CELL CENTERED FINITE VOLUME METHODS USING TAYLOR SERIES EXPANSION SCHEME WITHOUT FICTITIOUS DOMAINS

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 7, Number 1, Pages 1 9 c 010 Institute for Scientific Computing and Information CELL CENTERED FINITE VOLUME METHODS USING TAYLOR SERIES EXPANSION SCHEME WITHOUT FICTITIOUS DOMAINS GUNG-MIN GIE AND ROGER TEMAM Abstract. Te goal of tis article is to study te stability and te convergence of cell-centered finite volumes FV in a domain = 0, 1 0, 1 R wit non-uniform rectangular control volumes. Te discrete FV derivatives are obtained using te Taylor Series Expansion Sceme TSES, see [4] and [10], wic is valid for any quadrilateral mes. Instead of using compactness arguments, te convergence of te FV metod is obtained by comparing te FV metod to te associated finite differences FD sceme. As an application, using te FV discretizations, convergence results are proved for elliptic equations wit Diriclet boundary condition. Key Words. finite volume metods, finite difference metods, Taylor series expansion sceme TSES, convergence and stability, elliptic equations. 1. Introduction. Finite volumes FV are widely used bot in Engineering see e.g. [4], [10] and [13] and in Geopysical Fluid Dynamics GFD see e.g. [11], [1] and [8], because of teir local conservation property on eac control volume. From te matematical and numerical analysis points of view, tese metods are well studied for teir stability and convergence, using a variety of metods to compute te fluxes see e.g. [5], [6], [7], [9] and [14]. On a control volume in R, one simple way to compute te flux along a boundary is to start wit te difference of te given data at two cell centers divided by te lengt of te vector connecting tose cell centers and ten, taking te flux as te product of tat quantity and te lengt of te boundary, wic is te analog of te one dimensional case see [5], [6], [7] and [9]. However tis is not te best coice wen te unit normal on te boundary is not parallel to te vector connecting te two cell centers; to deal wit complicated meses in R, more efficient ways to compute te fluxes are needed. In tis article, we consider te cell centered FV by Taylor Series Expansion Sceme TSES, wic permits to compute te fluxes on a general quadrilateral mes in R see [4] and [10], and apply tem to quasi- but, non- uniform meses on ; we also intend to consider more general meses in te future. For te matematical analysis of te FV metod, one specific difficulty is due to te weak consistency of FV. Indeed te companion discrete FV derivative arising in te discrete integration by parts does not usually converge strongly to te corresponding derivative of te limit function see e.g. [6] or [9]. To overcome tis difficulty, discrete compactness arguments ave been used Received by te editors Marc 15, Matematics Subject Classification. 65N1, 65N5, 76M1, 76M0. Tis work was supported in part by NSF Grant DMS , and by te Researc Fund of Indiana University. 1

2 G. GIE AND R. TEMAM as in e.g. [6]. But ere instead we consider te finite differences FD associated wit te FV and compare te FV and FD spaces by defining a map between tem. Te convergence of te FV metod is ten inferred. Our work is organized as follows. In Section, we describe te cell centered FV setting by TSES witout using fictitious domains, but using instead flat domains at te boundary. In Section 3, we introduce an external approximation of H 1 0 using FV spaces V see [3] and [15], and sow tat te truncation error between a function in H 1 0 and its projection onto te FV space V tends to zero as te mes sizes decrease. Due to te weak consistency of te FV, we are not able at tis point to sow tat te external approximation of H 1 0 by te FV spaces is convergent. Instead, in Section 4, we present te FD metod associated wit tis FV metod and prove te stability and convergence of te external approximation of H 1 0 by te FD spaces Ṽ in Section 5. In Section 6, comparing te FV and FD spaces and tanks to te convergence of te FD, we obtain te convergence of te FV in te end. Finally, in Section 7, as an application, we demonstrate ow one can use te FV metod to approximate te solution of some typical elliptic equations wit Diriclet boundary condition, and, using our results, sow te convergence of suc an approximation via finite volumes to te solution of te original problem.. Te Finite Volume Setting. Te domain is = 0, 1 0, 1 in R. We set x 0 = x 1 = 0, x M+ 1 = x M+1 = 1, y 0 = y 1 = 0, y N+ 1 = y N+1 = 1 and we coose te nodal points x i+ 1, y j+ 1 for 1 i M 1, 1 j N 1,.1 0 = x 0 =x 1 < x 3 < < x M+ 1 = x M+1 = 1, 0 = y 0 =y 1 < y 3 < < y N+ 1 = y N+1 = 1. We define te control volumes on wic appear on Fig. 1,. K i,j = x i 1, x i+ 1 y j 1, y j+ 1, 1 i M, 1 j N, x i 1, x i+ 1 {y j}, 1 i M, j = 0, N + 1, {x i } y j 1, y j+ 1, i = 0, M + 1, 1 j N. Here, we ave cosen flat control volumes at te boundary to andle and enforce te boundary conditions. For 1 i M, 1 j N, te center of K i,j is.3 x i, y j = x i 1 + x i+ 1, y j 1 + y j+ 1 We set i = x i+ 1 x i 1, k j = y j+ 1 y j 1, 1 i M, 1 j N,.4 i+ 1 = x i+1 x i, k j+ 1 = y j+1 y j, 0 i M, 0 j N, and, for convenience, we also set.5 0 = M+1 = k 0 = k N+1 = 0. Ten we infer from.3-.5 tat.6 i+ 1 = 1 i + i+1, k j+ 1 = 1 k j + k j+1, 0 i M, 0 j N, and write te nodal points x i+ 1, y j+ 1 x i, x i+1, y j and y j+1 see Fig. :. as proper weigted averages of te points

3 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 3 y j+3/ y j+1 y j+1/ y j y j 1/ x i 1/ x i x i+1/ x i+1 x i+3/ Figure 1. Control volumes K i,j and tose centers x i, y j on. i i+1/ i+1 x i 1/ x i x i+1/ x i+1 x i+3/ Figure. Te order of te FV points in te x direction..7 x i+ 1 = i+1x i + i x i+1 i + i+1, 1 i M 1, y j+ 1 = k j+1y j + k j y j+1 k j + k j+1, 1 j N 1. It is interesting to notice and empasize te sequential order of te FV points x or y:.8 x i = x i i, x i+ 1 = x i 1 + i, x i+1 = x i 1 + i + 1 i+1, x i+ 3 = x i 1 + i + i+1. We now introduce te following function space:.9 V := and, for any u V, we write step functions u on suc tat u Ki,j = u i,j, 0 i M + 1, 0 j N + 1, and u i,j = 0, if i = 0, M + 1, or j = 0, N u = u i,j χ Ki,j, were χ Ki,j is te caracteristic function of K i,j. For 1 i M, 0 j N, we define te quadrilateral K i,j+ 1, solid line in Fig.

4 4 G. GIE AND R. TEMAM y j+3/ y j+1 y j+1/ y j y j 1/ x i x i+1/ x i+1 x i 1/ x i+3/ Figure 3. K i+ 1,j dased line and K i,j+ 1 domains of constancy for te FV derivative. tick solid line as 3:.11 K i,j+ 1 is te quadrilateral connecting x i 1, y j+ 1, x i, y j+1, x i+ 1, y j+ 1, and x i, y j, and, for 0 i M, 1 j N, we also define te quadrilateral K i+ 1,j 1 dased line in Fig. 3:.1 K i+ 1,j is te quadrilateral connecting x i, y j, x i+ 1, y j+ 1, x i+1, y j, and x i+ 1, y j 1. Te discrete FV derivative u = x u, y u for u V is obtained by TSES; see [4] and [10]. Here we sligtly modify te original TSES of [4] and [10] to ensure te consistency see 3.4 and 3.16 below and we set:.13 x u = u i+1,j u i,j i+ 1 u i+ 1,j+ 1 u i 1,j+ 1 on K i+ 1,j, 0 i M, 1 j N, i on K i,j+ 1, 1 i M, 0 j N, were we define te term u i+ 1,j+ 1 by a weigted average between te four neigbors i, j, i + 1, j, i, j + 1 and i + 1, j + 1: for 0 i M, 0 j N,.14 u i+ 1,j+ 1 = ik j u i+1,j+1 + i k j+1 u i+1,j + i+1 k j u i,j+1 + i+1 k j+1 u i,j ; i + i+1 k j + k j+1 note tat, due to.5 and.9, u i+1/,j+1/ is equal to 0 wen i = 0, M or j = 0, N. Te definition of y u is similar; we obtain y u from.13 by replacing x and by y and k, and intercanging te indices i and j. We define on V, te scalar 1 Because x0 = x 1/ = 0, K 1/,j is in fact a triangle, 1 j N. Te same is true of K M+1/,j, K i,1/ and K i,n+1/.

5 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 5 products, V and, V tat mimic tose of L and H0 1 : for u, v V, u, v V = u, v L = u i,j v i,j i k j,.15 wit.16 u, v V = x u, x v L + y u, y v L, x u, x v L = + k j i+ 1 u i+1,j u i,j v i+1,j v i,j k j+ 1 i u i+ 1,j+ 1 u i 1,j+ 1 v i+ 1,j+ 1 v i 1,j+ 1. Te corresponding norms V and V are defined as usual. We will need to impose some restrictions on te mes sizes i and k j. Here we begin wit te uniformity assumptions; furter ypoteses appear below in 5.9 and We set.17 = max i, = min i, k = max k j, k = min k j, ρ = max, k, ρ = min, k, and assume tat, as ρ 0, tere exit 0 < α x, α y < 1 suc tat.18 α x, k α y k, and, furtermore, comparing te x and y directions, we also assume tat,.19 k α y, α x k. If we set.0 α = min α x, α y, α = max α x, α y, ten we infer from.18 tat.1 M M 1 α x 1 α x Nk N 1 α y k 1 α y i 1 α x 1 α, k j 1 α y 1 α. We start wit te following easy lemma wic provides te discrete Poincaré inequality for te FV space. Lemma.1. For every u V,. u V α 1 u V. Proof. We consider u as in.10. For any 1 i M, 1 j N, since u 0,j = 0, we ave u i,j = u i,j u i 1,j + u i 1,j u i,j + + u 1,j u 0,j.

6 6 G. GIE AND R. TEMAM Terefore, by te Scwarz inequality, i 1 M u i,j i u l+1,j u l,j M u l+1,j u l,j. l=0 Ten, using.1, we find u V = Similarly, we find M M and ence, we obtain.. u i,j i k j 0 l M 0 l M M 0 l M l=0 u l+1,j u l,j k j u l+1,j u l,j l+ 1 u l+1,j u l,j i k j k j αx x u L α x u L u V α y u L, 3. External approximation of H 1 0 by FV. As we briefly mentioned in te introduction, ere we introduce an external approximation of a normed space V as a set consisting of a normed space F, an isomorpism ω of V into F and a family of triples {W, p, r }, in wic, for eac, W is a normed space, p is a linear prolongation operator of W into F and r is a restriction operator of V into W ; see [3], [], [15] and Fig. 4. Here we set V = H 1 0, F = L 3 and W = V, and define te maps ω, p and r as follows: 3.1 ωu = u, D x u, D y u, u H 1 0, p u = u, x u, y u, u V, and, for all v V = C0, 3. 1 vx, y dx dy, x, y K i,j, 1 i M, 1 j N, r vx, y = i k j K i,j 0, x, y K i,j, i = 0 or M + 1, or j = 0 or N + 1. Tanks to te Poincaré inequality., we ave V r ω W p F Figure 4

7 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS p u F = u L + u L = u V + u V 1 + c 0 u V, were c 0 = α is independent of i or k j ; te stability of te p follows. To prove te convergence of our FV sceme, we need to prove te two following properties see [3] and [15]: 3.4 C1 u V, p r u ωu in F as ρ 0, C If u V and p u ϕ in F weakly as ρ 0, ten ϕ ωv. Along te proof of tese properties, we will use repeatedly te following lemmas: Lemma 3.1. For any quadrilateral K wit barycenter x G, y G and area K, and for any function ϕ C K, we ave ϕ x, y dx dy = ϕ x G, y G + O K, K were K 3.6 O K ϕ C K K. We also introduce te useful interpolation lemmas: Lemma 3.. For any function ϕ C l were l is te line connecting te points ξ 1 and ξ in R, and for any point ξ l, we ave 3.7 ϕξ ϕξ 1 = ϕξ ξ ξ 1 + O ξ ξ 1, were 3.8 O ξ ξ 1 ϕ C l ξ ξ 1. Lemma 3.3. For any two-dimensional convex polygon K wit p vertices ξ i, 1 i p, p, we consider a point ξ in K, ξ = p i=1 λ iξ i were p i=1 λ i = 1 and λ i 0. Ten, for any function ϕ C K, we ave 3.9 λ i ϕξ i = ϕξ + O max ξ i ξ j. 1 i,j p were 1 i p 3.10 O max ξ i ξ j ϕ C 1 i,j p K max ξ i ξ j. 1 i,j p For any oter point η in K, 3.11 λ i ϕξ i = ϕη + O max ξ i ξ j. 1 i,j p 1 i p We omit te proofs of tese elementary lemmas; using te Taylor expansions, one can easily verify 3.5, 3.7 and 3.9. We obtain 3.11 by combining 3.7 and 3.9.

8 8 G. GIE AND R. TEMAM 3.1. Proof of C1 for FV. To prove C1, we first sow tat, for u V, 3.1 r u u strongly in L as ρ 0, We consider x, y K i,j and, using 3.5 for te set K i,j wit center x i, y j and area i k j, and also using 3.7, we write 1 r ux, y ux, y = ux, y dx dy ux, y i k j K i,j 3.13 ux i, y j + O ρ ux, y sup Du ρ + O ρ 0 as ρ 0, were O ρ means, trougout tis article, O ρ cρ, c independent of te mes sizes; ere, of course, c depends on te C norm of u. Hence, from 3.13, r u u in L as ρ 0 and 3.1 olds. To sow tat, for u V, 3.14 x r u D x u strongly in L as ρ 0, we consider two cases. Firstly, if x, y K i+ 1,j for some 0 i M, 1 j N, ten, using.4,.13 and 3.7 in te x direction along x i, x i+1 at x i+ 1, we ave 3.15 x r ux, y = D x ux i+ 1, y j + O ρ = D x ux, y + O ρ. Secondly, if x, y K i,j+ 1 for some i, j, we first notice tat, using 3. and 3.5, te term r u i+ 1,j+ 1 is obtained by te same average as in.14. Ten, applying 3.9 to u were K is te quadrilateral connecting x i, y j, x i+1, y j, x i+1, y j+1 and x i, y j+1, wit te weigted average x i+ 1, y j+ 1 in.7, we find tat, for 1 i M 1, 1 j N 1, 3.16 r u i+ 1,j+ 1 = u x i+ 1, y j+ 1 + O ρ. Ten, using also.13 for r u and 3.7 again in te x direction along x i 1, x i+ 1 at x i, we obtain tat, for x, y K i,j+ 1, 3.17 x r ux, y = D x ux i, y j+ 1 + O ρ = D x ux, y + O ρ. Hence, from 3.15 and 3.17, we see tat in bot cases 3.18 x r ux, y D x ux, y cρ, were te constant c related to te C norm of u is independent of x, y and ρ. Terefore, x r u D x u in L as ρ 0 and 3.14 olds. Te proof being te same for te y derivative, C1 is now proven. Remark 3.1. As we mentioned in te introduction, due to te weak consistency of te FV sceme, we cannot prove C for FV directly see [6] and [9]. Instead we introduce te corresponding associated FD sceme and prove C for it as well as te stability and C1 properties. Ten, comparing te FV space and te FD space, we will finally prove C for FV.

9 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 9 4. Te corresponding Finite Difference Setting. To define te FD mes associated wit te previous FV mes, we set x 0 = x 1 = ỹ 0 = ỹ 1 = 0 and x M+1 = x M+ 1 = ỹ N+1 = ỹ N+ 1 = 1. We also set x i = x i, ỹ j = y j for 1 i M, 1 j N. Ten, we define te FD nodal points x i+ 1, ỹ j+ 1 see Fig. 5: 4.1 x i+ 1 = 1 x i + x i+1, 1 i M 1, ỹ j+ 1 = 1 ỹ j + ỹ j+1, 1 j N 1. Togeter wit te order of te FV points in.8, it is also interesting to notice te sequential order of te FD points x or ỹ: 4. x i+ 1 x i = x i, = 1 x i + x i+1 = wit.8 = x i i i+1. Hence, comparing wit.8, we see tat 4.3 x i = x i < x i+ 1, x i+ 1 < x i+1 = x i+1, 1 i M 1, but te respective orders of x i+ 1 and x i+ 1 We set 4.4 may vary wit i. i = x i+ 1 x i 1, kj = ỹ j+ 1 ỹ j 1, 1 i M, 1 j N, i+ 1 = x i+1 x i, kj+ 1 = ỹ j+1 ỹ j, 0 i M, 0 j N, and compare te FV mes sizes or k and te FD mes sizes or k: 4.5 i = x i+ 1 x i 1 = x i i i+1 xi i i = x i 1 x 3 i 3 4 i i i+1 = 1 4 i 1 + i + i+1, i+ 1 = i+ 1. Due to 4.1, we also observe useful relations among te FD mes sizes: for i M, 4.6 i+ 1 + i 1 = x i+1 + x i x i + x i 1 = x i+ 1 x i 1 = i. Te K i,j are defined in te same manner as te K i,j in. by replacing x and y by x and ỹ; teir sides are i+ 1 and k j+ 1, and te geometric relation between te FV and FD grids appears on Fig. 6 in wic e.g. x i+ 1 < x i+ 1 but ỹ j 1 > y j 1 ; see 4.3. Te space of step functions for FD is given by, step functions ũ on suc tat 4.7 Ṽ := ũ Ki,j = ũ i,j, 0 i M + 1, 0 j N + 1,. and ũ i,j = 0, if i = 0, M + 1, or j = 0, N + 1

10 10 G. GIE AND R. TEMAM ~ y j+3/ ~ y j+1 ~ y j+1/ ~ y j ~ y j 1/ x i 1/ ~ ~ ~ x i x~ i+1/ x~ i+1 x i+3/ Figure 5. Te corresponding FD mes and sets K i,j rectangles, K i+ 1,j dased line and K i,j+ 1 tick solid line. ~ y j+3/ y j+3/ y j+1 y j+1/ y~ j+1/ y j ~ y j 1/ y j 1/ y j 1 y j 3/ ~ y j 3/ x x~ i 1/ i 1/ x x~ i+1/ x i+1/ x i+1 x ~ i i+3/ x i+3/ Figure 6. Te FVsolid lines and FDdased lines meses in. We also introduce te discrete FD derivative: for ũ Ṽ, ũ i+1,j ũ i,j i x ũ = ũ i+ 1,j+ 1 ũ i 1,j+ 1 on K i+ 1,j, 0 i M, 1 j N, i on K i,j+ 1, 1 i M, 0 j N,

11 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 11 wit 4.9 ũ i+ 1,j+ 1 = ũi+1,j+1 + ũ i+1,j + ũ i,j+1 + ũ i,j, 0 i M, 0 j N. 4 Te discrete derivative y ũ is defined similarly by replacing x, by ỹ, k and intercanging te indices i, j in 4.8. Te domains of constancy for ũ are te sets K i,j+ 1 for 1 i M, 0 j N and K i+ 1,j for 0 i M, 1 j N tat are defined in te similar way as in te FV case using x, ỹ instead of x, y see.11,.1 and Fig. 5. Te scalar products, Ṽ,, Ṽ and corresponding norms Ṽ, Ṽ are obtained from.15 by replacing u, v,, k by ũ, ṽ,, k. Exactly as in te FV case, we can prove te discrete Poincaré inequality for FD, wic occurs wit te same constant: Lemma 4.1. For every ũ Ṽ, 4.10 ũ Ṽ α 1 ũ Ṽ. 5. External approximation of H 1 0 by FD. We again consider te diagram in Fig. 4 wit now V = H0 1, F = L 3 and W = Ṽ. Te maps ω and p are te same as in 3., but we now define r v, for v V, as follows { v xi, ỹ j, x, y K i,j, 1 i M, 1 j N, 5.1 r vx, y = 0, x, y K i,j, i = 0 or N + 1, or j = 0 or M + 1. Similar to te FV case, te stability of te operators p follows from te Poincaré inequality in Lemma Proof of te property C1 for FD. Te proof is similar, and even simpler tan in te FV case; we recall it briefly. To sow tat, for u V, 5. r u u strongly in L as ρ 0, we consider x, y K i,j, and, using 3.7, write 5.3 r ux, y ux, y = u x i, ỹ j ux, y sup Du ρ + O ρ. Hence r u u in L as ρ 0 and 5. is valid. To sow tat, for u V, 5.4 x r u D x u strongly in L as ρ 0, we let x, y and consider two cases. Firstly, if x, y K i+ 1,j for some 0 i M, 1 j N, ten by using 4.8, 5.1, and 3.7 along x i, x i+1 at x i+ 1, 5.5 x r ux, y = D x u x i+ 1, ỹ j + O ρ = D x ux, y + O ρ. Note tat ũi+1/,j+1/ may not be zero for i = 0, M or j = 0, N, but may be small. Tis is not inconsistent wit te Diriclet boundary condition wic is well enforced by 4.7.

12 1 G. GIE AND R. TEMAM Secondly, if x, y K i,j+ 1 for some i, j, we observe tat, from 5.1, te term r u i+ 1,j+ 1 is given by te same average as in 4.9. Hence, applying 3.9 to u were K is te quadrilateral connecting x i, ỹ j, x i+1, ỹ j, x i+1, ỹ j+1 and x i, ỹ j+1, wit barycenter x i+ 1, ỹ j+ 1, we find tat, for 1 i M 1, 1 j N 1, 5.6 r u i+ 1,j+ 1 = u x i+ 1, ỹ j+ 1 + O ρ. From 4.8, 5.1 and 5.6, we now infer tat, for x, y K i,j+ 1, using 3.7 again along x i 1, x i+ 1 at x i, 5.7 x r ux, y = D x u x i, ỹ j+ 1 + O ρ = D x ux, y + O ρ. From 5.5 and 5.7, we see tat in all cases 5.8 x r ux, y D x ux, y O ρ 0 as ρ 0, and tus, x r u D x u in L as ρ 0 and 5.4 olds. Wit te same result for te y variable, te proof of C1 is complete. 5.. Proof of C for FD. To prove C, we impose anoter condition to our mes sizes, namely 5.9 i+1 i 1 sup = η 1 0 as ρ 0, i M 1 k j+1 k j 1 sup = η 0 as ρ 0. j N 1 k Note tat η 1 = η = 0 for a uniform mes and in te typically annoying case considered in [6] and [9] were i =, i+1 =. We want to verify C; so let us assume tat ũ Ṽ,, and tat, as ρ 0: 5.10 ũ ϕ 0, ũ ϕ 1, ϕ weakly in L. We ave to sow tat, for θ C0 R, 5.11 ϕ1, ϕ θdxdy = ϕ 0 Dθdxdy R R were ϕ is equal to ϕ in and to 0 in R \. Indeed if 5.11 is proven, ten ϕ 1, ϕ = Dϕ 0 wic implies tat ϕ 0 H 1 R, and tus ϕ 0 H0 1 wit ϕ 1, ϕ = Dϕ 0, since ϕ 0 vanises outside of. We set 5.1 I = I, x I y R = ũ θdxdy = ũ θdxdy. By 5.10, we promptly see tat I converges to te left-and side of Bot directions being andled similarly, to verify 5.11 and C, we only need to sow tat: 5.13 I x = x ũ θdxdy ϕ 0 D x θdxdy = ϕ 0 D x θdxdy, as ρ 0. R

13 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 13 Te regions of constancy K i+ 1,j and K i,j+ 1 of te discrete FD derivatives ũ are andled differently. Terefore, to obtain 5.13, we write 5.14 I x = I 1 + I, I 1 = x ũ χ Ki+ 1 θdxdy, I =,j x ũ χ Ki,j+ 1 θdxdy, and, after we simplify eac of I 1 and I, we will sow tat under te assumptions.18,.19 and 5.9, I x converges to te rigt-and side of 5.13 as ρ 0. We observe tat, for 0 i M, 1 j N, te areas of te quadrilaterals K i+ 1,j are 1 i+ 1 kj and teir centers are x i+ 1, ŷ j, were 5.15 x i+ 1 = x i+ 1, 1 i M 1, x 1 = 1 3 x 1, x M+ 1 = x M, ŷ j = 1 1 3ỹj + ỹ j + ỹ j+ 1, 1 j N; ence, using Lemma 3.1, we find tat i+ 1 K i+ 1,j θdxdy = k j θ x i+ 1, ŷ j + O ρ 3, were, due to 3.5, O ρ 3 is bounded by c θ C ρ 3 for a constant c independent of te mes sizes. We ten simplify I 1 by using 4.7 and 5.16: 5.17 I 1 = K i+ 1,j 1 ũ i+ 1 i+1,j ũ i,j θdxdy = by canging te indices and using u 0,j = u M+1,j = 0 = ũ { 1 1 i,j θdxdy θdxdy } i 1 i+ 1 = 1 K i 1,j K i+ 1,j { ũ i,j kj θ xi+ 1, ŷ j θ xi 1, ŷ } j + O ρ ũ Ṽ. Using 3.7 for θ along x i 1, x i+ 1 at x i, we write I 1 in 5.17 in te form: 5.18 I 1 = 1 ũ i,j D x θ x i, ŷ j i kj + O ρ ũ Ṽ. Now, to obtain te expression of I similar to 5.18, we consider te quadrilaterals K i,j+ 1, 1 i M, 0 j N wit areas 1 i kj+ 1 and centers x i, ŷ j+ 1 : 5.19 x i = xi + x i + x i+ 1, 1 i M, ŷ j+ 1 = ỹ j+ 1, 1 j N 1, ŷ 1 = 1 3ỹ1, ŷ N+ 1 = ỹ N.

14 14 G. GIE AND R. TEMAM Using Lemma 3.1 on K i,j+ 1, we find tat, for 1 i M, 0 j N, i K i,j+ 1 θdxdy = kj+ 1 θ x i, ŷ j+ 1 + O ρ 3, were O ρ 3 is bounded by c θ C ρ 3 as in Due to te definition of in 4.8, 4.9, using 3.7, we write I in 5.14 in te form: 5.1 I = K i,j i ũ i+1,j+1 + ũ i+1,j ũ i 1,j+1 ũ i 1,j θdxdy = canging indices for i and using 5.19 and 5.0 = 1 ũ i,j+1 + ũ i,j 8 k { j+ 1 θ x i+1, ŷ j+ 1 + O ρ ũ Ṽ = 1 8 θ xi 1 }, ŷ j+ 1 ũ i,j+1 + ũ i,j k j+ 1 D xθ x i, ŷ j+ 1 xi+1 x i 1 + O ρ ũ Ṽ, were, using 5.0 and treating boundary terms for i = 0, M, O ρ is bounded by c θ C ρ for a constant c independent of te mes sizes. For I in 5.1, we cange te indices for j, and use 3.9 and te analog of 4.6 in te y direction. As a result, we find I = 1 ũ i,j D x θ x i, y 5. 4 j x i+1 x i 1 k j + O ρ ũ Ṽ, were 5.3 y j = We also notice tat, for i M 1, 5.4 kj+ 1 ŷj+ 1 + k j 1 ŷj 1 kj+ 1 + k j 1 x i 1 x i+1 = using 5.15 for x i, and.6,.8, 4. and 4.5 = i { i i+ + i i }, and ence, using te assumption 5.9, 5. yields I = 1 ũ i,j D x θ x i, y 5.5 j i kj + O η 1 ũ Ṽ, were O η 1 is bounded by cη 1 for a constant c independent of te mes sizes. Now, since te area of K i,j is equal to i kj, from 5.14, 5.18 and 5.5, we can rewrite I x in te form: 5.6 I x = ũ Dθdxdy x + O η 1 ũ Ṽ,.

15 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 15 wit 5.7 D x θ = 1{ Dx θ x i, ŷ j + D x θ x i, y j } χ. Ki,j Ten, since θ is in C0 R, it is easy to see tat, as ρ 0, D xθ converges to D xθ strongly in L and we now conclude tat 5.8 I x ϕ 0 D x θdxdy as ρ 0; te proof of C for FD is complete. 6. A map between te FD and FV spaces. To prove te C property for finite volumes, we introduce te following map Λ : Ṽ V between te FD and FV spaces: 6.1 Λ ũ i,j χ Ki,j = ũ i,j χ Ki,j. Its inverse Λ 1 6. Λ 1 mapping V into Ṽ is defined by u i,j χ Ki,j = u i,j χ Ki,j. We now state and prove a lemma estimating te L norms of u Λ 1 u and of ũ Λ ũ. Lemma 6.1. For any u V and ũ Ṽ, we ave 6.3 u Λ 1 u L 30α 1 ρ u V, ũ Λ ũ L 30α 1 ρ ũ Ṽ. Proof. We only prove By te points ordering relations.8 and 4. see also Fig. 6, we see tat K i,j can only intersect its neigbors K i,j±1, Ki±1,j, K i±1,j±1 and tat u i,j u i,j±1, on K i,j K i,j±1, 6.4 Tus 6.5 u Λ 1 u = u Λ 1 K i,j u i,j u i±1,j, on K i,j K i±1,j, u i,j u i 1,j±1, on K i,j K i 1,j±1, u i,j u i+1,j±1, on K i,j K i+1,j±1, 0, on K i,j \ Ki,j±1 K i±1,j K i±1,j±1. u { i k j ui,j u i,j±1 + u i,j u i±1,j + u i,j u i 1,j±1 + u i,j u i+1,j±1 }. For te last two terms in te rigt-and side of 6.5, we write 6.6 u i,j u i 1,j±1 u i,j u i 1,j + u i 1,j u i 1,j±1, u i,j u i+1,j±1 u i,j u i+1,j + u i+1,j u i+1,j±1,

16 16 G. GIE AND R. TEMAM and ence 6.7 u Λ 1 K i,j u { 5 i k j ui,j u i,j±1 + u i,j u i±1,j By summing in i and j, we find 6.8 u Λ 1 u dxdy 5 { i k j ui,j u i,j±1 + u i,j u i±1,j + u i 1,j u i 1,j±1 + u i+1,j u i+1,j±1 } by canging indices 15 { } k j u i,j u i,j±1 + 5 wit.13 and k 30α 1 ρ u V ; + u i 1,j u i 1,j±1 + u i+1,j u i+1,j±1 }. u i,j+1 u i,j i + 10k 1 k k j+ 1 { k i u i,j u i±1,j } u i+1,j u i,j i follows. We pursue in our task of proving te property C for finite volumes, and now we want to compare u and Λ 1 u. We recall tat te domains of constancy of te FV derivatives x u, y u are te quadrilaterals K i+ 1,j and K i,j+ 1 ; for te finite differences, tose are te quadrilaterals K i+ 1,j and K i,j+ 1. Considering for instance K i+ 1,j, we notice tat tis quadrilateral may only intersect te quadrilaterals K i+ 1,j+s, s = 0, ±1 and K i+r,j± 1, r = 0, 1; see Fig. 7. To obtain te property C for FV, we impose an additional tecnical assumption on te mes, namely: 1 K k i,j+ 1 \ K i,j+ 1 K i,j+ 1 = η3 0 as ρ 0, 6.9 sup i M 1 j N 1 sup i M 1 j N 1 1 k K i+ 1,j \ K i+ 1,j K i+ 1 =,j η4 0 as ρ 0. We notice tat te areas of K i,j+ 1 and K i,j+ 1 are respectively 1 i k j+ 1 and 1 i k j+ 1, and terefore, tere exits 0 < ĥ i min i, i suc tat 6.10 K i,j+ 1 K i,j+ 1 = 1 ĥ i k j+ 1. Tanks to 6.10, we can rewrite te assumptions 6.9 in te form: 6.11 i ĥ i k j kj sup = η 3 0, sup i M 1 j N 1 k k j = η 4 0 as ρ 0,

17 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 17 ~ y j+3/ y j+3/ y j+1 y j+1/ y~ j+1/ y j ~ y j 1/ y j 1/ y j 1 y j 3/ ~ y j 3/ x x~ i 1/ i 1/ x x~ i+1/ x i+1/ x i+1 x ~ i i+3/ x i+3/ Figure 7. K i+ 1,j tick solid lines and nearby K i,j+ 1, Ki+ 1,j tick dased lines wic may intersect. and, due to te assumptions 5.9 and 6.11, we find tat 6.1 ĥ i+1 ĥ i 1 sup 3 i M kj+1 kj 1 sup 3 j N k ĥ i+1 i+1 = sup + i+1 i 1 3 i M η 1 + η 3, kj+1 k j+1 = sup + k j+1 k j 1 3 j N k k η + η 4. + i 1 ĥ i 1 + k j 1 kj 1 k We now state te following lemma wic is te last ingredient needed to sow te property C for FV. Lemma 6.. Under te assumptions.18,.19, 5.9 and 6.11, we ave tat, for any φ C0 R, and u V, x u x Λ 1 u φdxdy cη 1 + η 3 + η 4 u V, 6.13 y u y Λ 1 u φdxdy cη + η 3 + η 4 u V, for a constant c depending on φ, but independent of te mes sizes. Proof. We only prove te first inequality in We observe from.13 and 4.8 tat x u x Λ 1 u vanises on te sets

18 18 G. GIE AND R. TEMAM K i+ 1,j K i+ 1,j, but it may not on te sets K i,j+ 1 K i,j+ 1. On eac K i,j+ 1 K i,j+ 1, by using.13,.14, 4.5, 4.8 and 4.9, we find 6.14 were x u x Λ 1 u K i,j+ 1 K i,j+ 1 = 1 i ui+ 1,j+ 1 u i 1,j i ũi+,j+ 1 ũ i 1,j+ 1 = J 1 + J, J 1 = i+1k j u i,j+1 + i+1 k j+1 u i,j i i + i+1 k j + k j+1 i 1k j u i,j+1 + i 1 k j+1 u i,j, i i 1 + i k j + k j J = k ju i+1,j+1 + k j+1 u i+1,j i + i+1 k j + k j+1 k ju i 1,j+1 + k j+1 u i 1,j i 1 + i k j + k j+1 u i+1,j+1 + u i+1,j u i 1,j+1 u i 1,j i 1 + i + i+1. We now rewrite te terms J 1 and J in te form: 6.16 J 1 = = i+1 i 1 i 1 + i i + i+1 k j + k j+1 [k ju i,j+1 + k j+1 u i,j ] i+1 i 1 k j k j+1 i 1 + i i + i+1 k j + k j+1 u i,j+1 u i,j + J 1, J 1 = i+1 i 1 i 1 + i i + i+1 u i,j+1 + u i,j, and 6.17 [ 1 J = i + i+1 1 i 1 + i + i+1 [ 1 + i 1 + i + 1 i 1 + i + i+1 + k [ j k j+1 ui+1,j+1 u i+1,j k j + k j+1 i + i+1 [ k j k j+1 ui+1,j+1 u i+1,j = k j + k j+1 i + i+1 ] u i+1,j+1 + u i+1,j ] u i 1,j+1 + u i 1,j u i 1,j+1 u i 1,j i 1 + i u i 1,j+1 u i 1,j i 1 + i ] ] + J J i 1 i+1 = i + i+1 i 1 + i + i+1 u i+1,j+1 + u i+1,j i 1 i+1 + i 1 + i i 1 + i + i+1 u i 1,j+1 + u i 1,j. We can combine as follows te terms J 1 and J in 6.16 and 6.17: 6.18 i 1 i+1 i + i+1 i 1 + i + i+1 [u i+1,j+1 u i,j+1 + u i+1,j u i,j ] i 1 i+1 + i 1 + i i 1 + i + i+1 [u i 1,j+1 u i,j+1 + u i 1,j u i,j ].

19 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 19 Ten, using 6.16, 6.17 and 6.18, we can rewrite 6.14 in te form: 6.19 J 1 + J = K 1 + K + K 3, K 1 = K = i+1 i 1 k j k j+1 i 1 + i i + i+1 k j + k j+1 u i,j+1 u i,j, i 1 i+1 i + i+1 i 1 + i + i+1 [u i+1,j+1 u i,j+1 + u i+1,j u i,j ] i 1 i+1 + i 1 + i i 1 + i + i+1 [u i 1,j+1 u i,j+1 + u i 1,j u i,j ], K 3 = k { j k j+1 ui+1,j+1 u i+1,j u } i 1,j+1 u i 1,j. k j + k j+1 i + i+1 i 1 + i Using.16 and te assumption 5.9, we ten treat te term K 1 : 6.0 c c K i,j+ 1 K i,j+ 1 K 1 φdxdy i+1 i 1 u i,j+1 u i,j i+1 i 1 u i,j+1 u i,j c η 1 + O ρ u V, K i,j+ 1 K i,j+ 1 φdxdy were c is a positive constant depending on φ, but independent of te mes sizes. Using te estimate similar to 6.0, we also obtain 6.1 K i,j+ 1 K i,j+ 1 K φdxdy c η 1 + O ρ u V. For te term K 3, we use 3.5 on K i,j+ 1 K i,j+ 1 wit area 1 ĥ i k j+ 1 and barycenter x i,j+ 1, y i,j+ 1 : 6. K i,j+ 1 K i,j+ 1 φdxdy = 1 ĥ i k j+ 1 φ i,j+ 1 + O ρ 4, φ i,j+ 1 = φ x i,j+ 1, y i,j+ 1, were O ρ 4 is bounded by c φ C ρ 4. Ten, by.6 and 6., we write K 3 φdxdy = K 3 ĥ i k j+ 1 φ i,j+ 1 + O ρ u V, K i,j+ 1 K i,j+ 1 and, by canging te indices in 6.3, we find

20 0 G. GIE AND R. TEMAM 6.4 = K i,j+ 1 K i,j+ 1 +O ρ u V. We also find tat K 3 φdxdy k j k j+1 u i,j+1 u i,j { ĥ i 1 φ 8 i 1 + i 1,j+ 1 ĥ i+1 } φ i i + i+1,j+ 1 i+1 ĥ i 1 φ i 1 + i 1,j+ 1 ĥ i+1 φ i i + i+1,j+ 1 i and = 1 { + 1 { ĥ i 1 i 1 + i + ĥ i 1 i 1 + i ĥ i+1 i + i+1 } φi 1,j+ 1 φ i+1,j+ 1 ĥ i+1 i + i+1 } φi 1,j+ 1 + φ i+1,j+ 1, ĥ i 1 i 1 + i ĥ i+1 i + i = i ĥi 1 ĥ i+1 + ĥi 1 i+1 i 1 + ĥ i 1 ĥ i+1 i 1 i 1 + i i + i+1 = ĥ i 1 ĥ i+1 ĥ i 1 i+1 i 1 + i + i+1 i 1 + i i + i+1. Terefore, using 6.5 and 6.6, we can bound 6.4 by L 1 + L + L 3 were 6.7 { kj k j+1 L 1 = u i,j+1 u i,j [ ĥ i 1 ĥ i+1 ] + 16 i 1 + i i + i+1 L = L 3 = φi 1,j+ 1 φ i+1,j+ 1 }, k j k j+1 ĥ i 1 ĥ i+1 u i,j+1 u i,j φi 1,j i + + φ i+1 i+1,j+ 1, { kj k j+1 16 ĥ i 1 i+1 i 1 u i,j+1 u i,j i 1 + i i + i+1 φi 1,j+ 1 + φ i+1,j+ 1 }. We control te L i, 1 i 3, terms of 6.7: for a positive constant c independent of te mes sizes, 6.8 L 1 c u i,j+1 u i,j φ i 1,j+ 1 ρ c Dφ L u V ρ. φ i+1,j+ 1

21 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 1 Under te assumption 6.1, we use te analog of 6.0 and find 6.9 L c u i,j+1 u i,j ĥ i+1 ĥ i 1 c η1 + η 3 + O ρ u V. Under te assumption 5.9, te term L 3 can be easily treated as 6.0 and we finally obtain, from and , tat 6.30 K i,j+ 1 K i,j+ 1 x u x Λ 1 u φdxdy cη 1 + η 3 u V. Now, to treat x u x Λ 1 u on K i,j+ 1 \ K i,j+ 1 K i,j+ 1 Ki+ 1,j K i+ 1, we recall tat te quadrilateral,j Ki+ 1 or Ki+ 1,j \,j may only intersect te quadrilaterals K i+ 1,j+s, s = 0, ±1 and K i+r,j± 1, r = 0, 1, and similarly te K i,j+ 1 may only intersect te quadrilaterals K i+s,j+ 1, s = 0, ±1 and K i± 1,j+r, r = 0, 1; see Fig. 7. We ten, observe tat 6.31 x u K i+ 1 or,j x Λ 1 x Λ 1 u Ki+ 1,j 1 u i+1,j u i,j, u 1 Ki,j+ 1 4 { u i+1,j+1 u i 1,j+1 + u i+1,j u i 1,j }, x u 1 K i,j+ 1 8 { u i+1,j+1 u i+1,j + u i+1,j+1 u i,j+1 + u i+1,j u i,j If we set + u i,j+1 u i,j + u i,j+1 u i 1,j+1 + u i,j u i 1,j + u i 1,j+1 u i 1,j }. 6.3 K i,j = { K i,j+ 1 \ K i,j+ 1 K } { i,j+ 1 K i+ 1,j \ K i+ 1,j K } i+ 1,j, were te indices are 1 i M, 0 j N for K i,j+ 1, and 0 i M, 1 j N for K i+ 1,j, ten, tanks to 6.9 and 6.31, we find tat, for a constant c independent of te mes sizes, 6.33 x u x Λ 1 u φdxdy i,j Ki,j { x u + } x Λ 1 u φdxdy i,j K { x u + } x Λ 1 u k η 3 + η 4 i,j by canging indices in 6.31 and treating te boundary terms c η 3 + η 4 i,j { u i+1,j u i,j + u i,j+1 u i,j } c η 3 + η 4 u V.

22 G. GIE AND R. TEMAM Consequently, from 6.30 and 6.33, we obtain x u x Λ 1 u φdxdy = x u x Λ K i,j+ 1 K i+ 1,j x u x Λ 1 u u φdxdy φdxdy cη 1 + η 3 + η 4 u V, and tis is as desired. Remark 6.1. It is noteworty tat, since η 3 in 6.11 is not equal to zero for te case were i = and i+1 =, te assumption 6.11 is somewat more restrictive tan te assumption 6.1; in 6.1, ĥ i+1 ĥ i 1, 3 i M, are equal to zero wen i = and i+1 =. Remark 6.. For te case were i =, i+1 =, it is enoug to impose te condition 6.1 and, for suc a special case, we can prove Lemma 6. by using te discrete integration by parts. But ere we assume 6.11 to andle more complicated meses, for wic, te geometric complexity of te mes prevents us from using te discrete integration by parts. Now, tanks to Lemma 6., we deduce te convergence result of te FV in te following teorem: Teorem 6.1. Under te assumptions.18,.19, 5.9 and 6.11, te C property for te external approximation of H 1 0 by te FV spaces V olds true. Hence, wit 3.3 and 3.4 1, we conclude tat te FV approximation is stable and convergent. Proof. Consider {u } V suc tat p u ϕ = ϕ 0, ϕ 1, ϕ weakly in F : u ϕ 0 weakly in L, 6.35 x u ϕ 1 weakly in L, y u ϕ weakly in L. Ten, to prove te C property for te FV metod, we ave to verify tat ϕ 0 H0 1 and ϕ 1, ϕ = Dϕ 0. But, since te property C for te FD metod olds, it is sufficient to sow tat 6.36 Λ 1 u ϕ 0 weakly in L, x Λ 1 u ϕ 1 weakly in L, y Λ 1 u ϕ weakly in L. Property is true by Lemma 6.1 and te boundedness of u V ; 6.36 and are valid because of Lemma 6. and te boundedness of u V again. Hence we obtain te property C for te FV metod.

23 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 3 7. An application. Now, as an application of te convergence result for te FV metod, we briefly sow ow one can implement te FV metod to approximate te solutions of some typical elliptic equations wit Diriclet boundary condition, and prove te convergence results. We consider a general two dimensional Diriclet problem in te form: 7.1 { α a αβ x, y β u + α bα x, yu + gx, yu = fx, y in = 0, 1 R, u = 0 on, were te Einstein summation convention is understood for te Greek indices α, β = 1,. For simplicity, we assume tat, for eac α, β = 1,, 7. a α,β, g, f C 0, b α C, and, for te coercivity of te problem 7.1, we also impose te following properties on a αβ x, y, b α x, y and gx, y: a αβ x, yξ α ξ β κ 1 ξ, ξ = ξ 1, ξ R, αb α x, y + gx, y κ > 0, for suitable strictly positive constants κ 1 and κ. Te variational form of 7.1 is classical: 7.4 To find u V = H 1 0 suc tat au, v =< l, v >, v V, were 7.5 au, v = < l, v >= aαβ β u α v b α u α v + guv dxdy, fvdxdy. Note tat, tanks to te Lax-Milgram teorem, we obtain te existence and uniqueness of te solution u of 7.1 in V = H 1 0. To construct te FV approximation of 7.5, we use te spaces V introduced in Section as well as all te notations of te previous sections as needed. We first consider te convection term in 7.1 wic is te most problematic: we start by integrating tis term over a rectangle K i,j for fixed i, j, and use te Stokes formula: α bα u dxdy = b1 u, b u n i,j ds 7.6 K i,j K i,j = F i+ 1,j F i 1,j + F i,j+ 1 F i,j 1 j, were n i,j is te unit outer normal on K i,j and were F i+ 1,j and F i,j+ 1 are te fluxes along te parts of boundary {x i+ 1 } y j 1, y j+ 1 and x i 1, x i+ 1 {y j+ 1 } respectively. Since te unit outer normal n i,j of K i,j is ±1, 0 or 0, ±1, we can approximate tose fluxes in te following way: for 0 i M, 1 j N, 7.7 F i+ 1,j = yj+ 1 y j 1 b 1 x i+ 1, yux i+ 1, ydy = b 1 x i+ 1, y ju i+ 1,jk j,

24 4 G. GIE AND R. TEMAM and, for 1 i M, 0 j N, xi F i,j+ 1 = b x, y j+ 1 ux, y j+ 1 dx = b x i, y j+ 1 u i,j+ 1 i, x i 1 were 7.9 u i+ 1,j = iu i,j + i+1 u i+1,j, u = k ju i,j + k j+1 u i,j+1 ; i + i,j+ 1 i+1 k j + k j+1 moreover, for any u V, we write 7.10 u = u i+ 1,jχ K i+ 1,j + u i,j+ 1 χ Ki,j+ 1. Tanks to , for u, v V, multiplying 7.6 by v i,j and summing over 1 i M, 1 j N, we find tat 7.11 a u, v = a 1 u, v + a u, v, were, remembering tat v 0,j = v M+1,j = v i,0 = v i,n+1 = 0, a 1 u, v = { b1 x i+ 1, y ju i+ 1,j b 1x i 1, y ju } i 1,j vi,j k j 7.1 = a u, v = = b 1 x i+ 1, y ju i+ 1,j { vi+1,j v i,j } kj, { } b x i, y j+ 1 u i,j+ b 1 x i, y j 1 u i,j 1 vi,j i { } b x i, y j+ 1 u i,j+ 1 vi,j+1 v i,j i. Now, using 7.9, 7.11 and 7.1, we introduce te FV discrete variational problem of 7.1 in te form: 7.13 To find u V suc tat a u, v =< l, v >, v V, were 7.14 a u, v = a αβ α u, β v + a u, v + gu, v, < l, v >= f, v ; ere, is te usual L inner product over, and 1 = x and = y. Remark 7.1. For te sake of simplicity, in te FV discrete variational form 7.13 and 7.14, we only use te fluxes from te convection term of te original equation 7.1. One can also use te fluxes from te diffusion term in 7.1 and construct te corresponding FV discrete variational form; see e.g. [8] or [10]. However, in suc a case, more difficulties may occur in te analysis, e.g. in te computation for te uniform coercivity of te bilinear forms a. Before we start te analysis of te problem 7.13, we first state and prove a simple, but useful lemma:

25 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 5 Lemma 7.1. For any u in V, 7.15 u u L α 1 ρ u V. Proof. We use te points ordering in.8 and notice tat eac K i,j may only intersect K i,j± 1 and K i± 1,j ; see also Fig. 3. Ten, using.10, , 7.9 and 7.10, we observe tat, on eac K i,j, 7.16 u u Ki,j 1 α 1{ u i+1,j u i,j + u i,j u i 1,j + u i,j+1 u i,j + u i,j u i,j 1 }, and write 7.17 u u L = ence 7.15 follows. We now set α a = max α,β=1, u u K i,j i k j α ρ u V ; u i+1,j u i,j k j + α 1 sup a αβ, b = max α=1, sup b α u i,j+1 u i,j i k, g = sup and we promptly see tat te families {a } and {l } are uniformly continuous wit respect to te mes sizes: using 7.14 and 7.15, we find tat, for any u, v in V, 7.19 a u, v a α u β v L + b u L v L + g u v L a u V v V + b u L u V + g u V u V a + bc 0 + gc 0 + α 1 ρ u V v V, a + bc 0 + gc 0 + α 1 u V v V, were c 0 = α 1 is te discrete Poincaré constant in.; ence te family {a } is uniformly continuous. We also notice tat 7.0 < l, v >= f, v L f L v V ; due to te independence of f L on te mes sizes, and we ave te uniform continuity of te family {l }. Now, to obtain te uniform coercivity of a on V, we first establis a lemma for te term a : Lemma 7.. For any u in V, 7.1 a u, u 1 αb α u dxdy κ 3 ρ u V, for a positive constant κ 3 depending on b 1 and b, but independent of te mes sizes. g,

26 6 G. GIE AND R. TEMAM Proof. From 7.9, 7.11 and 7.1, using te Taylor expansion of b 1 x, at x = x i, we first write te bilinear form a 1 in te form: for u V, a 1 u, u = { b 1 x i+ 1, y ju i+ 1,j b 1x i 1, y ju i }u 1,j i,j k j 7. were 7.3 M 1 = M = M 3 = 1 M 4 = = M 1 + M + M 3 + M 4, { i+1 u i+1,j u i,j i 1u i 1,j u i,j } b 1 x i, y j u i,j k j, i + i+1 i 1 + i 1 b 1 x i, y j u i,j i k j, { i+1 u i+1,j u i,j + i 1u i 1,j u i,j } 1 b 1 x i, y j u i,j i k j. i + i+1 i 1 + i { [b1 x i+ 1, y j b 1 x i, y j 1 1b 1 x i, y j i ] u i+ 1,j [ b 1 x i 1, y j b 1 x i, y j + 1 1b 1 x i, y j i ] u i 1,j }u i,j k j. Since u 0,j = u M+1,j = 0, after canging te indices i in M 1, we find M 1 = { u i+1,j u i,j u i,j i+1 b 1 x i, y j 7.4 wit 7.5 M 1 = 1 M 1 = 1 = M 1 + M 1, We can bound te term M 1: M 1 α 1 b } u i,j u i+1,j u i+1,j ib 1 x i+1, y j i + i+1 1 k j { u i+1,j u i,j i+1b 1 x i, y j i b 1 x i+1, y j } k j, i + i+1 { u i+1,j u i,j i+1b 1 x i, y j + i b 1 x i+1, y j } k j. i + i+1 u i+1,j u i,j kj α 1 bρ u V, and, tanks to Lemma 3.3, we write te term M 1 in te form: M 1 = 1 u i+1,j u i,jb 1 x i+ 1, y jk j = 1 u i,j{ b1 x i+ 1, y j b 1 x i 1, y j } k j.

27 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 7 Ten, using te Taylor expansion of b 1 at x i again, we find 7.8 M u i,j 1 b 1 x i, y j i k j cρ u V. We also observe tat 7.9 M 3 4α 1 sup 1 b { } u i+1,j u i,j + u i 1,j u i,j u i,j i k j α sup 1 b u V u V ρ cρ u V, and, using te Taylor expansion and Lemma 7.1, we find 7.30 M 4 c u i+ 1,j u i 1,j u i,j k j ρ cρ u L u V cρ u V, for a positive constant c depending on b 1, but independent of te mes sizes. Terefore, combining and using te Poincaré inequality, we find 7.31 a 1 1 u, u 1b 1 u dxdy cρ u V. Similarly, one can easily verify tat 7.3 a u, u ence 7.1 follows by 7.31 and b u dxdy cρ u V ; Tanks to 7.3, 7.14 and 7.1, we finally obtain 7.33 a u, u κ 1 u V + κ u V κ 3 ρ u V, and, for sufficiently small ρ < κ 1 /κ 3, te uniform coercivity of te bilinear continuous forms a on V follows. Due to 7.19, 7.0 and 7.33, te Lax-Milgram teorem asserts tat, for ρ < κ 1 /κ 3, te equation as a unique solution u in V ; we say tat u is te FV approximate solution of To prove tat te FV approximate solution u converges to te exact solution u as te mes size decreases, we now introduce te following consistency lemma; ten, te convergence result will follow by te general convergence teorem in [3] see also [15]: Lemma 7.3. If te family v converges to v strongly in F as ρ 0, and if te family w converges to w weakly in F as ρ 0, ten 7.34 lim a v, w = av, w, ρ 0 lim ρ 0 a w, v = aw, v, lim < l, w >=< l, w >. ρ 0 Proof. From te ypoteses of Lemma 7.3, we ave 7.35 v, x v, y v v, x v, y v strongly in F = L 3,

28 8 G. GIE AND R. TEMAM and 7.36 w, x w, y w w, x w, y w weakly in F = L 3, and, by te property C of FV, we notice tat v, w are in V = H0 1. To verify , using 7.9, 7.10 and 7.1, we first write a 1 v, w in te form: 7.37 a 1 v, w = b 1v, x w Ki+ χ Ki+ 1 1,,j,j were 7.38 b 1 = b 1 x i+ 1, y jχ Ki+ 1,j + Using te Taylor expansion of b 1, one can easily verify tat 7.39 b 1 b 1 strongly in L as ρ 0, and, using 7.15 and 7.35, we also find 7.40 v converge to v strongly in L as ρ 0. b 1 x i, y j+ 1 χ K i,j+ 1. Moreover, from 5.18, 5.8, 6.34 and 7.36, we infer tat 7.41 x w K χ 1 i+ 1 Ki+ 1,j,j 1w weakly in L as ρ 0, and ence, using , 7.37 yields 7.4 a 1 v, w b 1 v, 1 w as ρ 0. Wit te same result in te y variable, we also find 7.43 a v, w b v, w as ρ 0, and ten, tanks to 7.11, 7.14, 7.35, 7.36, 7.4 and 7.43, we finally obtain For 7.34, from 7.9, 7.10 and 7.1, we write 7.44 a 1 w, v = w i+ 1,jχ K, i+ 1 b x,j v and, using 7.36, one can easily verify tat 7.45 w i+ 1,jχ K 1 i+ 1,j w weakly in L as ρ 0. Hence, from 7.35, 7.39 and 7.45, we find 7.46 a 1 w, v w, b 1 1 v as ρ 0, and similarly, 7.47 a w, v w, b v as ρ 0; ten, by 7.11, 7.14, 7.35, 7.36, 7.46 and 7.47, 7.34 follows. Finally, using 7.36, we promptly notice 7.48 < l, w >= f, w f, w =< l, w > as ρ 0; and te proof of Lemma 7.3 is complete.,

29 CELL CENTERED FINITE VOLUMES USING TSES WITHOUT FICTITIOUS DOMAINS 9 Now, wit te general convergence teorem in [3] and [15], we obtain te convergence of te FV approximate solution u to te exact solution u: Teorem 7.1. Under te ypoteses 7.19, 7.0, 7.33 and 7.34, te FV approximate solution u of converges strongly to te solution u of in F as ρ 0. References [1] K. Adamy and D. Pam, A finite volume implicit Euler sceme for te linearized sallow water equations: stability and convergence, Numer. Funct. Anal. Optim , no. 7-8, [] D. N. Arnold, Stability, consistency and convergence, lecture at te joint AMS-MAA meeting, Wasington, January 009. [3] J. Céa, Approximation variationnelle des problemes aux limites, Ann. Inst. Fourier Grenoble fasc., [4] T. J. Cung, Computational fluid dynamics, Cambridge University Press, Cambridge, 00. [5] J. Droniou and R. Eymard, A mixed finite volume sceme for anisotropic diffusion problems on any grid, Numer. Mat , no. 1, [6] R. Eymard, T. Gallouët and R. Herbin, Finite volume metods, Handbook of Numerical Analysis vol. VII, Nort-Holland 000, pp [7] R. Eymard, T. Gallouët and R. Herbin, A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meses in any space dimension, IMA J. Numer. Anal , no., [8] S. Faure, J. Laminie and R. Temam, Colocated finite volume scemes for fluid flows, Commun. Comput. Pys , no. 1, 1-5. [9] S. Faure, D. Pam and R. Temam, Comparison of finite voulume and finite difference metods and applications, Analysis and applications. V , no., pp [10] W. Huang and A. M. Kappen, A study of cell center finite volume metods for diffusion equations, 1998 electronic. [11] B. Macenauer, E. Kaas, and P. H. Lauritzen, Finite-Volume Metods in Meteorology, in [16]. [1] K. W. Morton and E. Süli, Finite volume metods and teir analysis, IMA J. Numer. Anal , no., [13] M. Svärd, J. Gong, J. Nordström, Stable artificial dissipation operators for finite volume scemes on unstructured grids, Appl. Numer. Mat , no. 1, [14] E. Süli, Convergence of finite volume scemes for Poisson s equation on nonuniform meses, SIAM J. Numer. Anal , no. 5, [15] R. Temam, Navier-Stokes equations. Teory and numerical analysis, Reprint of te 1984 edition. AMS Celsea Publising, Providence, RI, 001. [16] R. Temam and J. Tribbia, Computational metods for te atmospere and te oceans: special volume, Handbook of Numerical Analysis, V. 14, Editor: P. G. Ciarlet, Elsevier Science, 008. [17] R. Vilsmeier, F. Benkaldoun and D. Hänel, Editors, Finite volumes for complex applications II: Problems and perspectives, Hermes Science Publications, Paris, [18] P. Wesseling, Principles of computational fluid dynamics, Englis summary Springer Series in Computational Matematics, 9. Springer-Verlag, Berlin, 001. Te Institute for Scientific Computing and Applied Matematics, Indiana University, 831 East Tird Street, Bloomington, Indiana 47405, U.S.A. gugie@indiana.edu and temam@indiana.edu URL: ttp://mypage.iu.edu/ gugie/ and ttp://mypage.iu.edu/ temam/