently gave rse to the scheme on 3D prmal meshes wth quadrangular faces, as sketched n [38, Sec.2.2]; ths s a 3D analogue of the 2D constructon n [9, IX.B]. Let us pont out that the scheme of [38, Sec.2.2] can be rather naturally re-nterpreted as a DDFV scheme of knd (A). From the practcal pont of vew, the DDFV method (D) was successfully appled n [20] (for Leray-Lons ellptc problems or general meshes) and n [5] (for the Stokes problem). Numercal results of all these schemes [24, 46, 47, 20, 6] show good convergence propertes, even on strongly dstorted meshes. A drect comparson of these schemes wll be held through the 3D ansotropc benchmark [, 2, 23] and [42]. Accordng to the locaton of unknowns wrt the prmal mesh, one can qualfy the methods (A), (B) n the verson [47], and (C), as CeVe-DDFV methods (cell and vertex unknowns); and the method (D), as a CeVeFE-DDFV method (cell, vertex and face+edge unknowns). We have already ponted out (cf. [6, 23]) that dscretzatons orgnatng form (A),(B) (verson of [47]) and (C) concde, up to detals that are not of prme mportance. Whle (D) appears as a qute dfferent method (n partcular, the data structure used for mplementaton s dfferent), a strong connecton exsts when the prmal meshes have quadrangular faces. Indeed, the method (A) can be put n correspondence wth the verson [38] of the method (D) (upon splttng the dual mesh The meshes and dscrete dvergence of [46] concde (up to unnecessary restrctons) wth those of our paper; thus the dscrete gradents also concde, snce the dscrete dualty property that holds for both schemes. Yet the motvatons behnd the reconstructon of the dscrete gradent of [47] (see also [57]) and the ours are rather dfferent.

5 4 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL of (A) nto two meshes each of them recoverng the doman only once). Further, the method (C) on cartesan meshes (see Secton 2.3) rewrtes as a varant of (D), see the proof of [3, Prop. 3.7]..2. The goal of the paper. Ths paper gves a detaled account on the CeVe-DDFV scheme n ts verson (C), on general meshes, and t prepares the ground for the sequel [3] where we provde convenent tools for the mathematcal analyss of DDFV dscretzatons on three-dmensonal domans. We exhbt a consstent dscrete gradent operator whch possesses the same knd of ntegraton-by-parts property as the 3D schemes of [57], [5], [20] or as the 2D DDFV schemes (see [9, 30]). Ths means that the dscrete gradent operator T s dual to the usual fnte volume dscrete dvergence operator dv T on the double mesh, for approprately defned L 2 parngs of dscrete functons and of dscrete vector felds. A bref summary of our 3D DDFV constructon and of the dscrete dualty result, along wth an applcaton on Delaunay-Voronoï meshes detaled n [5], were gven n [4] by Karlsen and the frst two authors; wth respect to [4], the present paper gves a smpler (but equvalent) formula for the dscrete gradent, many generalzatons, examples and numercal tests. The reason we focus on the dscrete dualty features s that they make the fnte volume dscretzaton of numerous ellptc operators structure-preservng ; and they underle the convergence proofs (see, e.g., [30, 9, 3, 5, 20, 6] and the sequel [3] of ths paper). Note that dfferent varants of the dscrete dualty property hold for the schemes of [32, 3], of [33,34,37,35], of[7]. Forthe complementaryfntevolumes schemesasdescrbedn[40,2,0,4], a dscrete dualty property, completely smlar to the one shown n the present paper, s true n 2D (see [4] and Remark 4.6). For the mmetc fnte dfference (MFD) schemes, the dscrete dualty s bult nto the defnton of the schemes (so that the MFD schemes can be defned through only one of the two dscrete operators). Also note that the dscrete dualty property can be somewhat relaxed. Eymard, Herbn and Guchard ntroduced n [38] the noton of gradent scheme, ncludng some 3D DDFV schemes. Gradent schemes accordng to [38] fulfll an approxmate dscrete dualty property, property that s suffcent to nfer convergence at least for the lnear test cases as those consdered n Secton 5 of the present paper (see [38, Sec.2])..3. Bref descrpton of the operators. In the 3D DDFV framework (C) we postulate, ndependently one from another 2, qute natural defntons for both T (of a dscrete feld F ) and dv T (of a dscrete functon u T ) on a double mesh T. Namely (up to notatonal detals; cf. Secton 2.2), dv T F s obtaned by the Green-Gauss formula from the ntegraton of F n on the boundares of the volumes of the double mesh (ths s the core of any fnte volume approach); Damond wth three dual vertces x L Damond wth l=4 dual vertces x L Fgure. (a) A damond: the smplest case (b) The general case (here, l = 4 ) 2 at ths pont, our approach dffers n ts sprt from the one of mmetc fnte dfference (MFD) schemes

6 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 5 T u T s reconstructed pecewse constant per damond, by a well-chosen ponderaton of the vertex values; the reconstructon s consstent wth affne functons. Consder a generc damond D = D K,L bult on two prmal vertces,x L and on l dual vertces, =,...,l of the double mesh (see Fgure ). In ths case, the vector g =: D u T s assembled from ts projecton Proj D g on the drecton x L and from ts projecton Proj g on the plane contanng the face K L. Each of the projectons D s unquely determned from the values of u T at the vertces of D, as follows. Frst, the projecton Proj D g s obtaned from the values u K,u L at the ponts,x L n a straghtforward way: Proj D D u T = u L u K x L. x L Second, the projecton Proj g s reconstructed from the values x D K attached to the ponts n the face K L. If l = 3 (see Fgure (a)), there exsts a unque affne functon w on the face K L such that w( ) = u K, for =,2,3. Naturally, we mpose that Proj g = w. Thanks to a remarkable dentty (Lemma 5. and Corollary D 5.2 gven n Appendx A), ths can be rewrtten as follows (wth l = 3 and l+ := ): () Proj D D u T = l (u K m + u K ) [ n L K K L +], = where m K L s the area of the face K L, n K L s the unt normal vector to K L pontng from K to L, L s some fxed pont (a center) n the face K L, and K s the + mddlepont (the center) of the edge [, + ]. Further, when a damond spans l > 3 dual vertces,..., l (see Fgure (b)), an affne functon w nterpolatng the vertex values does not exst n general. Yet formula() stll makes sense; moreover, by Lemma 5. and Corollary 5.2, ts rght-hand sde stll yelds w whenever w exsts. Due to ths remarkable dentty 3, we postulate the choce () to defne Proj D Du T. The justfcaton of the remarkable dentty s the most techncal part of the constructon, and t s of ndependent nterest; we postpone t to Appendx A. To gve an example, 3D DDFV schemes on cartesan meshes are descrbed n Secton 2.3, and the assocated dscrete operators are wrtten down explctly..4. Outlne of the paper. Along wth the constructon tself, the key theoretcal results of the paper are those of Proposton 2.3 and Proposton 3.2. The lnk between consstency and dscrete dualty s based upon the reconstructon property of Lemma 5. (Appendx A) whch s of ndependent nterest. For smplcty, n Secton 2 we restrct our attenton to the case of convex prmal volumes, and mpose constrants on the choce of the centers of volumes and faces; these constrants can be relaxed (see Remarks 4., 4.2 and Example 4.3 n Secton 4 devoted to generalzatons). The proof of the dscrete dualty property gven n Secton 3 requres a rather nvolved notaton, ncludng a conventon on the orentaton of the damonds. The notaton s ntroduced n Secton 2 and n Lemma 3.3, and llustrated wth the help of fgures. One beneft from the notaton s that we are able to gve closed-form formulas for the dscrete gradent and for the dscrete dvergence n terms of vector products and mxed products n R 3. Let us pont out that the notaton n terms of dscrete operators, dscrete functons and felds and ther scalar products as ntroduced n Secton 2 s also amed to gude the reader through the convergence proofs presented n the sequel [3] of ths paper. Ths notaton stresses the far-reachng analogy between the contnuous framework and the dscrete DDFV framework. Whle convergence proofs and some numercal experments for several degenerate parabolc problems wll be brefly presented n [3], numercal tests on lnear ansotropc dffuson problems are the object of Secton 5. These tests are based on the 3D benchmark for lnear ansotropc 3 It s an analogue of the celebrated magcal formula of [32, Lemma 6.].

7 6 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL dffuson problems prepared by R. Herbn and the thrd author, [42]. Convergence of the method n thesetestcasesseaslyestablshedusngthedscretedualtypropertyandthetoolsof[30,9,3,3]. 2. Double meshes and the assocated gradent and dvergence operators Let Ω be a polyhedralopen bounded subset of R 3. In what follows, we ntroduce the notaton related to double fnte volume schemes; each pece of new notaton s gven n talc scrpt. The notaton s redundant n many cases, whch s convenent because the role of objects we ntroduce s often multfold. Most of the notatons are llustrated wth the help of fgures. Throughout the paper, a denotes the eucldean norm of a R 3 ; a b (respectvely, a b ) denotes the scalar product (respectvely, the vector product) of a, b R 3 ; a, b, c denotes the mxed product of a, b, c R 3. We use extensvely the geometrc meanng of these products. 2.. Constructon of double meshes. A partton of Ω s a fnte set of dsjont open polyhedral subsets of Ω such that Ω s contaned n ther unon, up to a set of zero three-dmensonal measure. A double fnte volume mesh of Ω s a trple T = ( M o,m,d ) descrbed below. We take M o a partton of Ω nto open polyhedra. We assume them convex. Each K M o s called control volume and suppled wth an arbtrarly chosen center ; for smplcty, we assume K. We call M o the set of all faces of control volumes that are ncluded n Ω. These faces are consdered as boundary control volumes ; for K M o, we choose a center K. We denote by M o the unon M o M o. We call vertex (of M o ) any vertex of any control volume K M o. In the case of conformal meshes, we call neghbours of K, all control volumes L M o such that K and L have a common face. In the non-conformal case, t s enough that K and L ntersect along a set of non-zero two-dmensonal measure. The set of all neghbours of K s denoted by N(K). Note that f L N(K), then K N(L) ; n ths case we smply say that K and L are (a couple of) neghbours. If K and L are neghbours, we denote by K L the nterface (face) (or ts part, n the nonconformal case) K L that separates K and L. Due to the convexty of K,L, the nterface K L s planar. A generc vertex of M o s denoted by ; t wll be assocated later wth a unque dual control volume K M. Each face K L s suppled wth a face center L whch should le n K L (the more general stuaton s dscussed n Remarks 4., 4.2). For two neghbour vertces and x L (.e., vertces of M o joned by an edge of some polygon K L ), we denote by L the mddlepont of the segment [,x L ]. Now f K M o and L N(K), assume,x L are two neghbour vertces of the nterface K L. We denote by T K;L the tetrahedron formed by the ponts ;L K,x K, L, L. Agenerctetrahedron T K;L K ;L s calledan element ofthe mesh and denotedby T (see Fgure4); the set of all elements s denoted by T. If s a vertce of T T, then we say that T s assocated 4 wth the volume K, and we wrte T K. Defne the volume K assocated wth a vertce of M o as the unon of all elements T T havng for one of ts vertces. The collecton M of all such K forms another partton of Ω. If Ω, we saythat K s a dual control volume and wrte K M ; and f Ω, we say that K s a boundary dual control volume and wrte K M. Thus M = M M. We call dual vertex (of M ) any vertex of any dual control volume K M. Note that by constructon, the set of vertces concdes wth the set of dual centers ; the set of dual vertces conssts of centers, face centers L and edge centers (mddleponts) L. 4 Because we have made the assumpton that xk L K L, the relaton T K smply means that T s ncluded n K. We wll develop a more general pont of vew n Remark 4.2. The same observaton apples to the notaton T K, T D, S D ntroduced later on.

8 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 7 Pcturng dual volumes n 3D s a hard task; an dea of how the dual volumes look lke can be nferred from a deformaton of the example gven n Secton 2.3 (see Fgure 3). If s a vertce of T T, then we say that T s assocated wth the volume K, and we wrte T K. We denote by N (K ) the set of (dual) neghbours of a dual control volume K, and by K L, the (dual) nterface K L between dual neghbours K and L. For an element T = T K;L, we denote by σ K ;L T the face of T contaned wthn the plane K L. The area of σ T s denoted by m T ; fnally, n S denotes the unt normal vector to σ T exteror to T. Smlarly, we denote by σ T the face of T contaned wthn the plane K L. The area of σ s denoted by T m,and T n s the correspondng exteror unt normal vector. T Fnally, we ntroduce the partton of Ω nto damonds. If K,L M o areneghbours, then the unonofthe convexhullof and K L wth the convex hull of x L and K L s called damond and denoted by D K L. In the sequel, to each damond we wll prescrbe an orentaton by fxng arbtrarly the orentaton of the segment [,x L ]. We denote by D the set of all damonds. Generc damond n D s denoted by D. Notce that D s a partton of Ω. We wll wrte T D to sgnfy that the element T T s ncluded wthn D (or, more generally, s assocated wth D ; see Secton 4). 2 volume K K orentaton 3 K 3 nterface K K volume K damond D K K 2 K K 3 K 3 subdamond S K K K 3 K 3 K 3 x 2 x, e, n, x Smplfed notaton n a damond x subdamond x 3 S K K K 3 K x x 3x x 3, x, x 3, Fgure 2. 3D neghbour volumes, damond, subdamond. Zoom on a subdamond. (see Fgure2) Wheneverthe orentatonofadamond D should be caredof, the prmalvertces defnng t wll be denoted by, n such a waythat the vector has the orentaton prescrbed for the damond. The orented damond s then denoted by D K K. We denote by e K the correspondng unt vector, and by d,k K, the length of x x,k K. We denote by K n the unt normal vector to K K K K such that n e K K K,K > 0. The normal vector n to K K K K beng fxed, ths nduces the orentaton n the correspondng face K K, whch s a convex polygon wth l vertces : we denote the vertces of K K by, [,l], enumerated n the drect sense. By conventon, we assgn l+ :=. We denote by e K,K the unt normal vector pontng from x + K towards +, and by d K,K, + the length of +. In order to lghten the notaton n an orented damond (and n ts subdamonds, ntroduced n Secton 3 below and pctured n Fgures 2,4), we wll drop the K s n the subscrpts and denote the objects ntroduced above by x, x, e,, d,, n, and by x, e,+, d,+ whenever D K K s fxed. We also denote by x the mddlepont x,+ K K of the segment [x +,x + ], and by x,, the center K of K K. For a damond D = D K K, we denote by Proj D the operator of orthogonal projecton of R 3 on the lne < e K,K > ; we denote by Proj D the operator of orthogonal projecton of R3 on the plane contanng the nterface K K.

9 8 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL We denote by Vol(A) the three-dmensonal Lebesgue measure of A whch can stand for a control volume, a dual control volume, an element, a damond, etc.. Remark 2.. Damonds permt to defne the dscrete gradent operator (see (4),(5) below). In turn, elements (and, even more convenently, the subdamonds ntroduced later on) permt to defne the dscrete dvergence operator (see (2),(3) and (2) below). In the context of 2D double schemes, ntroducng damonds s qute standard (see e.g. [9, 30]). Subdamonds are hdden n the 2D constructon : they actually concde wth damonds. The above defntons are llustrated and generalzed n Remark 4.2 at the end of the paper Spaces of dscrete functons and felds, dscrete dvergence and gradent operators. A dscrete functon on Ω s a set w T = ( w Mo,w M ) consstng of two sets of real values w Mo = (w K ) K M o and wm = (w K ) K M. The set of all such functons s denoted by RT. A dscrete functon on Ω s a set w T = ( w Mo,w M ;w Mo,w M ) ( wt ;w T) consstng of w Mo = (w K ) K M o, wm = (w K ) K M, w Mo = (w K ) K M o, w M = (w K ) K M. Thesetofallsuchfunctonssdenotedby R T. Incaseallthecomponentsof w T = ( w Mo,w M) are zero, we wrte w T R T 0. A dscrete feld on Ω s a set FT = ( FD )D D of vectors of R3. The set of all dscrete felds s denoted by (R 3 ) D. If F T s a dscrete feld on Ω, we assgn F T = F D whenever T D. On the set (R 3 ) D of dscrete felds F T, we defne the operator dv T [ ] of dscrete dvergence by (2) dv T : F ( T (R 3 (dvk ) D dv T FT = F ) T, ( dv K ) ) F T R T, K Mo K M where the entres dv KFT, dv K FT of the dscrete functon dv T FT on Ω are gven by (3) dv KFT = m TFT n T, dv K FT = Vol(K) T K Vol(K ) m F T K T T n. T Formulas (3) correspond to the standard procedure of fnte volume dscretzaton appled on each part of the double mesh T. A more explct formula (2) for the dscrete dvergence s gven n (2) and (5) n Secton 3. By constructon, t s clear that the dscrete dvergence operator s consstent. Its consstency (n the weak sense; cf. [3, Prop. 3.()]) can also be nferred from the one of the dscrete gradent operator (see Proposton 2.3) and from the dscrete dualty property that s the object of the next secton. On the set R T of dscrete functons w T on Ω, we defne the operator T [ ] of dscrete gradent (4) T : w T R T T w T = ( D w T) D D (R3 ) D where the entry D w T of the dscrete feld T w T correspondng to D = D K K s gven by (5) Proj D ( D w T ) = w w e,, d, D w T s s.t. Proj ( 2 l D Dw T )= l = n,, x, x, x,+ x + (w+ w ) [ n, x,,+] x. wth w = w K, w = w K, w = w K, etc. (we use the smplfed notaton n the damond D = D K K, as depcted n Fgure 2). Remark 2.2. In (5), the prmal mesh M o serves to reconstruct one component of the gradent, whch s the one n the drecton e,. The dual mesh M serves to reconstruct, wth the help =

10 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 9 of the formula (26) of Appendx A, the two other components whch are the components n the plane contanng K K. The above choce of the dscrete gradent s explaned n the Introducton; the specfc choce of Proj D( D w T ) stems from the analyss of Appendx A. Now we pont out the crucal fact that (5) and Corollary 5.2 n Appendx A mply the consstency of our gradent approxmaton: Proposton 2.3. Let w,w,(w,+ )l = be the values at the ponts x,x,(x,+ )l =, respectvely, of an affne on D = D K K functon w. Then D w T concdes wth the value of w on D. Now let us gve a more explct representaton of D w T. Lemma 2.4. The defnton (5) of D w T s equvalent to the representaton l { x (6) D w T x, x, x =, x,+ x + 6Vol(D) (w w ) n, +2(w+ x = x n w ) [ x x x,,+] } x., Notce that n formula (6), we can substtute the mxed product x x, x, x, x,+ x + by 6Vol(S K K K K + ), where S K K K K + s the subdamond ntroduced later on (see Fgure 4). Proof. Frst, notce that f p = Proj D ( D w T ), p = Proj D ( D w T ) are gven, we have (7) D w T = p + e, ( p p ) e, n, n,. Consder the contrbuton of the term (w + w amounts to (w + w ) e, n, ) [ n, x,,+] x n p nto formula (7). It { [ n, x ], x ( e, n,+, ) n, ( [ n, } x, x ).,+] e, Usng the well-known BAC mnus CAB dentty ( b a) c ( c a) b and the fact that n, [ n, ] x, x,+ = x, x (w +, we transform the above expresson nto w ) [,+ e e,, n, x, x,+] = (w + w ) [ x x n, x x x, x,+]. Further, notce that and that ( x x n, ) n,, x, x,+, x x + = x x, x, x,+, x x +, l x x, x, x, x,+ x + = 6Vol(D). = e Fnally, the contrbuton of p nto (7) amounts to, p e, n, n, = w w x x n, n,. Combnng the obtaned denttes, we get (6) from (7). w w d, e, n, n, = 2.3. Cartesan DDFV meshes n 3D. Let us gve one very smple yet mportant concrete example of a 3D DDFV mesh as descrbed above. It s sutable for parallelepped domans and ther unons. For the sake of smplcty, we take the unt cube Ω = [0,] 3 and partton t nto N 3 prmal cubc volumes of edge N (nether the cubc form, nor the unformty of the meshes s mportant; the constructon generalzes to non-unform cartesan meshes). Then one easly sees that damonds are octahedrons bult on two prmal cubes centers x, x and on the square nterface K K K K between them. If one chooses for x the center of symmetry of K K K K, then the nteror dual volumes are also cubes of the same edge N centered at the vertces of the prmal mesh that do not le on Ω. The boundary dual volumes are ether /8th of the standard cube (at the corners of Ω ) or the quarter-cubes (at the edges of Ω, exceptng the corners) or the half-cubes (on the faces of Ω, exceptng the edges).

11 0 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL Dual volume K Four prmal cells and a dual cell A prmal nterface and ts damond Prmal volume K NE Damond D K,K NW SW Interface K K SE Prmal volume K Prmal volume K Fgure 3. Cartesan DDFV mesh n 3D and an assocated damond To gve the entres of the assocated dscrete gradent, for the sake of beng defnte we consder a damond D = D K K wth the nterface K K parallel to the horzontal plane xoy and wth x x pontng up n the drecton Oz (see Fgure 3). We take the conventon that the Ox axs ponts from West to East and the Oy axs ponts from South to North. Then wemark the vertces of the square K L counterclockwse (wth respect to the orentaton gven by Oz and x x ) by SW, SE, NE and NW, wth an obvous cartographc nterpretaton of the notaton. Wth ths notaton, applyng the reconstructon formula (5) one fnds that the entry D w T of T w T s computed as (8) D w T = w K w K /N k + 2( wk NE w K NW /N + w K w ) SE K SW + ( wk NE w K SE /N 2 /N + w K w ) NW K SW j /N (recall that d K,K = d K NE,K =...= NW N ); here the trple (, j, k ) s the canoncal bass of R 3. Further, let K be an nteror prmal volume. For the sx damonds that ntersect K, we ntroduce the specfc notaton Dabv(K),Dblw(K), D E (K),D W (K) and D N (K),D S (K). For nstance, D S (K) contans the south-ward face of K (the exteror unt normal vector to K D S (K) s the vector j ), and Dabv(K) contans the upper face of K (the correspondng exteror normal vector s k ). Wth ths notaton, applyng formula (3) one fnds that the entry dv KFT of dv T F T s computed as dv KFT = { /N 3 N 2( F DE(K) F DW(K)) + N 2( F DN(K) F DS(K)) j+ N 2( F D abv (K) F } D ) k blw (K). The formula for the entres of dv K FT has an entrely smlar form. The above meshes and dscrete operators can be compared to the 2D constructons of [7, 8] and to the 2D and 3D complementary volumes constructons of [39, 40, 55]. As the schemes of [8] and of [39], the above DDFV scheme tends to avod preferred orentatons (cf. the dscusson n [39]). 3. The dscrete dualty property Let us defne the convenent multplcaton of dscrete functons/dscrete felds and state our man result. Recall that R T s the space of all dscrete functons on Ω. For w T,v T R T, set ] (9) [w T, v T = Ω 3 Vol(K) w Kv K + 2 K M o 3 K M Vol(K ) w K v K ;

12 t s clear that 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I [ ], s a scalar product on Ω RT. Remark 3.. Notce that, contrarly to the 2D DDFV method, the role of the prmal and the dual meshes n 3D CeVe-DDFV methods s not symmetrc; ths asymmetry s also reflected by Remark 2.2. In R d, the prmal mesh would account for d of the product, and the dual mesh, for d d (see [5]). Note that n the 3D CeVe-DDFV scheme developed by Perre et al. (see [57, 24, 25, 23]), the weghtsnthescalarproduct(9)areboth equalto 3 ; but, becausethedualmeshn[57,24,25,23] covers twce the doman Ω R 3, ts weght s doubled wth respect wth the weght of the prmal mesh. Ths s smlar to what happens n our formula (9). As to the 3D CeVeFE-DDFV scheme developed by Coudère and Hubert (see [9, 20, 2]), a trple mesh s nvolved, and the assocated scalar product takes the contrbutons of each of the meshes wth the equal weghts 3. Recall that (R 3 ) D s the space of all dscrete felds on Ω. For FT, G T (R 3 ) D, set (0) { FT, G } T Vol(D) F D G D ; Ω D D { } t s clear that, Ω (R3 ) D. Proposton 3.2. The dscrete dvergence and gradent operators dv T, T defned n Secton 2 are lnked by the followng dualty property : () w T R T 0 F [ ] { } T (R 3 ) D dv T FT, w T = FT, T w T. Ω Ω Before turnng to the proof, let us ntroduce the partton of Ω nto subdamonds (cf. [5, 4]), as used n the proof, and gve two more formulas for the dscrete dvergence operators (namely, (2) and (5) below). If K,L M o are neghbours, and,x L are neghbour vertces of the correspondng nterface K L, then the unon of the four elements 5 T K;L,, K ;L TK;L L,K TL;K K,L, and TL;K s called L,K subdamond and denoted by S K L. In ths way, each damond K L DK L gves rse to l subdamonds ( l beng the number of vertces of K L ). We denote by S the set of all subdamonds. Generc suddamond n S s denoted by S. Each subdamond s assocated wth a unque nterface K L, and thus wth a unque damond D K L. We wll wrte S D to sgnfy that S s assocated wth D ; n ths case, we set FS = F D for a dscrete feld F. Further, f (resp., ) s a vertce of S, then we wrte S K (resp., S K ). Lemma 3.3. Formulas (3) can be rewrtten under the form dv K FT = ( ) ǫk S FS, x 2Vol(K), x,+, x x +, S K (2) dv K FT = S FS, x x, x 2Vol(K, ) x.,+ S K ( ) ǫk In formulas (2), we mean that each subdamond S assocated wth K (resp., wth K ) has the form S = S K K, wth some K K K,K,K,K + ; the notaton under the sgn refers to + (see Fgure 2). Because K may concde ether wth K or wth K (smlarly, K S = S K K K K + may be K or K + ), the sgn selectors { { 0, f (3) ǫ K := K = K, ǫ S, f K = K 0, f K = K := S K, f K = K + are ntroduced n (2). Proofof Lemma 3.3 : Fx a subdamond S S, S = S K K. The subdamond contrbutes K K + twce to each of the formulas (3) correspondng to K = K, to K = K, to K = K, and to 5 Ths defnton should be generalzed f the constrant xk K s dropped; see Remark 4..

13 2 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL element T K ;K K ;K 3 2 K 3 3 K n, nterface σ S (part of K K+) angle α, e, n S n, x subdamond n S S K K K K x x 0000 x,, e 000, x 0 x0 nterface σ S (part of K K+) nterface σ S (part of K K ) Fgure 4. Element. Subdamond: σ S,σ S,σ S and ther normal vectors. K = K + ; each contrbuton s brought by two of the four elements T T that consttute S. We denote the four elements by T := T K ;K, K ;K T + + := TK ;K K + ;K + and T := T K ;K, K ;K T + + := TK ;K K + +.,K It s convenent to splt S = S K K K K + nto the lower and the upper parts S := T T + and S := T T + (see Fgure 4). Each of the two parts contans one flat porton of the nterface K K +, and they are separated by the common face K K S. We denote these faces as follows: σ S := K K S, σ S := K K + S, σ S := K K + S. The areas of σ S, σ, S σ are denoted by m S S, m, S m, respectvely; one calculates them S from the areas of faces of the elements T, T, T +, T +. We denote by n S, n, S n the unt normal vectors to σ S S, σ, S σ, respectvely, wth the S orentatons that are chosen so that to satsfy (4) n S = n, and n, n S,, x, x 0,,+ n, n S,, x, x 0.,+ Thanks to the constrants x K, x K and to the orentaton choce n S x x = d, n, e, > 0, vector n S ponts from K to K ; thus by defnton (3) of ǫ K S, the vector ( )ǫk S n S s the unt normal vector to σ S K exteror to K, n the case K = K and also n the case K = K. Thus for K = K and for K = K, for all element T S such that T s assocated to K, we have n T = ( ) ǫk S n S. Smlarly, havng chosen x, K K we ensured that n S, n S pont from K to K +, thanks to(4) and tothe fact that the vertcesof K K arenumberedaccordngto the orentaton of n,. By (3), ( ) ǫk S n, S ( )ǫk S n are the unt normal vectors to the flat portons S σ, S σ of S K, exteror to K. Thus for K = K and for K = K +, for all element T S (resp., T S ) such that T K, we have n T = ( ) ǫk S n S (resp., n T = ( ) ǫk S n S ). (5) Now we see that (3) rewrtes under the form dv KFT = m SFS ( ) ǫk S ns, Vol(K) S K dv K FT = ( FS ( ) ǫk S m Vol(K S ) n S +m S S ) n. S K

14 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 3 Fnally, notce that the above defntons and the standard propertes of the nner product of R 3 yeld m S n S = 2 x, x,+ (6) x x +, m S n S +m S n = S 2 x x, x, x +,+ 2 x, x x, x =,+ 2 x x x, x.,+ Usng equaltes (6), from (5) we deduce (2). Ω Proof of Proposton 3.2 : The proof s by drect calculaton, usng the summaton-by-parts procedure. ] Denote the product [ dv T FT, w T by Z. Frst, we put together the terms n Z correspondng to adjacent couples of prmal and dual volumes. Ths amounts to make the summaton over all subdamonds S = S K K (see Fgures 2, 4 for the notaton n K K S ); ghost terms correspondng to the boundary volumes [ ] can be added, because w T s zero on the boundary volumes. + Wth the expresson (9) of,, usng formulas (2) of Lemma 3.3, takng nto account the Ω sgns selectors conventon (3) whle summng by parts, we fnd Z = ( S S,S=S K K F S 3 (w K w ) [ x, x x,+ +] x + 3 (w + w ) [ K + Z D := S D,S=S K K K K + ( F S Vol(S K K K +) K x x ] ) x, x,+ w w x x n, n, + 3 (w + w ) [ x x x,,+] ) x. = S S,S=S K K K K + Next, we put together the terms correspondng to the subdamonds S assocated wth the same damond D ; snce F S = F D n ths case, ths amounts to make the summaton over all damonds D = D K K (see Fgures 2, 6 for the notaton n D ). We get Z = F D D D Z D wth ( Vol(S K K K +) K w w x x n, n, + 3 (w + w ) [ x x x, x,+] ). equals [ Vol(D) Dw T ] T. Accordng { to the} ex- dv T FT, w T = Z = FT, T w T. Thanks to Lemma { 2.4, } the above expresson Z D presson (0) of,, we have justfed that Ω Ω Ω 4. On the choce of the face centers L and other generalzatons Here we provde a seres of remarks that dscuss and generalze the above constructon. Remark 4.. (an overvew of the dfferent constrants) Gven a partton of Ω nto convex dsjont open subsets K, one can always construct a double mesh T satsfyngthe constrantson the choceof, L, L mposed n Secton 2. But f one wants to use some smpler constructons such as the classcal Voronoï dual mesh n Example 4.3 below, some of the below generalzatons are needed. The barycenter (mddlepont) choce L = 2 (x + L) on the edges cannot be relaxed. Ths choce s n the heart of the consstency property of Proposton 2.3 (see the proof of Lemma 5.). The constrant L K L can be relaxed, and ths generalzaton s mportant. In partcular, t may be convenent (as e.g. n Example 4.3 below) to let L be the pont of ntersecton of the plane contanng the nterface K L wth the lne passng through the centers,x L. Even f K and the volumes are convex, the ntersecton pont can fall outsde K L. In fact, the property L K L s only needed to ensure that the subdamond volume, Vol(S) = 6 x x, x, x, x,+ x +, s postve. Yet, ths postvty restrcton does not appear n Lemma 5., Corollary 5.2 and Proposton 2.3. Let us stress that all the formulas n terms of vector and mxed products gven n ths note can be used wthout changes, f

15 4 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL L belongs to the plane contanng K L (see n partcular Remark 5.3). But t becomes necessary to generalze the notons of K and Vol(K ), allowng for subdamonds and elements wth negatve volume. We througly llustrate the stuaton n Remark 4.2 below. The requrement that belong to K can be relaxed, whch s mportant as well. The defnton of damonds and subdamonds becomes a bt more complcated n ths case, because some elements T T may have negatve volume; the stuaton n entrely smlar to that of Remark 4.2 below. In order to have Vol(D) > 0, t suffces to guarantee that the normal to K K vector n, (whch, by defnton, forms an acute angle wth x x ) pont from K to K. To ths end, the Delaunay property s requred n Example 4.3. The convexty constrant on the prmal volumes K can be relaxed. E.g., f each volume K s star-shaped wth respect to some pont, the constructon goes on wthout any change. More generally, we can even admt non-planar faces, by separatng them nto planar parts. Remark 4.2. (on the constrant L K L and ts relaxaton) () Under the assumptons that K for all K, L K L for all K L, the set of all elements T s a partton of Ω, and each of the parttons M o, M, D of Ω s obtaned by combnng elements. More exactly, we have K = T K;L K ;L, where the unon runs over all L N(K) and all K,L whch are neghbour vertces of the polygon K L. Smlarly, K = T K;L K ;L, where the unon runs over all K,L such that T K;L K ;L makes sense. Fnally, D K L = ( ) T K;L K ;L TK;L L ;K TL;K K ;L TL;K = L ;K S K L K, L where the unon runs over all couples {K,L } of neghbour vertces of the nterface K L. When an element T T contrbutes to the constructon of K, we say that T s assocated wth K, and wrte T K. We therefore have K = T K T, and Vol(K) = T K Vol(T). Analogous meanng s gven to the notaton T K, T D, and S D, T S ; e.g., (7) D = S, and Vol(D) = Vol(S). S D S D In each case, the relaton smply means the ncluson. () Now, for one example where L / K L, let K L be a trangle wth vertces denoted by,x L,x M, wth obtuse angle at x L ; let L be the center of crconscrbed crcle of the trangle whch therefore falls outsde K L (ths stuaton occurs n Example 4.3). Instead of the decomposton D K L = S = S K L K L SK L L M SK L M, K we now have S D D K L = ( S K L ) K L SK L L M \ K L S M. K But f (wth the notaton of Fgures 2, 4) we keep the formula (8) Vol(S) = 6 x x, x, x,+, x x + for the volume of S, we see that Vol(S K L M K ) becomes negatve, and cancellatons lead to Vol(D K L )= Vol(S K L )= Vol(S). K L ) + Vol(SK L L M ) Vol(SK L M K ) =Vol(SK L K L )+Vol(SK L L M )+Vol(SK L M K S D We see that the set-theoretc relaton n (7) looses ts sense, but the formula for Vol(D) keeps workng. Smlarly, a prmal volume K s a set of ponts of Ω that can be obtaned by the operatons, \ from the elements T assocated wth K, and Vol(K) = S K Vol(T) ; the (sgned) volume of T can be computed by a formula smlar to (8). Let us pont out that sgn(vol(t)) = sgn(vol(s)) when T S, and Vol(S) = T S Vol(T).

16 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 5 The general stuaton s the same as n the above example. Notce that nether T, nor S form a partton of Ω ; but each one forms a sgned partton of Ω n the sense that (9) sgn(vol(t))l T(x) = and sgn(vol(s))l S (x) = a.e. on Ω T T S S (here l A ( ) stands for the characterstc functon of a set A Ω ). The stuaton wth dual volumes K can be more ntrcated: K may n general consst of a postve and a negatve part, to whch no set-theoretcal meanng can be gven 6 ; but we can gve the sense of Vol(T) to T K Vol(K ). In ths case, let us call K a generalzed dual volume. Here a constrant appears on the choce of the famly ( L ) K L of the face centers: one should keep Vol(K ) > 0, n order that (9) be a scalar product. Smlar nterpretaton can be gven to the dscrete dvergence formulas (2). In (2), the normal flux of FS through σ S s automatcally taken wth the same sgn as sgn(vol(s)), and the cancellatons n the expresson S N(K) m S F S ( ) ǫk S ns make t be equal to the normal flux of the feld F through K. Notce that the relaton S K should be understood n the sense that S T for some T K. Smlarly, for a (possbly generalzed) control volume K, the flux through ts (possbly generalzed) boundary K sthe sumofthesgnedcontrbutonsofthe normalfluxesthrough σ S,σ S S wth S K. Formulas (2) take care of ths conventon. () Clearly, these conventons should affect the dscretzaton of source terms on the mesh T. In the case () above, one naturally defnes the projecton of f L (Ω) on the space R T of dscrete functons by f ) T = (( f K ) K Mo,( f K) K wth f M K = Vol(K) f, fk = f. But K Vol(K ) K n the case (), we should rather generalze these formulas and wrte ( ) (20) fk = Vol(K ) sgn(vol(t)) f = Vol(T) T K T K T T Ksgn(Vol(T)) f T when K sageneralzedvolume. The sgnedpartton property (9) s aclue tothe consstency of such projecton operator. The dscrete functonal analyss propertes gven n [3] can be proved also n ths generalzed framework, under some addtonal restrctons such as Vol(K ) > 0, Vol(D) > 0. Example 4.3. (Delaunay-Voronoï meshes) Let the prmal mesh M o of Ω R 3 be such that each K M o s a polyhedre admttng a crcumscrbed ball wth center (for nstance, a tetrahedron), and assume that all neghbour volumes K, L satsfy the standard Delaunay condton. It follows that each face K L s an nscrptable polygon. Take for the dual mesh M, the standard Voronoï mesh constructed from the vertces of the prmal mesh. Ths defnton of M enters our framework, wth the followng choce: the center L of a face K L s the center of ts crcumscrbed crcle; the center L of an edge K L s ts mddlepont. Ths constructon does not guarantee that L K L nor that K ; thus we make appeal to the generalzatons of the above Remarks 4., 4.2. Ths Delaunay-Voronoï double scheme possesses the orthogonalty property requred for the approxmaton of entropy or renormalzed solutons of (may be, degenerate) dffuson PDEs. See [46, 47, 5], for examples of the use of ths scheme. Remark 4.4. Note that n order to get the mesh of Example 4.3, one can also reverse the constructon procedure. Startng from a gven set of ponts, one constructs the Voronoï mesh whch wll play the role of the dual mesh M. A tetrahedrcal prmal mesh M o s obtaned by jonng aproprately the vertces of M ; a slghtly dfferent conventon on boundary volumes s needed n ths case. 6 We guess that ths problem does not occur n Example 4.3, thanks to the Delaunay condton: the negatve part of K s completely cancelled by ts postve part, as n the case of the prmal volumes.

17 6 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL Remark 4.5. Note that the dscrete dualty property of Proposton 3.2 s sutable for dscrete functons satsfyng the homogeneous Drchlet condton on Ω. For dfferent boundary condtons, see e.g. [9, 5, 6]. The case of the homogeneous Neumann boundary condton s brefly dscussed n [3]. Remark 4.6. In 2D, the scheme descrbed n [4] s much smpler than the DDFV scheme and stll possesses the dscrete dualty property; the dscrete dualty for ths scheme follows drectly from the reconstructon property of the Appendx below. Ths smpler scheme s well known for the case one starts wth a trangulaton of Ω R 2 (ths s essentally the complementary volumes scheme, cf. [2, 40, 55, 39, 0, 59]). Elements of the trangulaton play the role of damonds of the DDFV scheme; n partcular, dscrete gradent s reconstructed as beng constant per trangle. The dual Voronoï mesh of the trangulaton s the fnte volume mesh (these are the complementary fnte volumes, n the termnology of the papers [40, 55, 39, 59]) on whch one consders constant per volume dscrete functons. In the ltterature, generalzatons of such schemes to 3D were consdered (see n partcular [39]). To the authors knowledge, the dscrete dualty property for the 3D complementary fnte volume scheme only holds for very partcular mesh geometres (e.g., unform tetrahedral or rectangular meshes); see the dscusson of [5, Appendx B]. 5. Numercal experments for lnear dffuson problems For Ω R 3, consder the followng lnear dffuson problem: (2) dva(x) u = f, u Ω = u, where f s an L 2 source term, (A(x)) x Ω s a measurable (pecewse smooth, n many applcatons) famly of unformly bounded and coercve 3 3 matrces and u s a suffcently regular boundary condton (when u 0, the boundary condtons are taken nto account at the centers of boundary volumes, cf. [9]). The analogous 2D problem was the example on whch many 2D strateges of fnte volume dscretzaton (ncludng the DDFV method) were tested, see n partcular [4]. Here we present some of the analogous results n the 3D case, n three stuatons. More ample results and comparson to other methods wll be presented n the 3D benchmark [, 42]. Let P T denote the projecton on the DDFV mesh T (.e., the components of P T f are the mean values of f L (Ω) per prmal and per dual volumes, cf. Remark 4.2()). Let P T denote the projecton on the damond mesh D. The heterogenety of the matrx A s taken nto account by usng the damond-wse projecton A T := P T A ; smlarly, we use f T = P T f as the dscrete source term. Boundary data ū are taken nto account usng the projecton P T on boundary volumes: P T := ( P T,P T ). For a fully practcal dscretzaton of A and f (whch are contnuous n all our tests), for every element (recall that damonds, prmal volumes and dual volumes of a DDFV mesh are unons of elements, see Fg. 4) we take the mean value of the four vertces of the element. Therefore, for the case of homogeneous boundary condton, gven a DDFV mesh T of Ω the method wrtes as: fnd u T R T 0 soluton of dv T[ A T T u T] = f T. More generally, the Drchlet boundary condtons are prescrbed at the centers of the boundary volumes M o and M usng P T ū ; n ths case, u T = ( u T,P T ū ) wth u T unknown. In each of the test cases descrbed below, we start wth a gven mesh of the unt cube Ω (cartesan, tetrahedral, hexagonal, prsmal) usng t as the prmal mesh for the DDFV constructon; we choose the barycenters for the centers of these prmal volumes and of ther faces. It should be notced that the dual mesh s never actually constructed, all the nformaton s read from the prmal mesh usng n partcular (6). The matrx and the source term of the lnear system for the method are assembled per damond. Thanks to the dualty property, one can easly prove that the resultng scheme s well-posed and convergent. It s also classcal to prove an order h convergence n the H 0 norm usng e.g.

18 3D DDFV DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS. I 7 the tools developed n [9, 20, 3]. Whenever the (unform) dscrete Poncare nequalty can be justfed, one derves an order h convergence n the L 2 norm for the soluton. Ths s the case f l = 3 (.e. the prmal faces are trangles), but n general one can construct counterexamples to the dscrete Poncaré nequalty. Yet numercally, the value of l does not seem mportant, and the order h s a rather pessmstc one. Measure of errors and convergence orders. To put the dscrete and the exact solutons at the same level, we use the projecton P T u e of the exact soluton and the assocated dscrete gradent reconstructon T P T u e, where P T = (P T ; P T ). The L 2 norms of the errors e T :=u T P T u e and T e T := T u T T P T u e are [ ] measured n terms of the scalar products, on Ω RT and ndcators we use are relatve errors defned, respectvely, as ( [[ ErrL 2 et, e T]] /2 := [[ ]) and ErrGradL 2 := PT u e, P T u e {, } Ω on (R3 ) D : the error ( { T e T, T e T} { T P T u e, T P T u e }) /2. In all the tests, the convergence orders are reported wrt the number of unknowns (#unkw). Laplacan, unform cartesan meshes and non-conformal checkerboard meshes In ths case, we frst take the unform cartesan DDFV mesh of Secton 2.3 (see Fg. 3) and we take the exact soluton u e (x,y,z) = sn(2πx)sn(2πy)sn(2πz) of the homogeneous Drchlet problem ( f s calculated accordngly). It should be noted that the soluton s smooth and both the mesh and the problem are symmetrc wrt reflexons n each drecton. Therefore error analyss based on cancellatons n Taylor expansons (cf. [8] for the more dffcult p -laplacan case n 2D ) allows to prove the order h 2 convergence for the soluton tself (measured n the L 2 norm) and for the dscrete gradent (measured n the L 2 norm). The numercal results confrm these orders as optmal ones; ndeed, the followng table gves results and convergence orders: #Cubes #unkw ErrL 2 Order ErrGradL 2 Order 6x6x E E+00-9x9x E E x2x E E x8x E E x24x E E x32x E E Then we consder the same problem dscretzed on non-conformal cartesan meshes; the meshes are made of cubes of sze h of whch every second cube s refned nto four cubes of sze h/2 (the so-called checkerboard meshes). The correspondng results are: #Cubes #unkw ErrL 2 Order ErrGradL 2 Order E E E E E E E E E E Here we clearly observe the superconvergence order h 2 for the soluton n L 2 norm whch would be much more dffcult to justfy theoretcally; at the same tme, the order of the dscrete gradent convergence n the L 2 norm falls down to h. Mldly ansotropc permeablty A, cartesan, tetrahedral and Kershaw meshes In ths test, the permeablty matrx A does not vary n space, but t s ansotropc: A

19 8 B. ANDREIANOV, M. BENDAHMANE, F. HUBERT, AND S. KRELL The exact soluton s gven by the formula u e (x,y,z) = +sn(πx)sn(π(y + 2 ))sn(π(z + 3 )), the data u and f are calculated accordngly. We use three dfferent knds of prmal meshes. Frstly, on the same cartesan meshes as n the prevous case (see Secton 2.3 and Fg. 3) we get the followng results: #Cubes ErrL 2 Order ErrGradL 2 Order 6x6x E E-0-9x9x E E-0.5 2x2x2 0.87E E x8x E E x24x E E x32x E E Here, the h 2 convergence for the gradent s lost and h 3/2 convergence s observed, cf. [4] (clearly, the ansotropy the permeablty matrx A destroys some of the symmetres that led to superconvergence n the case of laplacan). Yet an almost h 2 convergence n L 2 for the soluton tself s stll observed. Fgure 5. Meshes (prmal): tetrahedral mesh on the left, Kershaw mesh at the center, and prsmal mesh on the rght Secondly, we use tetrahedral prmal meshes (see Fg. 5, left). They do not lead to the Delaunay- Voronoï DDFV meshes of Example 4.3, because of the barycentrc choce for the cell and face centers; cf. Hermelne [46] for the choce of crcumcenters, on slghtly dfferent test cases. Actually, n the below test we have used tetrahedral meshes that do not respect the Delaunay property. Let us remnd that our constructon does not mpose any specfc choce for cell and face centers; only the edge centers must be fxed to be the mddleponts. The barycentrc choce of cell and face centers leads to rather regular dual meshes. Tetrahedral mesh corresponds to l = 3, whch s the smple case of the gradent reconstructon formulas that can be understood as straghtforward affne nterpolaton n K L. On the correspondng DDFV meshes, we get the followng results: #Tetras #unkw ErrL 2 Order ErrGradL 2 Order E E E E E E E E E E E E-0.02 One sees that here, the convergence order for the soluton gradent falls down to h, whle superconvergence s stll observed for the L 2 norm of the soluton tself. Thrdly, we use the so-called Kershaw meshes (see Fg. 5, center) wth hexahedral faces 7. In ths case, l takes the value 4 ; here (5) s the (non-evdent) choce of order one-consstent dscrete 7 these meshes, generated by K. Lpnkov, are proposed n the 3D benchmark [42] (see and can be vsualzed at the same Web locaton