DISCRETE DUALITY FINITE VOLUME SCHEMES FOR LERAY-LIONS TYPE ELLIPTIC PROBLEMS ON GENERAL 2D MESHES

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1 ISCRETE UALITY FINITE VOLUME SCHEMES FOR LERAY-LIONS TYPE ELLIPTIC PROBLEMS ON GENERAL 2 MESHES BORIS ANREIANOV, FRANCK BOYER AN FLORENCE HUBERT Abstract. iscrete duality finite volue schees on general eshes, introduced by Hereline in [24] and oelevo & Onès in [3] for the Lalace equation, are roosed for nonlinear diffusion robles in 2 with non hoogeneous irichlet boundary condition. This aroach allows the discretization of non linear fluxes in such a way that the discrete oerator inherits the key roerties of the continuous one. Furtherore, it is well adated to very general eshes including the case of non-conforal locally refined eshes. We show that the aroxiate solution exists and is unique, which is not obvious since the schee is nonlinear. We rove that, for general W, source ter and W, boundary data, the aroxiate solution and its discrete gradient converge strongly towards the exact solution and its gradient resectively in aroriate Lebesgue saces. Finally, error estiates are given in the case where the solution is assued to be in W 2,. Nuerical exales are given, including those on locally refined eshes. Keywords. Finite-volue ethods, Error estiates, Leray-Lions oerators. AMS Subject Classification.. Introduction. 35J65-65N5-74S.. Nonlinear ellitic equations. In this aer, we are interested in the study of a finite volue aroxiation of solutions to the nonlinear diffusion roble with non hoogeneous irichlet boundary conditions: { div ϕz, uez = fz, in,. ue = g, on, where is a given bounded olygonal doain in R 2. We first recall the usual functional fraework ensuring that the roble above is well-osed. Let ], [ and =. The flux ϕ : R2 R 2 in equation. is suosed to be a Caratheodory function which is strictly onotonic with resect to ξ R 2 : ϕz, ξ ϕz, η, ξ η >, for all ξ η, for a.e. z. H We also assue that there exist C, C 2 >, b L, b 2 L such that ϕz, ξ, ξ C ξ b z, for all ξ R 2, a.e. z, H 2 ϕz, ξ C 2 ξ + b 2 z, for all ξ R 2, a.e. z. H 3 These assutions ensure that u div ϕ, u is a Leray-Lions oerator, and in articular the a G L 2 ϕ, G L 2 is continuous..2 Theore.. Under assutions H,H 2,H 3, for any source ter f W, and any boundary data g W,, the roble. has a unique solution ue W,. Laboratoire de Mathéatiques, Université de Franche-Coté, 6 route de Gray, 253 Besançon Cedex, France Laboratoire d Analyse, Toologie et Probabilités, 39 rue F. Joliot Curie, 3453 Marseille Cedex 3, France

2 Reark.. Note that, in view of the nuerical aroxiation of the source ter, it is necessary to suose that f L. This is not a restriction of our aroach. Indeed, in order to treat the case of general source 2, ter f W,, it is ossible to write f = f + div f with f L and f L so that the roble. is equivalent to div ϕz, ue = f z, where ϕ : z, ξ ϕz, ξ + f z. It is easily seen that, if ϕ satisfies H -H 3, so does the new flux ϕ. It is worth noticing that the coule f, f is not unique and each choice will lead to a different aroxiation of the original equation see [5]. As a consequence, fro now on we always assue at least that f L..2. Exales. Our fraework includes classical ellitic oerators like the linear anisotroic Lalace equation div Az ue = f,.3 Az being a uniforly coercive syetric atrix-valued a, or the -lalacian div ue 2 ue = f..4 One can also encounter, for instance in the odelling of non-newtonian fluids flows in a orous ediu, equations like div kz F z + ue 2 F z + ue = f,.5 where F is a vector-valued a and k a ositive scalar a bounded fro below. Notice that F is not necessarily a gradient, so that this roble ay not reduce to the -lalacian.4 through a change of variables. The odels resented in [2] are even ore general, since the flux ϕ deends also on the unknown ue. In [22], the authors also roose such nonlinear ellitic roble for the study of glacier flows. In this case, the flux ϕ deends only on ξ but in an ilicit way. We recall the key technical lea which ilies the onotonicity and continuity roerties of the two nonlinear odel robles above see [5]. Lea.2. For any ], + [ and δ, there exists C, C > such that for any n N we have ξ 2 ξ η 2 η, ξ η C ξ η 2+δ ξ + η 2 δ, ξ, η R n, ξ 2 ξ η 2 η C ξ η δ ξ + η 2+δ, ξ, η R n..3. Finite volue aroach. Finite eleents aroxiation of robles like. are now quite classical see for instance [5, 8, 2, 26]. Nevertheless, it is also natural to consider finite volue ethods for these robles. Indeed, finite volue ethods allow ore flexibility on the geoetry of the eshes and ensure the local consistency of the nuerical fluxes inside the doain. Furtherore, this kind of discretization is well-adated if one adds a convective ter in the roble.. The nonlinearity and the ossible anisotroy of the flux ϕ with resect to ue akes it difficult to aroxiate the roble by standard cell-centered finite volue ethods as resented, e.g., in [8], since only the noral coonent of ue on the interface between two adjacent volues can be easily aroxiated on conforal eshes. In the case of linear anisotroic Lalace equation, soe studies are available in the literature for instance in [, 4, 2], with various aroaches. To our knowledge, three kinds of gradient reconstruction were roosed for the finite volue aroxiation of the fully non-linear equation. The one of [4] alies for the -lalacian 2

3 on eshes that are dual to triangular ones, oreover, the triangular eshes should be close enough to the structured esh. The one of [] alies on rectangular eshes. Finally, for general grids it was recently roosed in [7] to handle fluxes on the edges of a control volue as new unknowns, and reconstruct the discrete gradient, constant er control volue, using these fluxes. In all cases, the crucial feature is that the suation-by-arts rocedure erits to reconstitute, starting fro the one-diensional finite differences u K u L, the whole two-diensional discrete gradient see [4, Lea 8, Proosition 4] and [, Proosition 2.5, Lea 2.7]. The coercivity and onotonicity roerties of the continuous ellitic oerator are then inherited by its discrete finite volue counterarts. For instance, the variational structure of the -lalacian oerator can be inherited by its discrete analogues. We consider in this aer the class of finite volue schees introduced by Hereline in [24], by oelevo, Onès in [3] for the Lalace equation and in [] for other linear equations like the Stokes or the iv-curl syste. More recisely, we show that the ethod can be successfully extended to the case of the nonlinear diffusion equation. we are interested in while reserving the ain features of the continuous roble. In [], these schees are called iscrete uality Finite Volue FV for short since the discrete gradient and discrete divergence oerators are dual one fro each other see Lea 4. below. The equation is aroxiated siultaneously on two interrelated eshes: the rial and the dual esh. The nuber of variables and of equations doubles coared to usual cell-centered FV schees, but the gradient aroxiation the one already used by Coudière and al. [] becoes sile and quite efficient. Furtherore, the ethod is well-suited to alost arbitrary eshes since few geoetrical constraints are iosed to the rial and dual control volues. Indeed, non convex control volues, non atching triangulations or locally refined eshes can naturally be handled by this ethod and also fulfill the assutions needed for the convergence analysis given in this aer..4. Outline. This aer is organized as follows. The fraework of FV eshes, the discrete gradient and the finite volue schee associated with equation. are described in Section 2. In Section 3, we resent the ain results of discrete functional analysis necessary for the theoretical study of the finite volue ethod. These results include the discrete Poincaré inequality Lea 3.2, the study of the ean-value rojection of functions on the eshes Proosition 3.5 and Corollary 3. and finally a discrete coactness result siilar to the Rellich theore Lea 3.6. Existence and uniqueness of a discrete solution of the schee as well as a riori estiates are given in section 4 Theore 4.4. The structure roerties of Leray-Lions oerators being inherited in the fraework of FV schees, the ethod we use is siilar to the one for the continuous roble.. Section 5 is devoted to the roof of Theore 5. which states the convergence of the aroxiate solution in case of general data g W, and f W, see Reark.. Notice that the strong convergence of the discrete gradient of the aroxiate solution towards the gradient ue of the exact solution is obtained in L 2. In Section 6, we study the stability roerties of the aroxiate solution with resect to the data f and g Proosition 6.. Finally, in Section 7, we rove error estiates for the discrete gradient in L 2 in the case where the exact solution lies in W 2,, which is a usual assution for the error analysis Theore 7.. The convergence rate obtained is sizet for 2 and sizet for < 2. These rates are the sae than the one obtained in [, 26] for different schees. As an exale, this result ilies the first order convergence in the case of the anisotroic Lalace equation which Lischitz coefficients. Note that error estiates for general solutions of the -lalacian equation with source ter in L were obtained in [2], aking use of the intrinsic Besov regularity of continuous and discrete solutions, in the case of structured rectangular eshes. It is an oen question how to generalize this Besov aroach to the unstructured FV schees. In Section 8, we rovide soe nuerical results which show in articular that, in the truly nonlinear case, the ethod behaves better than what is exected even for non conforal locally refined eshes. In the concluding Section 9, we discuss the extension of our study to soe fully ractical variants of the finite volue schee and to even ore general eshes than the ones described in Section The finite volue ethod. 3

4 x L x L L K x K L K x K x K P ext, K M x K P ext, K M esh M esh M M esh Fig. 2.. Exale of a FV esh 2.. efinition of the esh. We call T a trile M, M, of eshes on, defined as follows. The esh M is a set of disjoint olygonal control volues K such that K =. We denote by M the set of edges of the control volues in M included in, which we consider as degenerate control volues. We associate to M, M a faily of oints P. The set P = P int P ext is coosed of one oint er control volue K M called x K P int and one oint er degenerate control volue K M called x K P ext : P int = {x K, K M}, P ext = {x K, K M}. Let P denote the set of vertices of the esh M. The set P can be decoosed into P = P int P ext where P int = and P ext see Figure 2.. The sets M and M are two failies of dual control volues defined as follows. To any oint x K P int res. x K P ext we associate the olygon K M res. K M whose vertices are {x K P/x K K, K M} res. {x K } {x K P/x K K, K M M} sorted with resect to the clockwise order of the corresonding rial control volues. For all adjacent control volues K and L, we assue that K L is a segent that we call an edge of the esh M and that we denote by σ = K L. Let E be the set of such edges. The corresonding notations σ = K L and E refer to the dual esh M M. Even though ore general situations can be handled, we concentrate in this aer to eshes satisfying the following assution. Main assution. We assue either that all the rial control volues K M are star-shaed with resect to x K either that all the dual control volues K M are star-shaed with resect to x K. 4

5 Reark 2.. This hyothesis is not so restrictive and is fulfilled for instance in the case of a rial Voronoï esh associated to the vertices x K, or in the dual case of a elaunay conforal triangulation of the doain see [9]. For each coule σ, σ E E such that σ = K L = x K, x L and σ = K L = x K, x L, can we introduce the quadrilateral diaond cell σ,σ whose diagonals are σ and σ, as shown in Figure 2.2. Notice that the diaond cells are the union of two disjoint triangles and can be non convex. Furtherore, if σ E, then the quadrilateral σ,σ degenerate in a single triangle. The set of diaond cells is denoted by and we have = As a consequence of the ain assution above, we easily see that the interior of the diaond cells are disjoint. For silicity, we will also assue that the interiors of the dual control volues are all disjoint. Nevertheless, it is ossible to coe with articular eshes with overlaing dual control volues see [3] but it would be necessary to introduce soe ore notations Notations. For any control volue K M, we define K, the easure of K. E K, the set of edges for K M and abusively the edge σ = K for K M. K = { σ,σ /σ E K }. ν K, the outward unit noral vector to K. d K, the diaeter of K. For any degenerate control volue K M, ν K stands for the outward unit noral vector to. In the sae way, for a dual control volue K M M, we set K, the easure of K. E K, the set of edges for K M M. K = { σ,σ /σ E K }. ν K, the outward unit noral vector to K. d K, the diaeter of K.. x K σ = K L x K σ = K L ν x K σ = K L x K ν τ τ σ = K L x L α x L x L x L Fig Notations in a diaond cell σ,σ For a diaond cell σ,σ, recall that x K, x K, x L, x L are the vertices of σ,σ and note : σ, the length of σ, σ the length of σ and the easure of the diaond cell. τ, the unit vector arallel to σ, oriented fro x K to x L. ν, the unit vector noral to σ, oriented fro x K to x L. τ, the unit vector arallel to σ, oriented fro x K to x L. ν, the unit vector noral to σ, oriented fro x K to x L. 5

6 α, the angle between τ and τ. d, the diaeter of σ,σ. It can haen that the oint x K does not belong to K if K is a triangle and x K its circucenter for instance. As usual in that case see [8] it is necessary to assue that ν K oints fro x K towards x L, that is ν K = ν with the notations above. Siilarly we assue that ν K = ν. In the case of a triangular esh M in which is the circucenter of K, this assution is known as the elaunay condition. x K 2.3. Unknowns and boundary data. The finite volue ethod associates to all rial control volues K M, an unknown value u K and to all dual control volues K M, an unknown value u K. We denote the aroxiate solution on the esh T by u T = u K K M, u K K M. 2. The sace of all discrete functions u T in the sense of definition 2. is denoted by R T. In this aer we deal with non-hoogeneous irichlet boundary condition. We describe here the way the boundary data will enter the schee. Note first that = K = K. K M K M For any boundary data g W, we introduce its discrete counterart by defining for each K M, a value g K and for each K M, a value g K. The faily g K K M, g K K M is denoted by gt and is also associated with a iecewise constant function in L as follows g T 2 K M K g K + 2 K M K g K, where here and in the sequel, we denote by E the characteristic function of any set E. We consider the discrete ean-value boundary data denoted by Pg T = P M g, P M g defined by P M g = gs ds, P M g = gs ds. 2.2 σk σ K K M σk σ K K M Here def def σ K = Bx K, ρ K, and σ K = Bx K, ρ K 2.3 and ρ K and ρ K are ositive nubers associated to the esh T and such that σ K K, σ K K. Finally, introduce nubers ρ K and ρ K for any K M res. K M such that def def B K = Bx K, ρ K, B K = Bx K, ρ K. These balls are only introduced in order to rove the convergence of the schee but not to rove the error estiates. In articular, they do not enter the definition of the schee. Of course, soe assutions are needed on the radii ρ K and ρ K as stated in the next aragrah Regularity of eshes. We note sizet the axiu of the diaeters of the diaond cells in. The following bounds follow: σ sizet, σ E; σ sizet, σ E ; K π sizet 2, K M; K π sizet 2, K M ; 2 sizet 2,. 6

7 We introduce now a ositive nuber that quantifies the regularity of a given esh and is useful to erfor the convergence analysis of the finite volue schees. We first define α T to be the unique real nuber in ], π 2 ] such that def sin α T = in sin α, 2.4 that is the inial angle between the diagonals of the diaond cells in the esh. Let us introduce the nuber N T def = su # { K s.t. x K B K } + su # { } K s.t. x K B K x x + su # { s.t. x B K, K, K M } x + su # { s.t. x B K, K, K M }, 2.5 x where Ê denotes the convex hull of any set E R2. In the usual case where the rial control volues, the dual control volues and the diaond cells are convex and where B K K, B K K then N T can be bounded by a function of the axial nuber of edges er rial and dual control volues. Since we do not iose any convexity assution on the eshes, we need to control this nuber N T in the convergence analysis. We can now introduce regt def = ax α T, N T, ax d ax K M M, ax K, ax dk K M ρ K + ρ K d K d K d K, K M K dk, ax K M M x K corner dk ax K M ζ x K =corner ρ K + ζ ρ K d K ρ K, ax K M K d K + ρ K, d K d, ax K M K d K, d 2.6 where, for any s, we have ζ s = s, if < 2, ζ s = s 2, if > 2, ζ 2 s = s log s. 2.7 This secial treatent of the corner vertices of the esh is a urely technical assution which is used in the convergence rate analysis see Lea 7.2. Let us oint out that regt essentially easures: how flat the diaond cells are. how large is the difference between the size of a rial control volue res. a dual control volue and the size of a diaond cell as soon as they intersect. Convention. In any estiate given in this aer, the deendence of the constants in regt is ilicitly assued to be non-decreasing. Furtherore, the deendence of the constants on the doain is often oitted. Reark 2.2. For conforal finite volue eshes see [8], it is assued that α = π 2 for any diaond, so that α T = π 2. In our case, not only this orthogonality condition is relaxed but also M can resent atyical edges, non convex dual control volues and non convex diaond cells see Figure 2.3. Reark 2.3. The boundedness of regt ioses only local restriction on the esh. It is easy to construct a faily of locally refined esh such that regt is bounded indeendently on the level of the refineent. Figure 2.4 rovides a very sile exale of such a construction. 7

8 Atyical edge α π 2 Non-convex dual control volue Fig Non conforal eshes M Fig Local refineent allowed by the boundedness of regt 2.5. iscrete gradient. We consider the discrete gradient introduced by Coudière and al. in [] and alied by Hereline [24] and oelevo and Onès [3] in the fraework of FV schees described above. For a given discrete irichlet data g T as defined above, the discrete gradient oerator T g T can be defined as follows : for any u T R T, T g T u T is the function, constant on each diaond cell σ,σ, given by T g T ut = g T ut, with g u T, τ T g u T, τ T 8 = u L u K σ = u L u K, σ 2.8

9 where σ,σ is noted when no confusion can arise. The deendence on g T only aears when σ,σ intersects, in which case we relace the values of u K or u K by g K or g K, for the oints x K or x K located on the boundary. Reark that g T u T can be exressed in the ν, ν basis in the following way : g T ut = ul u K ν + u L u K ν sin α σ σ thanks to the following lea. Lea 2.. Using the notations of Section 2.2 and Figure 2.2 in articular the orientation conventions, for any vector ξ R 2 we have sin α ξ = ξ, τ ν + ξ, τ ν. 2.9 In the fraework of FV schees on general eshes, this lea is the crucial arguent which ensures the coercivity and onotonicity roerties of the finite volue aroxiates of Leray-Lions tye oerators; coare to [4, Lea 8] and [, Proosition 2.5], which only work due to the articular geoetry of eshes. Reark 2.4. Our notation for the discrete gradient can be easily handled thanks to the following roerty: for any discrete boundary data g T and gt 2 and for any discrete functions ut and ut 2, we have g T u T + g u T 2 T 2 = g T +gt 2 u T + u T In articular, if T denotes the zero vector of R T then g T u T g u T = 2 T g T, u T R T. T gt The schee. iscrete uality Finite Volue schees are obtained, as in Hereline [24] or in oelevo and Ones [3], by integrating equation. on both control volues K M and dual control volues K M : K fz dz = = K div ϕz, uez dz = ϕs, ues, ν K ds K ϕs, ues, ν K ds, σ,σ K σ fz dz = div ϕz, uez dz = ϕs, ues, ν K ds K K K = ϕs, ues, ν K ds. σ σ,σ K 2. Let us introduce for any diaond the satial aroxiation ϕ : R 2 R 2 of the flux ϕ defined by ϕ ξ = ϕz, ξ dz. 2.2 Other aroxiations of ϕ on each diaond are ossible, we will discuss one of the in Section 9. On each diaond, we aroxiate ϕ, ue, using the discrete gradient oerator T P T g introduced in section 2.5, by ϕ P. Note that the choice of a constant value for the discrete flux ϕ T ξ on each diaond is necessary in gut the calculations using Lea 2.. The FV finite volue schee then reads a K u T def = σ,σ K σ ϕ P T gut, ν 9 = K f K, K M, 2.3

10 a K u T def = σ,σ K σ ϕ P T gut, ν = K f K, K M, 2.4 where f K res. f K denotes the ean value of the function f on K res. K. It is convenient for the analysis given below to introduce a notation for this kind of rojections on the set of discrete functions. efinition 2.. For any integrable function v on, we set Pv T def = P M v, P M v, where P M v def = vz dz and P M v def = vz dz BK B K K M BK B K K M. We call Pv, T the ean-value rojection of v on the sace R T. We also introduce the ean-value rojection on the control volues P v T def M = PM v, P v, where P M v def = vz dz, and K K K M M P v def = vz dz. K K K M The finite volue schee above can now be written under a coact for where a g u T defi = a g u T = P T f, 2.5 a K u T K K, a K u T. 2.6 K K We ostone to section 9 a discussion concerning soe variants of the roosed schee, in articular, with resect to the choice of the discretization of the data f and g. 3. iscrete functions and their roerties. 3.. Sobolev saces on the boundary of olygonal doains. We need to recall briefly the definitions and ain roerties of the Sobolev saces defined on and related trace theores. A colete study of these toics can be found, for instance, in [23]. efinition 3.. Let α ], [, and [, + [. We define W α,, to be the sace of functions g L such that g W α, def = g L + gx gy x y α dλx dλy <, x y where dλ is the natural length easure which can be defined on the boundary see [27]. We recall that the trace oerator γ is continuous fro W, onto W, and that there exists a linear continuous lift oerator R : W, W, such that γ R = Id, Rg W, C g W,, 3. where C deends only on and. For any g W we denote by W, g the closed subset of W, of all functions whose trace on is equal to g. Let us denote by Γ,..., Γ k the sides of. Since each Γ i is a segent we can define naturally the saces W +α, Γ i by h W +α, Γ i h W, Γ i, and T hx T hy dλx dλy x y α <, x y Γ i Γ i

11 where T h stands for the derivative of h in the direction of that we call tangential gradient. Since is not sooth enough, it is not ossible to define the sace W +α, but we can introduce the following sace W +α, = { g W,, g Γi W +α, Γ i }, endowed with its natural nor. We recall that the trace oerator γ is continuous fro W 2, onto a finite codiensional subset of W 2, which can be described recisely we do not need this descrition here and we refer to [23] for further develoents on this toic. We also recall that for any > 2 the ebedding of W, in the Hölder sace C, 2 holds true and that we have the following shar estiate. Lea 3.. Let > 2, there exists a constant C deending only on such that, for any connected subset σ of and any g W, we have gz gz C z z 2 σ σ gx gy dλx dλy, z, z σ. 3.2 x y x y Proof. The ebedding of W, in the Hölder sace C, 2 is given by the Morrey theore. In articular, there exists C > such that 3.2 holds for the unit segent σ =], [ R and any g W, ], [. It is now easy, using a linear change of the variables, to see that 3.2 holds with the sae constant C for any σ. u T 3.2. Basic notations and results. Whenever it is convenient, we associate to the discrete function = u K K M, u K K M the iecewise constant function where u M = K u K, u M = K M K M K u T 2 nor of u M, u M, u T. We denote by, the inner roduct on R T u T, v T = 2 um + u M, 3.3 u K. As a consequence, one can define for any r [, + ] the L r given by Ku K v K + K u K v K, 2 K M K M which stands for a discrete L 2 inner roduct, whereas the usual Euclidean inner roduct on R T by, : u T, v T = u K v K + v K. K M K M u K is denoted Let us finally state the discrete version of the Poincaré inequality. This result is classical in the case = 2 see for exale [3, 8]. When 2, it is roved in a slightly different context in [4], without any geoetrical assutions on the esh. In the FV fraework, we need to assue a lower bound on α T defined in 2.4. To begin with, let us oint out the fact that there exists C > deending only on and regt such that for any g W, we have Pg T L P M g L + P M g L C g, 3.4 W, where Pg, T P M g and P M are defined in 2.2. As a consequence, for any sequence of eshes T n such that regt n is bounded and sizet n we have P Mn g g, in n L, and P M n g g, in n L, 3.5

12 for any g W,. The roof of this result is not given since it is a straightforward adatation of the arguents used in Section 3.3 for the study of the ean-value rojection oerators for functions defined in the whole doain. Reark that we need to take g W, in 3.4 and 3.5 because of the articular treatent of the corner oints for 2 in 2.2 and 2.6. We can now state and rove the ain result of this section. Lea 3.2 iscrete Poincaré inequality. Let T be a esh of. There exists a constant C, only deending on, on the diaeter of and on regt, such that for any u T R T and any g W,, we have u T L u M L + u M L C T P TguT L + g W,, 3.6 where Pg T is defined in 2.2. Proof. We start as in the roof of the discrete Poincaré inequality given in [4] see also [9], taking into account the boundary conditions. It follows, using 3.4, u M L = K u K C σ σ u K u L + C K g K C K M σ,σ σ σ σ,σ σ K M u K u L u K + u L + C g σ W., Using the definition 2.6, we see that there exists C deending on regt such that σ σ C, σ σ C K, and σ σ C L. As a consequence, we can use the Hölder inequality to get u M L C u K u L σ,σ σ + C g W,. By definition of the discrete gradient, we have u K u L σ = P, τ T gut P, TguT so that, with 2.6 and 3.4, the estiate of the nor of u M follows. The estiate of the contribution of u M is estiated in the sae way. Notice that, using the sae arguent than in [4], we can rove that the constant C in 3.6 deend only on dia and α T in the case where all the diaond cells are convex and g = Proerties of the ean-value rojection oerators. In the convergence analysis of our schee we will have to use soe discrete aroxiation of test functions lying in W,. The natural rojection since these test functions ay not be continuous when < 2 is the ean-value rojection see efinition 2.. We give below the ain roerties of such a rojection onto the set of discrete functions in our fraework. To begin with, we give the following crucial result, which is siilar to [6, Lea 7.2], [8, Lea 3.4], generalized to the case of non convex control volues and 2 see also [7, Lea 6.]. We do not give the roof which is a straightforward extension of the roofs that one can found in the references above. Lea 3.3. For any q, there exists a constant C deending only on q such that for any bounded set P R 2 with ositive easure, any segent σ R 2 and any v W,q R 2 we have v P v σ q σ P σ P vx vy q dx dy C dia P σ q+ v z q dz, σ P cp σ 2

13 Case I Case II K x K σ L x L x L x K x K x L L x L σ x K Fig. 3.. Boundary dual control volues and diaonds where v P denotes the ean value of v on P, v σ the ean value of v on the segent σ, and P σ is the convex hull of P σ. Lea 3.4 Mean-value rojection bounds. Let T be a esh on and q [, + ]. There exists C deending on q and regt such that. for any v L q, we have { P T v L q P M v L q + P M v L q C v L q, P T v L q P M v L q + P M v L q C v L q for any v W,q we have T P T gpt v L q C v L q, 3.8 where g = γv is the trace of v on. Proof.. This oint is straightforward consequence of the Jensen inequality and of 2.5 and Recall that T P v q TgPT L C P v T K Pv T L q σ q + P T v K P T v L σ For interior control volues, we have Pv T K Pv T L σ Pv T K v σ σ + Pv T L v σ σ, 3.9 Pv T K Pv T L Pv T K v σ + Pv T L v σ, where v σ = σ σ vs ds and v σ = σ σ vs ds. Lea 3.3 can be alied to each of the ters in the σ σ right-hand side of 3.9. The case of boundary control volues can also be reduced, as shown in Figure 3., to estiates of differences between the ean values on the balls B K and the ean values on edges. But as for K M and K M, Pv T K, Pv T K are ean value of the function v on soe edges, we need to insert ean values on aroriate balls. Thanks to 2.6 we have σ diak B K CregT d K CregT in σ, σ, 3 q.

14 diak B K CregT d K CregT in σ, σ, and so that and the clai is roved. T P v q TgPT L C q CregT BK, CregT BK, + vz q dz + vz q dz K B K L B L K B K vz q dz + 2C vz q dz + K B K M K N T C vz q dz, L B L K M vz q dz K B K vz q dz Proosition 3.5 Convergence of the ean-value rojections. Let T be a esh on and q [, + ]. There exists C deending on q and regt such that P T v v L q P M v v L q + P M v v L q C sizet v L q, v W,q, 3. P T v v L q P M v v L q + P M v v L q C sizet v L q, v W,q, 3. T P T gpt v v L q C sizet v W,q, v W 2,q, g = γv. 3.2 We ostone the roof of this result to Section Corollary 3.. Let q [, + [ and T n n a sequence of eshes such that sizet n and regt n is bounded. Then, we have For any v L q, all the sequences P Mn v n, P M n v n, P T n v n, P Mn v n, P M n v n and P T n v n converge towards v in L q. For any v W,q, the sequence T n g n P T n v n converge towards v in L q 2. Proof. The two clais of the corollary can be shown in the sae way. Let us give, for instance, the roof of the second oint. For any v W,q, by density of W 2,q in W,q, for any ε > there exists v ε W 2,q such that v v ε W,q ε. We denote its trace by g ε = γv ε and its ean-value rojection by gn ε = Pg T ε. Thanks to Lea 3.4 and Proosition 3.5 we have T n g n P T n v v L q T n g n P T n v T n gn ε PT n + v ε v L q v ε L q + T n gn ε PT n C v v ε L q + C sizet n v ε W,q Cε + C sizet n v ε W,q. v ε v ε L q The real nuber ε being fixed, for n large enough we have sizet n v ε W,q ε so that we obtain T n g n P T n v v L q 2Cε, and the result follows. 4

15 3.4. A coactness result. As usual, in the convergence analysis of finite volue schees see [8] for instance one needs to rove a discrete coactness result, which is a discrete counterart of the Rellich coactness theore. Lea 3.6 iscrete coactness. Consider a sequence of eshes T n n such that sizet n tends to zero and regt n is bounded. Let g W, be the boundary data and g n = P T n g its ean-value discretization on the esh T n. Let u Tn R Tn be a sequence satisfying the discrete Wg, bound Then, there exists u Wg, such that, u to a subsequence, T n g n u T n L C, n N. 3.3 u T n T n g n u T n n u in L, u weakly in n L 2. Notice that we will rove in fact that, u to a subsequence, u Mn n and u M n n both converge in L but in general their two liits can be different. Proof.. For any n N, consider v Tn = u Tn P T n Rg, where R is the lift oerator satisfying 3.. By Lea 3.4, we know that T n Rg is bounded in L 2 so that, using the bound 3.3 we deduce that g n P T n T n vt n L C, n N. Hence, by Lea 3.2, the sequence v Tn functions w Tn defined by is bounded in L. Let us now consider the sequence of discrete w Tn K = v Tn K v Tn K, K M n, w Tn = K vtn K v Tn, K K M n, and extended by outside. This sequence of functions is of course bounded in L R d and vanishes outside a bounded subset of R 2. For any x, η R 2, and any edge σ = K L we define {, if σ [x, x + η], ψ σ x, η =, elsewhere. Hence, with the notations of Section 3.2 and by Lea.2, we have for any x R 2, w M n x + η w M n x σ=k L σ=k L ψ K Lx, η w L w K Now we reark that ψ σ x, η dx σ η so that we have R 2 C σ ψ K Lx, η v L v K σ v L + v K. w Mn + η w Mn L R 2 CregT n η v Tn v L + v K CregT n T n vt n L 5 v L + v K.

16 The last factor in this inequality can be treated, as in [4], as follows: v L + v K C K M Hence, there exists C > such that for all η R 2 and n we have K v K + CregT n sizet n T n vt n L. w Mn + η w Mn L R 2 C η. Thanks to the Kologorov theore, we deduce that there exists a subsequence of w Mn k k which converges towards a function w L R 2 which vanishes outside. The definition of w Tn reads as v Mn x = T w Mn x, x, where T is the nonlinear a defined by T ξ = ξ ξ. By Lea.2, we know that T is -Hölder continuous so that we have v Mn T w L C w Mn w L, which roves that v M n k k converges strongly in L. We can now aly the sae technique to the subsequence v M n k k defined on the dual eshes. We deduce, using 3.3, that there exists a function v L such that v Tn n v in L. By Corollary 3., we know that P T nrg tends to Rg in L, so that we finally have u Tn n def v + Rg = u in L It reains to show that u Wg, and that the discrete gradient weakly converges. Thanks to the bound 3.3, there exists χ L 2 and a subsequence which is still indexed by n such that T n g n u Tn χ weakly in n L Let ψ C 2. Using 3.4 and 3.5, we have def I n = T n g n u T n n z, ψz dz + χz, ψz dz + n u T n n zdiv ψz dz uzdiv ψz dz. 3.6 By definition of the discrete gradient we have T n g n u T n n z, ψz dz = gn u T n n, ψ, 3.7 where ψ = ψz dz. For each diaond = σ,σ let us introduce ψ σ = ψs ds, ψ σ = ψs ds, σ σ σ σ 6

17 and finally ψ uniquely defined by ψ, ν = ψ σ, ν, ψ, ν = ψ σ, ν. The test function ψ being sooth enough we have, using Lea 2., ψ ψ ψ ψ σ + ψ ψ σ sin α Tn 2regT n sizet n ψ L. Coing back to 3.7 we deduce T n g n u T n n z, ψz dz = g n u T n n, ψ + g n u T n n, ψ ψ, 3.8 and using 3.8 and the bound 3.3, we see that the second ter tends to zero as n goes to infinity. As far as the first ter is concerned, we use 2.9 to obtain g n u T n n, ψ = ul u K σ σ ν + u L u K ν, 2 σ ψ σ = u K σ ψ, ν K u K σ ψ, ν K 2 2 K M σ E K K M σ E K + ψ, ν K g K σ ψ, ν K. 2 2 K M P T n g K σ E K σ P T n K M σ E K σ We recall here that the two boundary ters above have different fors since the eleents of M are degenerate control volues whereas the eleents M are lain dual control volues located near the boundary of the doain. Thanks to the definition of ψ we have g n u T n n, ψ = 2 K M u K σ E K + 2 ν K ds σψs, 2 K M u K K M P T n g K ν K ds Kψs, 2 σ E K P T n K M σ ψs, ν K ds g K σ E K σ σ ψs, ν K ds. Let us ehasize the fact that in the last ter, only the edges σ which are not on the boundary of the doain are taken into account. Hence, using Stokes forula in the first two ters and in the last one, it follows g n u T n n, ψ = u K div ψz dz u K div ψz dz 2 K 2 K M K M K + P T n g K ν K ds 2 K M Kψs, P T n g K div ψz dz 2 K M K + g K ψs, ν K ds 2 σ P T n K M = u T n 2 σ E K σ n z div ψz dz + P T n K M g K 7 K div ψz dz P T n g ψz, ν ds

18 Notice that the last ter tends to zero thanks to 3.4 since K div ψz dz C ψ sizet n 2. Gathering all the coutations above and using the roerty 3.5, we find that I n defined in 3.6 converges towards gsψs, ν ds, so that we finally roved that, for any ψ C 2 we have χz, ψz dz + uz div ψz dz = gs ψs, ν ds. This roves that u W, with u = χ and that γu = g. 4. Proerties of the schee. In this section we show that the finite volue schee 2.5 inherits fro the roerties of the continuous roble.. In articular, we show the existence and uniqueness of a solution to this schee. In a second aragrah we concentrate on the very iortant, in view of any alications such as.3-.5, variational case. 4.. The general case. Let us begin with a basic lea which exress the duality, through the discrete suation-by-arts rocedure, of the discrete gradient and discrete divergence oerators on FV eshes. Let us recall that the nonlinear a a g defining the schee is introduced in 2.6. Lea 4. Suation by arts. For any u T, v T R T R T, we have a g u T, v T = ϕ P, TguT v T. Proof. Perforing the suation-by-arts fro the definition 2.5 of a g we deduce a g u T, v T = a K u T v K K M = 2 2 v K σ K M σ,σ K K M a K v K σ K M σ,σ K u T v K ϕ P T gut, ν K ϕ P, ν TguT K. Reorganizing the suation over the set of diaonds, we get using the definition 2.9 of the discrete gradient a g u T, v T = σ ϕ P, ν v 2 TguT K v L = 2 2 σ,σ σ,σ σ,σ Thanks to Lea 2., we conclude that a g u T, v T = 2 σ σ σ = σ,σ ϕ P T gut, ν v K v L ϕ P, TguT v T, τ ν + v T, τ ν. σ σ sin α ϕ P T gut, v T ϕ P T gut, v T. 8

19 It is now ossible to rove the coercivity of the nonlinear a a g fro R T into itself. Lea 4.2 Coercivity. Assue that the flux ϕ satisfies H 2, H 3 and let T be a esh on. There exists C > deending on C, C 2 and regt such that for any g W,, f L and any u T R T, we have a g u T P Tf, u T P T Rg C T P TguT L C g + W f + b, L L + b 2. L Proof. By Lea 4., we have for any v T R T a g u T P Tf, u T v T = ϕ P, TguT u T v T PT f, u T v T ϕ P, TguT P + TguT ϕ P, TguT P TgvT We derive thanks to assutions H 2,H 3 and to the inequality 3.7 P M f L um v M L P M f L um v M L. a g u T P Tf, u T v T C T P TguT L b L C T P TguT L + b 2 L T P TgvT L C f L v um M L + u M v M L. Using the Young inequality and the discrete Poincaré inequality Lea 3.2 alied to u T v T and g = we deduce a g u T P Tf, u T v T C T P L C b T gut L C b 2 L C f L C T P TgvT L. The clai is then roved by taking v T = P T Rg and by using the continuity of the oerator R given in 3. and the estiate 3.8. Lea 4.3 Monotonicity. Assue that the flux ϕ satisfies H. For any esh T on and any distinct eleents u T and v T of R T, we have a g u T a g v T, u T v T >. Proof. By Lea 4., and using 2., it follows a g u T a g v T, u T v T = ϕ P ϕ TguT P, TgvT P TguT P. TgvT Thus, the clai derives fro assution H. We can now rove the ain result of this section, that is the existence and uniqueness of the aroxiate solution. Theore 4.4. Assue that the flux ϕ satisfies H, H 2, H 3. For any f L and g W, and any esh T on, the finite volue schee 2.5 adits a unique solution u T R T. Furtherore, there exists C > deending only on C, C 2 and regt, such that the following estiate holds T P TguT L C g + f + b W, L L + b L 9

20 Proof. The continuity of the a u T a g u T follows fro.2. Thanks to the coercivity roerty Lea 4.2 and the Poincaré inequality Lea 3.2 we can use one of the classical consequences of the Brouwer fixed oint theore see [25] to obtain the existence of a solution of the schee. The uniqueness of the solution follows readily fro the strict onotonicity of the a a g Lea 4.3. Finally, since a g u T = P f, T the estiate 4. is a straightforward consequence of Lea The otential case. Let us ay secial attention to the case where the flux ϕ derives fro a convex otential Φ: { ϕz, ξ = ξ Φz, ξ, for all ξ R 2 and a.e. z, 4.2 Φz, =, for a.e. z. For instance, the -lalacian.4 derives fro Φz, ξ = ξ, the anisotroic lalacian.3 fro Φz, ξ = 2 Azξ, ξ for a syetric atrix A and the general odel.5 fro Φz, ξ = kz ξ +F z. Rearking that we have ϕ ξ = ξ Φ ξ, where Φ is naturally defined by Φ ξ = Φz, ξ dz, we can define on R T the discrete energy J g,t associated to the schee by: J g,t u T = Φ P u T, P T f = Φz, T TguT P dz u T, P f T. T gut Proosition 4.5 Variational structure of the schee. Assue that ϕ has the for 4.2 and satisfies H, H 2, H 3, then J g,t is a strictly convex coercive functional. Furtherore, the schee 2.5 is the Euler-Lagrange equation associated to the iniization roble for J g,t. More recisely, we have J g,t u T, v T = a g u T P Tf, v T, u T, v T R T. The roof is straightforward using Lea 4.. Corollary 4.. Under assutions H,H 2, H 3 and 4.2, the solution u T R T of the schee 2.5 is the unique iniizer of the functional J g,t on the set R T. The ractical coutation of the aroxiate solution can take advantage of the articular structure 4.2, for instance, by using the Polak-Ribière nonlinear conjugate gradient ethods. In fact, for the coutations shown in Section 8, we used a siilar saddle-oint enalized forulation of the discrete roble to the one roosed by Glowinski and Marrocco in [2] for the P finite eleent aroxiation of the -lalacian. This forulation allows the coutation of the iniizer of J g,t through a lagrangian algorith which aears to be uch ore efficient than nonlinear conjugate gradient ethods. 5. Convergence of the schee. The ai of this section is to rove the convergence of the solution of the finite volue schee given by Theore 4.4 towards the solution ue to the continuous roble.. More exactly, we rove the strong convergence of both coonents u M, u M of u τ to the aroxiate solution in L, the strong convergence of the discrete gradients towards ue in L 2, and the strong convergence of the discrete fluxes towards ϕ, ue in L 2. This last convergence is crucial in the alications since the flux ϕ, ue is often an iortant hysical quantity that one ay want to coute recisely. For instance, in the context of the odelling of non-newtonian flows in a orous ediu, this flux is nothing but the velocity of the fluid. Theore 5.. Assue that the flux ϕ satisfies H, H 2, H 3 and consider a faily of eshes T n such that sizet n tends to zero and regt n is bounded. For any f L, g W,, the solution u T n to 2

21 the schee 2.5 on the esh T n converges towards the solution ue of the roble. as n goes to infinity. More recisely, if we note to silify g n = P T n g, we have u T n n ue strongly in L, T g n n u T n n ue strongly in L ϕ, T g n n u T n ϕ, ue strongly in L. n Moreover, the two sequences u M n n and u M n n both converge towards ue in L. Proof. As usual see for instance [6, 7, 25] the key-oint of the roof is to take advantage of the onotonicity roerties in order to ass to the liit in the nonlinear ters this is known in the literature as the Minty- Browder arguent, see [25].. Using the estiate 4. and thanks to assution H 3, we see that the failies of functions u T n, T g n n u T and z ϕz, T g n n u T n z are bounded in L, L 2, L 2 resectively. Hence by the discrete coactness result of Lea 3.6, there exists a function u Wg, such that u to a subsequence, and a function ζ Let w Cc = u T n T n g n u T n L 2 such that and take P T n a gn u T n P T n n u in L, u weakly in n L 2, 2 ϕ, T g n n u T n ζ weakly in L. 5.2 n w as a discrete test function in the schee 2.5. By Lea 4. it follows f, P T n w = ϕz, T g n n u T n z, T P T n n wz dz PT f, P T n w. We can ass to the liit in this equality using 5.2 and Corollary 3.. We get fzwz dz = ζz, wz dz. 5.3 By density, we deduce that for any function v Wg,, we have fzuz vz dz = ζz, uz vz dz Thanks to the onotonicity of the schee Lea 4.3, we have a gn u T n a gn P T n v, u T n P T n v. 5.5 Let us ass to the liit as n, in this inequality. First, using the definition of the schee 2.5 and Corollary 3., we find a gn u T n, u T n P T n n v = PT f, u T n P T n v fzuz vz dz. n Using Lea 4. and 2.2, we can write a gn P T n v, u T n P T n v = ϕz, T n g n P T n 2 vz, T g n n u T n z T n g n P T n vz dz.

22 Using Corollary 3. and the roerty.2, we see that the function ϕ, T g n n P T n v converges strongly in L 2 towards the function ϕ, v. As a consequence, fro the weak convergence of g T n n u T n towards u it follows that a gn P T n v, u T n P T n v converges to the integral ϕz, vz, uz vz dz. Hence, taking the liit as n goes to infinity in 5.5 gives fzuz vz dz ϕz, vz, uz vz dz, for all functions v Wg,. By 5.4 it follows that ζz ϕz, vz, uz vz dz Let us take in 5.6 v = u ± tw with w Cc and t >, dividing by t we get ± ζz ϕz, u ± t w, wz dz. When t tends to zero, using.2 we obtain that for all w Cc ζz ϕz, u, wz dz =. We conclude using 5.3 that div ϕ, u = div ζ = f. Thus u Wg, is nothing but the unique solution ue of the roble.. Finally, the uniqueness of ue also guarantees that the convergence of u T n towards ue, in the sense of 5., holds without extracting a subsequence. 4. Let us show the strong convergence roerties of the discrete gradients following the techniques develoed in [7, 6]. Let us note G n z def = T g n n u T n z, H n z def = T g n n P T n uez and Ψ n z def = ϕz, G n z ϕz, H n z, G n z H n z. By assution H, we know that Ψ n. Furtherore, the first art of the roof above shows that the left-hand side ter in 5.5 tends to zero which reads, by Lea 4., as Ψ n z dz. 5.7 n Hence, Ψ n n tends to in L. Furtherore, by Corollary 3., H n n converges towards ue in L 2. Thus, there exists a set E such that \E has a zero Lebesgue easure and a subsequence, always indexed by n, such that Ψ n z and H n z uez for any z E. We can also assue that H, H 2 and H 3 hold for any z E. First of all, using H 3 and H we have Ψ n z C 2 G nz C H n z b z 2b 2 z H n z + G n z, z E. 5.8 For z E fixed, the sequence H n z n is bounded and Ψ n z n tends to. By 5.8, we deduce that G n z n is a bounded sequence in R 2. Moreover, if G is the liit of any subsequence G nk z k, we have = li k Ψ n k z = ϕz, G ϕz, uez, G uez. Using the onotonicity assution H, we deduce that G = uez. Thus, we deduce that for any z E, the whole sequence G n z n converges towards uez in R 2. 22

23 In addition, 5.7 and the already established convergences ily that ϕz, G n z, G n z dz ϕz, uez, uez dz. n Furtherore, by H 2, for all n N we have ϕ, G n, G n b L. As in [6, Lea 5], together with the a.e. convergence of G n this ilies the strong L convergence of the sequence ϕ, Gn, G n. The coercivity assution H n 2 ilies the equi-integrability of the sequence G n n. Since we have already roved that G n n weakly converges towards ue in L 2, using the Vitali theore we deduce that the sequence G n n which is in fact a subsequence of the initial sequence strongly converges towards ue in L 2. At the resent stage, we have roved that G n n is relatively coact in the strong toology of L 2 and that ue is its unique accuulation oint. Thus, the whole sequence G n n converges strongly in L 2 towards ue. Finally, the strong convergence of the fluxes ϕ, G n towards ϕ, ue coes fro the roerty We now have T nut n P T n ue = T g n n u T n T g n n P T n ue in L 2 as n. Thanks to the discrete Poincaré inequality Lea 3.2 we deduce that the two sequences u M n P Mn ue n and u M n P M n ue n tend to zero in L. The last clai of the theore follows by using Corollary 3. with v = ue. Note that the roof of the strong convergence of discrete gradients and fluxes can be notably silified, if a stronger onotonicity assution for the flux ϕ is assued: If < 2, there exist C 3 > and b 3 L such that for all ξ, η R 2 R 2 and alost every z, ϕz, ξ ϕz, η, ξ η C 3 ξ η 2 b 3 z + ξ + η 2. H a If > 2, there exists C 3 > such that for all ξ, η R 2 R 2 and alost every z, ϕz, ξ ϕz, η, ξ η C 3 ξ η. H b These assutions exress soe kind of Hölder continuity of the inverse [ϕz, ] of the flux ϕ; they will be needed for our subsequent results on stability of discrete solutions and on error estiates. Notice that ost of the usual exales like those given in Section.2 satisfy these stronger assutions see Lea.2 and [5, 8, 26]. 6. Stability with resect to the data. In this section we address the roble of the continuous deendence of the aroxiate solution with resect to the data. More recisely, we show that, as for the continuous roble., the discrete gradient of the solution to the finite volue schee is Hölder continuous with resect to the source ter f and the boundary data g uniforly with resect to the esh. This roerty is iortant because it ensures, for instance, that the nuerical ethod is stable with resect to the fully ractical coutation of the discretization of the data through quadrature forulae. Notice that the coutations below will also be useful in the roof of the error estiate theore in section 7. Fro now on, we need to assue soe kind of Hölder regularity with resect to ξ for the flux ϕz, ξ. More recisely, we consider the following assutions: If < 2, there exists C 4 > such that for all ξ, η R 2 R 2 and alost every z, ϕz, ξ ϕz, η C 4 ξ η. H 4a If > 2, there exist C 4 > and b 4 L 2 such that for all ξ, η R 2 R 2, and alost every z, ϕz, ξ ϕz, η C 4 b4 z + ξ 2 + η 2 ξ η. H 4b 23

24 We want to oint out that, once ore, these assutions are classical in this context and are satisfied for the various exales given in Section.2 see Lea.2 and [8, 2]. Furtherore, these new assutions do not involve regularity of ϕ with resect to the sace variable z. This allows, for instance, the resence of satial discontinuities in the coefficients of the roble we are studying. Proosition 6.. Let T be a esh on and f, f 2 L, g, g 2 W,. Under assutions H 2, H 3, H a-h b and H 4a -H 4b, if u T and u T 2 are the solutions of the schee 2.5 corresonding resectively to the data Pg T, Pf T and Pg T 2, Pf T 2, then we have T P TguT T P Tg2uT 2 L C M 2 f f 2 L + M 2 3 g g 2 3 W, + M 2 2 f f 2 2 L g g 2 2 W,, if < < 2, T P T gut T P T g2ut 2 L C f f 2 + g L g 2 W, + M 2 g g 2 W,, if > 2, where C deends only on regt, C i i 4, b i i 4, and M is defined by M = C + g + g W, 2 + f W, + f L 2. L Proof. Let us introduce v T = P T Rg and v2 T = P T Rg 2, R being the lift oerator see 3.. Testing the two schees with u T vt ut 2 + vt 2, we obtain by Lea 4. = a g u T a g2 u T 2, u T v T u T 2 + v2 T PT f P f T 2, u T v T u T 2 + v2 T = ϕ P TguT ϕ P Tg2uT 2, u T v T u T 2 v2 T PT f P f T 2, u T v T u T 2 + v2 T., Using 2., we obtain ϕ P TguT ϕ P Tg2uT 2, P TguT P Tg2uT 2 = ϕ P TguT ϕ P Tg2uT 2, P TgvT P Tg2vT 2 + PT f P f T 2, u T v T u T 2 + v2 T

25 Case < 2. Assution H a gives ϕ P TguT ϕ P Tg2uT 2, P TguT P Tg2uT 2 = = ϕz, P TguT ϕz, P Tg2uT 2, P TguT P Tg2uT 2 dz ϕz, T P TguT z ϕz, T P Tg2uT 2 z, T P TguT z T P Tg2uT 2 z C By the Hölder inequality, we get T P TguT T P Tg2uT 2 dz Hence, we have = b 3 z + T P T gut z + T P T g2ut 2 z 2 T P T gut T P T g2ut 2 b 3 + TP T gut + TP T g2ut dz T P T gut z T P T g2ut 2 z 2 dz. b 3 z + T P T gut + T P T g2ut T P T gut T P Tg2uT 2 2 b 3 + T P TguT + T P Tg2uT 2 2 b 3 z + T P T gut + T P Tg2uT 2 dz T P TguT T P Tg2uT L b C 3 L + T P TguT 2 L Thanks to assution H 4a, 6. gives ϕ P TguT ϕ P Tg2uT 2, P TguT P Tg2uT T P Tg2uT 2 2 L 2 dz ϕ P T gut ϕ P T g2ut 2, P T gut P T g2ut C T P TguT T P Tg2uT 2 L T P TgvT T P Tg2vT 2 L + C v M v M 2 L + v M v M 2 L + u M u M 2 L + u M u M 2 L f f 2 L. Cobining the last two inequalities and using the Poincaré inequality, we get the result. Case > 2. Using assution H b, we have ϕ P TguT ϕ P Tg2uT 2, P TguT P Tg2uT 2 = ϕz, P TguT ϕz, P Tg2uT 2, P TguT P Tg2uT 2 dz P C TguT P Tg2uT 2 dz = C T P TguT T P Tg2uT 2 L dz

26 enote by b 4 the ean value of b 4 on. By H 4b and the Young inequality, 6. ilies C T P TguT T P Tg2uT 2 L P TguT P Tg2uT 2 P TgvT P Tg2vT 2 b 4 + P T gut 2 + P T g2ut 2 2 The discrete Poincaré inequality and 3.8 then lead to PT f P f T 2, u T v T u T 2 + v2 T T P TguT T P Tg2uT 2 L T P TgvT T P Tg2vT 2 L b 4 + L 2 T P TguT 2 L + T P Tg2uT 2 2 L + C u M u M 2 L + u M u M 2 L f f 2 L + C v M v2 M L + v M v2 M L f f 2 L. T P TguT T P Tg2uT 2 L C g g 2 b W 4, L 2 + T P T gut 2 L + T P Tg2uT 2 2 L + g g 2 + f W, f 2. L The clai follows thanks to the estiate 4. alied to u T and u T Error estiates for W 2, solutions. We conclude the study of the convergence of the solution to the finite volue schee 2.5 by roviding an error estiate in the case where the exact solution of the roble. lies in the sace W 2, and the flux ϕ is sooth enough with resect to the satial variable z. More recisely, we consider in this section the following additional assutions on ϕ If < 2, there exist C 5 > and b 5 W, 2 such that for all ξ R 2 and alost every z, z 2, ϕz, ξ ϕz, ξ C 5 + ξ z z + b 5 z b 5 z. H 5a If > 2, there exist C 5 > and b 6 L such that for all ξ R 2 and alost every z, ϕ z, ξ z C 5 b6 z + ξ. H 5b Reark 7.. In the following result we will assue that ue lies in W 2, so that ue W, 2. Hence, using the revious assutions, we see that the a z ϕz, uez, lies in W, 2 if 2, and in W, 2 if ], 2[. These regularity roerties will justify all the coutations in the following roof. In articular, assutions H 5a -H 5b do not allow to consider non regular data f W, through the aniulation of Reark.. Let us coent on these assutions in the case of the exales given in section.2. For the anisotroic Lalace equation.3, assution H 5a is fulfilled as soon as the a A is Lischitz. In the nonlinear 26

27 exale.5, the assutions above are satisfied if the a k is Lischitz and if the vector-field F lies in W, 2. Our ain result is the following. Theore 7.. Assue that the flux ϕ satisfies not only H 2, H 3, but also H a-h b, H 4a -H 4b, H 5a -H 5b. Let f L and assue that the solution ue to. belongs to W 2,, which ilies that g W 2,. Let T be a esh on. There exists C > deending on ue W 2,, on regt, on the nors of the functions f, g, b i i 6, i =,..., 6 in their natural saces and on C i i 5, such that { ue u M L + ue u M L + ue T P TguT L C sizet, if 2, 7. ue u M L + ue u M L + ue T P TguT L C sizet, if > 2. Recall that J.W. Barrett and W.B. Liu roved in [5], in the articular case of the -lalacian on a convex doain, that if f L and if < 2, then ue belongs to H 2 and then to W 2,, so that the assution in the revious theore is fulfilled. On the other hand, when > 2, there exist solutions of. with f L which are not in W 2, but in Besov sace B +,. In this last case, otial error estiates were obtained in [2], in the fraework of cartesian eshes. 7.. Center-value rojection of continuous functions. In the roof of the convergence result Theores 5. we have shown that the difference of the discrete gradient of the aroxiate solution and the discrete gradient of the ean-value rojection of the exact solution Pue T tends to. Using the roerties of the ean-value rojection of any function in W, Corollary 3., we were able to conclude to the convergence of the discrete gradient towards the exact gradient ue. We are now in the case where ue is assued to be in W 2,, in articular, ue is Hölder continuous. Hence, it is ossible to define a ore natural rojection of this function on R T by sily taking the values of the function at the control oints x K and x K. This choice aears to be well adated to the coutations below. Let us state soe of the roerties of this new rojection oerator. efinition 7. Center-value rojection on the esh T. For any continuous function v on, set P T c v = vx K K M, vx K K M. We call P T c the center-value rojection of v on the sace R T of discrete functions. In the sae way, any g W 2, is Hölder continuous and we can consider its central-value discretization on the boundary P T c g = g K, g K to be defined by g K = gx K, K M, g K = gx K, K M iscrete boundary data. As stated before, we use P T c ue to coute the consistency error of our schee. Hence, since the boundary data g enters the schee through its ean-value rojection Pg, T it is needed to evaluate the contribution in the error of the difference P T c g Pg T between the two ossible discretizations of the boundary data. Lea 7.2. Let T be a esh on.. For any > 2, there exists C deending on and regt such that for any g W, we have T P T g PT c gt L C g W, For any >, there exists C deending on and regt such that for any g W 2, we have T P T g PT c gt L CsizeT g fw 2,

28 Proof. We want to estiate G def = P for any diaond cell near the boundary of since in the Tg PT c gt other cases this ter is zero. Using the definition of Pg T given in 2.2, 2.3 and the one of P T c g given in 7.2, we see that we have two kinds of ters to estiate in each diaond cell: Ter along the direction of τ. If K M case I in Figure 3., we have G, τ = gx K gs ds, 7.5 σ σk σ K whereas in the case II in Figure 3., this ter is zero. Ter along the direction of τ. Two situations ay occur a shown in Figure 3.. In the case II, we have and in the case I, G, τ = gx L gs ds, 7.6 σ σl σ L G, τ [ = gx K ] [ gs ds gx L gs ds]. 7.7 σ σk σ K σl σ L. The first oint is a consequence of the ebedding of W, in the Hölder class C, 2. Indeed, each of the ters in can be treated in the sae way. For instance, by Lea 3., the ter 7.5 is bounded as follows G, τ su gz gz σ z,z σ K 2 σ K gx gy dλx dλy σ σ K σ K x y gx gy CregT dλx dλy. σ K σ K x y Suing over the boundary diaond cells, we get the estiate Let us rove the second oint. We consider here the ters of the for 7.5. Let us suose that σ K =] h K, h K [ {} and x K =, recall that σ K is defined in 2.3 and is chosen in such a way that x K is located at the iddle of the edge σ K. We can write Now since σ a a 2t 2 h K G, τ = hk 2h K h K T gtxx dx dt = 2t 2 h K thk th K T gss ds dt. a T gys ds = T gy s ds = for any a >, integrating in y [ th K, th K ] we get a thk th K T gss ds = 4t 3 h 2 K = 4t 3 h 2 K thk thk th K thk thk th K 28 th K T gs T gys ds dy th K T gs T gy s y s ds dy. s y

29 It follows that G, τ Finally we have C t σ thk thk T gs T gy 2th K 2 s y s ds dy dt th K th K s y thk thk T gs T gy t σ 2th K 2 th K th K s y s y s ds dy dt thk thk th K 2 T gs T gy th K th K s y ds dy dt hk hk t 2 T gs T gy ds dy dt. h K h K s y s y Ch 2 K G, τ CsizeT CsizeT K M k i= Γ i σ K Γ i CsizeT g f W 2,. T gs T gy σ K s y T gs T gy s y ds dy s y ds dy s y Provided that neither x L nor x K are corners of, we have as reviously, in the case 7.6, G, τ CsizeT T gs T gy ds dy σ L σ L s y s y and in the case 7.7, G, τ CsizeT σ K σ L σ K σ L T gs T gy s y ds dy s y. If x L, for instance, is a corner of the doain, say the corner between the edges Γ j and Γ k, we estiate searately the contributions of σ L Γ j and σ L Γ k. More recisely, for < 2 we use the ebedding of W 2, Γ i in C,2 Γ i for i {j, k}, to find that G, τ CsizeT g 2 W f., In the case > 2, we recall that, thanks to 2.6 and 2.7, σ L Γ j and σ L Γ k are of size CsizeT 2 2. We use the ebedding of W 2, Γ i in C, Γ i to conclude. Finally, in the case = 2 we use the ebedding of H 3 2 Γ i in the set of Log-Lischitz functions and the definition 2.6 of regt with Proerties of the center-value rojection. We su u in this section the roerties of the center-value rojection oerator which are used in the estiate of the consistency error of our finite volue schee. Lea 7.3 Center-value rojection estiates. Let T be a esh on. There exists a constant C >, deending only on and regt, such that for any function v in W,, denoting by g = γv its trace, we have 29

30 . For 2, T P T c gpt c v L C v L ; 2. If, in addition, v W 2,, v T P T c gpt c v L CsizeT v W,, for >, 7.8 T P T c gpt c v L C v L + sizet v W,, for < < Proof. Let be a diaond cell, we use the notations of Figure 2.2. For a sooth function v and for all z we get by first-order Taylor exansion of v, and vx L vx K = v tz + tx L x L z dt v tz + tx K x K z dt 7. vx L vx K Using 2.9 and Lea 2., we get sin α P T c gpt c v = ν σ + ν σ = v tz + tx L x L z dt v tz + tx L x L z dt v tz + tx L x L z dt v tz + tx K x K z dt. 7. v tz + tx K x K z dt v tz + tx K x K z dt. 7.2 Let us define the quantity I K,σ and, by aroriate erutations of the subscrits, I L,σ, I K,σ, I L,σ as def I K,σ = v tz + tx K x K z dt σ dz. Averaging 7.2 over and using 2.6, we obtain Pc TgPT c v CregT I K,σ + I L,σ + I K,σ + I L,σ. Using the change of variables z z = tz + tx K, we can now bound each of the four ters in this inequality. For instance, we have I K,σ v tz + tx K x K z dt dz, σ C v tz + tx K dz dt C t 2 t vz dz dt, 3

31 x K x K t tx L + tx K t x L x L Fig. 7.. The rescaled diaond t where t is the rescaled diaond as shown in Figure 7.. Notice that t is included in. We have t = t 2 ; since > 2, by Hölder inequalities we obtain I K,σ C t 2 vz dz t C t 2 dt C vz dz b vz dz b. With siilar calculations for I L,σ, I K,σ, I L,σ, we deduce that Pc TgPT c v C vz dz. b t dt We conclude by suing this estiate over the diaonds set that T Pc TgPT c v L = Pc TgPT c v C v dz C v dz, b using the fact that the nuber N T defined in 2.5 is bounded by regt. Assue that v C 2, the clai will follow by density. Let be a diaond cell; the Taylor exansions 7.,7. can be relaced by the to second-order ones, so that σ vx L vx K = vz, τ + σ σ vx L vx K = vz, τ σ + σ σ t 2 v tz + tx L x L z, x L z dt t 2 v tz + tx K x K z, x K z dt, t 2 v tz + tx L x L z, x L z dt t 2 v tz + tx K x K z, x K z dt, 3

32 for any oint z. Using 2.9 and Lea 2., we get sin α P T c gpt c v vz = ν σ ν σ + ν σ ν σ t 2 v tz + tx K x K z, x K z dt t 2 v tz + tx L x L z, x L z dt t 2 v tz + tx K x K z, x K z dt t 2 v tz + tx L x L z, x L z dt. 7.3 As in the roof of the first oint, we take the average of 7.3 over. It follows that it is sufficient to control by eans of 2 v the four siilar quantities II, K,σ II L,σ, II K,σ, II L,σ where, for instance, b def II K,σ = t 2 v tz + tx K x K z, x K z dt dz, σ σ t 2 v tz + tx K x K z, x K z dt dz. The Jacobian deterinant of the change of variables z z = tz + tx K equals t 2. Hence, II K,σ d2 t 2 dt 2 vz dz. σ Since 2 >, using 2.6, we find II K,σ d 2 σ 2 vz dz, CregT d b b 2 vz dz. b With siilar calculations for II L,σ, II K,σ, and II L,σ, we have Pc TgPT c vz vz dz CregT d 2 vz dz, 7.4 b and the clai is roved by suing 7.4 over the diaonds set. The estiate 7.9 is a straightforward consequence of 7.8. Corollary 7.. Let T be a esh on. There exists a constant C >, deending only on and regt, such that. For any > 2 and any v W,, denoting by g = γv its trace, we have T P T gpt c v L C v W, and T P T c gpt v L C v W,. 2. For any > and any v W 2,, denoting by g = γv its trace, we have T P TgPT c v L C v L + CsizeT v W,, T P v c TgPT L C v L + CsizeT v W,, T P v T TgPT Pc TgPT c v L CsizeT v W,. 7.5 Proof. By 2., we have T P T gpt c v = T P T c gpt c v + T P T g PT c gt. Using the results of Leas 7.2 and 7.3, we deduce the first estiate of the first oint. The other estiates are roved in the sae way, using Proosition

33 7..3. Convergence of the ean-value rojection. In this section, we return to the roof of Proosition 3.5. Let us rove 3.. The case q = + is straightforward so that we only treat the case of finite values of q. We first reark that vx P M vx q dx C vx Pv T K q dx. Furtherore, using Jensen s inequality we get vx Pv T K q dx vx vy q dx dy K BK K B K C vx v σ q + v σ vy q dx dy, BK K B K C vx vs q ds dx + C K vx vs q ds dx, σ K σ σ BK B K σ where σ E K and v σ = vs ds. Thanks to Lea 3.3 and to 2.6, we get σ σ vx Pvx T q dx Cd q K vz q dz. K B K K Hence, 3. follows since the the nuber N T defined in 2.5 is bounded by regt. The sae arguent let us show 3. Let us now sketch the roof of 3.2. We follow the sae lines as that of 7.8 in Lea 7.3. Note that it is crucial that the ean-value rojection oerator P T averages v over balls centered at x K and x K. Assue first that v C 2. Let y B K, y 2 B L and z and use the second-order Taylor exansion, we have vy 2 vy = vz, y 2 y + K M K t 2 v tz + ty 2 y 2 z, y 2 z dt t 2 v tz + ty y z, y z dt. Take the average of this relation with resect to y B K and y 2 B L and integrate in z. In articular, we oint out that vz, y 2 y dy dy 2 = vz, x L x K = σ vz, ν. BK BL B L B K Proceeding as for the estiate of II K,σ in the roof of Lea 7.3, we find out that vy 2 dy 2 q vy dy vz, ν σ BK B K BL dz B L BL t 2 v tz + ty 2 y 2 z, x K z q dt B L σ dz dy 2 + BK t 2 v tz + ty y z, x K z q dt B K σ dz dy CregT sizet q 2 v q dz + CregT sizet q 2 v q dz. B K B L We conclude using 2.5 and 2.6. The general case of v W 2,q follows by density. Finally, the case v W 2, follows fro the liit q +. 33

34 7.2. Consistency error of the schee. As usual, for the error analysis of finite volue ethods see e.g. [8], the consistency error which has to be studied is the error on the nuerical fluxes across each of the edges and dual edges in the esh. We first give the recise definition of these ters, then we state the various estiates needed to rove Theore 7. in section 7.3. efinition 7.2 Pointwise consistency error. For any diaond cell, we define the ointwise consistency error in by R z = ϕ P T gpt c ue ϕz, uez, z. The ointwise consistency error R can be slit into three different contributions R grad, Rbound, and R ϕ. They originate, resectively, fro the errors due to the aroxiation of the gradient, to the discretization of the boundary data, and to the aroxiation with resect to the satial variable of the flux ϕ, ue : R z = R bound + R grad + R ϕ z, 7.6 where R bound = ϕ P TgPT c ue ϕ R grad = ϕ Pc TgPT c ue R ϕ z = P T c gpt c ue, ϕ z, uez dz, ϕ z, uez dz ϕ z, uez. Recall that σ and σ are the diagonals of = σ,σ. Let us introduce the following consistency errors on the nuerical fluxes: def def R σ,k = R σ,l = def R s, ν ds, R σ = R σ,k = R σ,l 7.7 σ σ def def R σ,k = R σ,l = R s, ν ds def, R σ = R σ = R,K σ,l. 7.8 σ σ Notice that these integrals ake sense since the a R is sooth enough to give a sense to its traces on edges see Reark 7.. In order to control R σ and R σ, let us estiate searately the different ters in the right-hand side of 7.6. Proosition 7.4 Error due to the discrete gradient. Assue that ϕ satisfies H 4a -H 4b and that ue W 2,. For any esh T on, there exists a constant C >, deending only on, C 4 and regt, such that in the case < 2, in the case > 2, R grad CsizeT 2 ue L ; R grad C sizet 2 ue L + sizet b 4 L ue L 2 ue L. 34

35 Proof. Let εz def = Pc TgPT c ue uez be the error of aroxiation of the gradient. Case 2. Using the definition 2.2 of ϕ, by assution H 4a and the Jensen inequality we have Using the estiate of ε given in 7.4, we get R grad C εz dz C εz dz. R grad Cd 2 uez dz, b and the clai is roved by suing this inequality over the diaonds set and using 2.5 and 2.6. Case > 2. We use 2.2 and the assution H 4b to obtain R grad C C b 4 z + uez 2 + P T c gpt c ue 2 εz dz b4 z + uez 2 + εz 2 εz dz. Using the ebedding of W, into L and the Hölder inequality, we deduce R grad C b 4 z 2 2 dz + ue 2 L Using once ore 7.4 to estiate ε, and suing R grad εz dz + C εz dz. over the diaonds, we conclude the roof. Proosition 7.5 Error due to the boundary data. Assue that ϕ satisfies H 4a -H 4b and that g W 2,. For any esh T on, there exists a constant C >, deending only on, C 4 and regt, such that in the case < 2, R bound CsizeT g 2 W f ;, in the case > 2, R bound CsizeT g fw 2, b 4 L 2 + ue 2 L + sizet 2 2 ue 2 L. Proof. We just have to use assutions H 4a and H 4b and the estiates 7.4 and 7.5. We define Rσ ϕ and R ϕ σ to be the resective contributions of Rϕ z to R σ and R σ, that is Rσ ϕ = R ϕ s, ν ds, and R ϕ σ = R ϕ σ σ s, ν ds σ. 7.9 σ Proosition 7.6 Error due to the aroxiate flux. Assue that ϕ satisfies H 4a -H 4b and H 5a - H 5b. For any esh T on, there exists a constant C >, deending only on, C 4, C 5 and regt, such that 35

36 in the case < 2 Rσ ϕ + R ϕ σ CsizeT + ue W, + b 5 L ; in the case > 2 Rσ ϕ + R ϕ σ CsizeT ue W + b, 4 2 L 2 + b 6 L. Proof. Let us give the roof for the ters involving the edges σ; the ters involving the dual edges σ are estiated in the sae way. First, by definition of R ϕ z, for each z we have R ϕ z ϕz, uez ϕz, uez dz ϕz, uez ϕz, uez dz + If < 2, assutions H 4a and H 5a yield R ϕ z b 5 z b 5 z dz + C + uez z z dz + C Averaging this inequality over the edge σ and suing over the diaonds set give ϕz, uez ϕz, uez dz. uez uez dz. Rσ ϕ C b 5 z b 5 s ds dz σ σ + d + uez dz + σ uez ues ds dz. σ Alying Lea 3.3 to b 5 and ue, we obtain Rσ ϕ CsizeT + b 5 L + ue L + 2 ue L. If > 2, thanks to assutions H 4b and H 5b, we see by the chain rule that the a ψ : z ϕz, uez belongs to W, 2 and that ψ L C 2 ue L + b 4 2 L 2 + b 6 L + ue L. Alying the notations and the result of Lea 3.3 to the function ψ, we deduce that Rσ ϕ = ψ σ ψ Cd ψz dz, b and the clai follows by suing R ϕ σ over the diaond cells. 36

37 7.3. Proof of Theore 7.. We are now in the osition to rove the error estiate 7. stated in Theore 7.. First of all, we have ue T P T gut L ue T P T c gpt c ue L + T P T g PT c gt L + T P T gpt c ue T P T gut L. 7.2 Thanks to Lea 7.3, the first ter is controlled by CsizeT ue W, and thanks to Lea 7.2, the second ter is controlled by CsizeT g fw 2. Therefore, in order to rove Theore 7. it is sufficient, to estiate the last ter in 7.2. To this end, let us rove the following inequalities: Case < 2. T P TgPT c ue T P TguT L CsizeT + ue W + b 2, 5 ue 2 2 W + f + b 2, L L 2 L 2 + b 2 L + b 3 2 L ; 7.2 Case > 2. T P TgPT c ue T P TguT L CsizeT ue W 2, + b 4 2 L 2 + b 6 L For > 2, taking u T = PT c ue, u T 2 = ut and g = g 2 = g in forula 6.3, we obtain Siilarly, for < 2 we use 6.2 to obtain T P T gpt c ue T P T gut L C a gp T c ue a g u T, P T c ue u T. T P TgPT c ue T P 2 2 TguT L b C 3 L + T P 2 TguT L + T P TgPT c ue 2 L a g P T c ue a g u T, P T c ue u T. Set I def = a g P T c ue a g u T, P T c ue u T. Let us exress I through the consistency errors R σ,k and R. σ,k Integrating equation. over the control volues K and the dual control volues K leads to 2., which is the exact counterart of the finite volue schee This coutation is valid since z ϕz, uez is sooth enough as seen in Reark 7.. Subtraction of these equations, together with the definitions 7.6, 7.7 and 7.8, yield a K u T a K P T c ue = σ,σ K σ R σ,k, K M, a K u T a K P T c ue = σ R, σ,k K M. σ,σ K Therefore, I = σ R σ,k uex K u K + σ R σ,k uex u. K K K M σ,σ K K M σ,σ K Let us rewrite I using the conservativity roerty of the fluxes in 7.7 and 7.8, the definitions 2.8 and 37

38 2.9 of the discrete gradient, and the suation-by-arts Lea 4.. We get I = σ E σ=k L = σ,σ sin α T sin α T σ R σ,k uex K u K uex L + u L + σr σ,k uex K u K uex L + u L σ E σ =K L σ σ Rσ,K P T c ue u T, τ + R σ,k P T c ue u T, τ σ,σ + sin α T σ,σ R σ R σ Hence, we have for < 2, T P TgPT c ue T P TguT L C b 3 and for 2, σ,σ R σ T P T gpt c ue T P T gut L. 2 L T P TgPT c ue T P TguT L C σ,σ + P T c ue u T σ,σ + T P TgPT c ue 2 L R σ σ,σ R σ σ,σ R σ P T c ue u T + T P 2 TguT L + C + R σ σ,σ R σ, 7.23, 7.24 where C deends only on regt and the other quantities allowed in the stateent of Theore 7.. Cobining 7.6, 7.7, 7.8 and 7.9 with the estiates shown in Proositions 7.4, 7.5 and 7.6, we deduce 7.2 and 7.22 fro 7.23 and 7.24 resectively. This ends the roof of Theore Nuerical results. In this section, we illustrate our theoretical study by showing the results of soe nuerical exerients. We suose that is the square ], [ 2. We will consider two kinds of analytic radial solutions for which the corresonding boundary data and source ter are couted exlicitly in order to test the accuracy of the ethod. For any real araeter α R, we define u αz = z α, u 2 αz = ex z 2 α 2 We coare the results obtained on two kinds of eshes. The esh is a faily of rectangular locally refined eshes obtained by successive global refineents of the original esh shown in Figure 8.. The esh 2 is a faily of standard unstructured triangulations of. 38.

39 Fig. 8.. Picture of the esh and the esh 2 In all the figures below, we lot in a logarithic scale the L relative error defined by ue ut L ue L and the W, error defined by ue T g u T T L as functions of the size of the esh sizet to lot but also as a ue L function of the nuber of unknowns for the discrete roble N botto lot. Let us oint out that for the locally refined esh, the size of the control volues in the refined zone equals.5 sizet. 8.. Anisotroic Lalace oerator. We consider here the anisotroic Lalace oerator.3 with a diffusion tensor Az = z 2 + 2z 2 z z 2 z 2 z z 2 2z 2 + z2 2, which is diagonalisable in a rotating frae around the origin with eigenvalues and 2. For sooth solutions, we observed the first order convergence in the H nor given by Theore 7. for = 2 and the second order convergence in the L 2 nor this suerconvergence is roved for the Lalace equation in [3]. We only resent here the results obtained for the radial function u.5 which is not in H 2 but only in H 3 2 +ε for any ε >. We find here a convergence rate of.4 in the H nor and.5 in the L 2 nor. This convergence rate is the one exected, at least for the usual Lalace oerator and cell-centered finite volue schees on adissible eshes see [6] Fully non-linear oerators. First, we consider the odel.5 for = 3., kz = and F z = z 2, z for any z = z, z 2. Notice that F is not a gradient field. The exact solution we used is u.35. The coefficient α =.35 is chosen just greater than 4 3 to ensure that u.35 is not uch ore sooth than W 2,. First of all, we observe an order of.73 in L 3 nor and an order of.98 in W,3 for both kinds of eshes. Thus, the theoretical convergence order given by Theore 7. is essiistic just like for any other studies in this field see [, 5, 26, 8]. To our knowledge, very few otial error estiates for nonlinear diffusion robles are available in the literature. Soe of the can be found, with in [26] for the P finite eleent aroxiation of the -lalacian and in [3] for the FV aroach on cartesian eshes. Nevertheless, an iortant feature is that the convergence rate is not sensitive to the resence of non conforing control volues in the esh. Furtherore, in the second lot, we observe that, a nuber of unknowns being fixed, the esh that is the one refined near the singularity gives better results than the non refined triangular esh 2. This eans that the finite volue schee resented in this aer for nonlinear equations, can be successfully used in conjunction with local refineent ethods in order to save soe CPU tie without loss of recision. Of course, the analysis of ossible a osteriori error indicators and adatative refineent techniques would be of great interest in this fraework. We finally collected in table 8. the nuerical convergence orders obtained for various values of on the exact solution u.35 on the triangular esh 2. We observe that the convergence order decreases as increases. Notice 39