Development and analysis of higher order finite volume methods over rectangles for elliptic equations

Transcription

1 Advances in Computational Matematics 19: Kluwer Academic Publisers. Printed in te Neterlands. Development and analysis of iger order finite volume metods over rectangles for elliptic equations Ziqiang Cai Jim Douglas Jr. and Moongyu Park Department of Matematics Purdue University 1395 Matematical Science Building West Lafayette IN USA Received 9 October 001; accepted 11 September 00 Communicated by C.A. Miccelli Currently used finite volume metods are essentially low order metods. In tis paper we present a systematic way to derive iger order finite volume scemes from iger order mixed finite element metods. Mostly for convenience but sometimes from necessity our procedure starts from te ybridization of te mixed metod. It ten approximates te inner product of vector functions by an appropriate critical quadrature rule; tis allows te elimination of te flux and Lagrange multiplier parameters so as to obtain equations in te scalar variable wic will define te finite volume metod. Following tis derivation wit different mixed finite element spaces leads to a variety of finite volume scemes. In particular we restrict ourselves to finite volume metods posed over rectangular partitions and begin by studying an efficient second-order finite volume metod based on te Brezzi Douglas Fortin Marini space of index two. en we present a general global analysis of te difference between te solution of te underlying mixed finite element metod and its related finite volume metod. en we derive finite volume metods of all orders from te Raviart omas two-dimensional rectangular elements; we also find finite volume metods to associate wit BDFM tree-dimensional rectangles. In eac case we obtain optimal error estimates for bot te scalar variable and te recovered flux. Keywords: finite volume metod iger order rectangular element AMS subject classification: 65N30 1. Introduction Numerical metods for partial differential equations are divided into tree general categories: finite difference metods FDMs finite element metods FEMs and finite volume metods FVMs. Finite difference and finite element metods ave received muc more attention tan finite volume metods and as a result ave been developed to a iger degree of sopistication tan finite volume tecniques; tus tere are wellknown iger order finite difference and finite element metods. In general finite dif- e researc of Professor Cai was sponsored in part by te National Science Foundation.

2 4 Z. Cai et al. / Higer order finite volume metods for elliptic equations ference metods are easier to implement tan finite element metods but finite element metods are more easily adapted to general geometries of te underlying domain on wic te differential problem is formulated and to te practical treatment of inomogeneous pysical properties of te media. Finite volume procedures are usually easier to implement tan finite element procedures and offer most of te advantages of flexibility in geometry of tese metods; in a sense finite volume metods lie in between te oter two tecniques in concept and implementation. ere as been one fundamental disadvantage for finite volume metods in comparison to finite difference and finite element metods. Essentially all current finite volume metods are low order metods based on approximating te solution of te differential equations by piecewise-constant functions. Consequently te finite volume approximation can converge to te solution of te differential problem globally only at a rate proportional to te diameter of te elements in te partition of te domain on wic te problem is set. In some cases it as been possible to sow tat certain points can be identified at wic te approximate solution converges at a more rapid rate tan te global rate; tis penomenon is called superconvergence. In te most commonly used finite volume metod based on rectangular elements for elliptic boundary value problems convergence at a rate O takes place at cell centers under some supplementary conditions. en second order convergence can be establised for a piecewise multilinear interpolation of tese cell-center values again under proper ypoteses. In [1] Russell and Weeler establised a relationsip between a mixed finite element metod on rectangular meses and cell-centered finite differences for diffusion problems wit coefficient matrix being diagonal. Specifically tey obtained a positive-definite cell-centered finite difference metod by using te lowest order Raviart omas R 0 space see [0] applying an appropriate quadrature rule and eliminating te flux. is work as been extended to triangular meses and to a full tensor coefficient matrix see e.g. [1]. For similar work see also te recent paper by Baranger et al. [6] were cell-centered finite difference procedures were considered to be finite volume metods. On general triangular meses te R 0 space usually leads to quite complicated finite difference stencils see [1] since piecewise-constant pressures on two adjacent triangles are not enoug to represent te flux across teir common edge. ese papers are based on te lowest order Raviart omas space and consequently lead to essentially low order metods. e object of tis paper is to present examples of a systematic way for deriving iger order finite volume metods. Our derivations are similar to tose mentioned above toug we find it convenient to start wit te ybridization of te underlying mixed finite element metod. en we approximate te weigted inner product of vector functions by an appropriate quadrature rule so as to diagonalize te resulting matrix A ; tis allows te immediate elimination of te flux variables. Finally we employ te consistency relations across interfaces to permit te elimination of te Lagrange multipliers introduced in te ybridization process to obtain a system of equations in te degrees of freedom for te scalar variable in te mixed formulation of te elliptic problem; tis will be our finite volume procedure. It will be clear tat te resulting

3 Z. Cai et al. / Higer order finite volume metods for elliptic equations 5 metod maintains local conservation of watever is being transported by te flux variable element-by-element as does te underlying mixed finite element metod. e coice of te quadrature rule is critical in our approac as it was in [1]. Our guidelines are tat te matrix A corresponding to te vector inner product is diagonal and tat te numerical integration does not decrease te rate of convergence of te original mixed finite element metod. Finite volume scemes can be derived in numerous ways see e.g. [ ] but te point of our derivation is tat te same metod can be applied to different mixed finite element spaces to obtain a variety of finite volume procedures. In tis paper we briefly review te standard lower order finite volume metod based on te rectangular R 0 mixed finite element space and ten turn first to te derivation of a iger order procedure based on te two-dimensional Brezzi Douglas Fortin Marini space of index two BDFM see [8] of rectangular elements. It is wort noting tat te derivation tecnique used ere and in [1] naturally leads to te employment of te armonic average of discontinuous diffusion coefficients wic is pysically correct and often is not introduced by oter derivations of finite volume scemes. e difference between te solution of te BDFM mixed metod and its related finite volume metod is discussed as a solution of a perturbed mixed metod. e analysis is given in a generalized fasion tat allows us to treat a variety of mixed metods and teir related finite volume metods by analyzing te effect of te quadrature rule used in deriving te finite volume procedure. As a consequence of tis analysis we indicate finite volume scemes related to te Raviart omas Nedelec rectangular mixed finite elements in two or tree dimensions and sow tat te resulting finite volume metods retain te global order of convergence of te RN metods. We also treat te treedimensional BDFM case were optimal order error estimates are obtained and some iger order BDFM k elements were it seems inevitable tat te error in te flux variable at least cannot obtain te optimal order of accuracy associated wit te underlying mixed metod. We defer discussing simplicial metods to future work. is paper is organized as follows. We end tis section by introducing some notation. In section we introduce te diffusion equation an equivalent first-order system and its mixed variational formulation. In section 3 te mixed finite element approximations based on te R 0 and BDFM are described along wit teir ybridizations; sections 4 5 and 6 are devoted to te derivation of finite volume metods from te two mixed metods. Global error estimates for finite volume metods interpretable as pertubations of mixed finite elements scemes are considered in section 7 along wit te application of tese estimates to te BDFM -based metod. wo- and tree-dimensional RN-based finite volume metods are indicated and analyzed in section 8. e treedimensional BDFM -based metod is derived and analyzed in section 9. e BDFM 3 case is discussed in section 10; it is an example tat so far as not been sown to lead to a satisfactory finite volume sceme derivable by te tecnique of tis paper.

4 6 Z. Cai et al. / Higer order finite volume metods for elliptic equations 1.1. Notation We use te standard notation and definition for te Sobolev space W mp B tat consists of functions wose partial derivatives are L p -integrable up to order m; its standard associated norm and seminorm are denoted by mpb and mpb respectively. For p = H m B = W m B is a Hilbert space and its inner product is denoted by mb. We omit te subscript B from te notation wen B =. Form = 0 H m B coincides wit L B. In tis case te inner product and norm will be denoted by B and 0B or and 0 wen B = respectively. Let V = Hdiv; = { v L : v L } ; V is a Hilbert space under te norm Set W = L and V = V W. v V = v 0 + v 0.. e elliptic problem its mixed formulation and preliminaries Consider te omogeneous Diriclet probem: { K p = f in.1 p = 0 on were te symbols and stand for te divergence and gradient operators respectively; K L and f L are given real-valued functions and te domain is te unit square Assume tat te diffusion coefficient K is bounded below and above by positive constants; i.e. tere exist positive constants K 0 and K 1 suc tat for almost all x y. We introduce te flux variable 0 <K 0 Kxy K 1 u = K p in.. Set cxy = K 1 x y; ten0<k 1 1 = c 0 cxy c 1 = K 1 0. en te mixed form of.1 is given by cu + p = 0 in u = f in.3 p = 0 on. e mixed weak formulation of problem.1 is obtained by multiplying te first and second equations of.3 by v V and q W respectively and integrating te two

5 Z. Cai et al. / Higer order finite volume metods for elliptic equations 7 equations over ; tus te weak solution of.3 is found by seeking up V V W suc tat { cu v vp= 0 v V.4 uq=f q q W. e analysis below will be carried out assuming te Diriclet boundary condition imposed above; owever Neumann or Robin boundary conditions can be treated analogously. 3. e mixed finite element metod and its ybridization Let be te partition of into squares wit te side lengt = N 1 : = N ij ij=1 were ij =[i 1 i] [j 1 j] =[x i 1 x i ] [y j 1 y j ]. e extension of te metods in tis paper to te connected union of rectangular elements is straigtforward. Let V V W be an admissible finite-dimensional subspace of V see e.g. [10]. en te mixed finite element approximation in V is te solution u p V = V W of { cu v vp = 0 v V 3.1 u q=f q q W. We find it convenient to localize te continuous mixed problem.3. By doing so we can point out exactly te difference between te finite volume metod we associate wit te underlying mixed finite element space and te reduction to a symmetric positive-definite algebraic system for te Lagrange multipliers introduced by Fraeijs de Veubeke [17] in is ybridization of.3 and analyzed in detail by Arnold and Brezzi [3]. For eac element let be an adjacent element wit a common edge e =.LetE be te set of all internal edges and denote restrictions of u and p to te element by u and p respectively. en solving te system cu x + p x = 0 x u x = f x 3. px = 0 x is equivalent to solving.3 if te consistency conditions { p e p e = 0 e E 3.3 u n e + u n e = 0 e E old were n and n are te unit outward vectors normal respectively to and.

6 8 Z. Cai et al. / Higer order finite volume metods for elliptic equations Let Set V = Hdiv; W= L V ={v : v V } W ={q : q W }. r t = rt ds. Now test 3. against v V and q W: { cu v p v + p v n = u q = f q. Note tat toug p was required to belong just to L it is actually sufficiently smoot tat its trace on is clearly defined; owever for all commonly used mixed finite element spaces p is discontinuous across internal edges in E. But 3.4 can be used as motivation for localizing 3.1 as follows. Let us replace te value of p on e in te discrete analogue of 3.4 by a single-valued Lagrange multiplier λ e on e wic can be defined as follows. Let and let Now let M e = { v n e : v V } e ={λ e M e : e E } = { λ: e E λ e }. V = Ṽ W were Ṽ ={v: v V }. Note tat continuity of te normal component of te flux across te interfaces is not imposed on functions in Ṽ ; but for any v Ṽ te flux consistency condition can be enforced by requiring tat v n µ \ = 0 µ. 3.5 us 3.1 can be localized or ybridized by seeking a triple u p λ V suc tat cu v vp + v n λ \ = 0 v V u q = f q q W 3.6 u n µ \ = 0 µ. In tis paper we begin by briefly considering te lowest order Raviart omas element R 0 see [0] and ten concentrating on te Brezzi Douglas Fortin and

7 Z. Cai et al. / Higer order finite volume metods for elliptic equations 9 Marini element see [8] of index two BDFM. Wit appropriate quadrature rules applied to te inner product cu v te former leads to te standard low order FVM and te latter to a second-order FVM tat is te first object of tis paper. Later we consider several oter mixed finite element spaces. Let P k be te set of all polynomials on of total degree not greater tan k; letp k z for z = x or y be te set of all polynomials of one variable z of te degree not greater tan k. Also let P kl = P k x P l y. For te R 0 mixed finite element space { V = v = v1 v Hdiv; : v 1 P 10 v P 01 } W = { q L : q P 0 } wile for te BDFM space V = { v Hdiv; : v [ P \ { y }] [ P \ { x }] } W = { q L : q P 1 }. It is well known tat bot te R 0 and te BDFM spaces satisfy te inf sup condition see e.g. [10] and are admissible mixed finite element spaces. Since { P0 e for R 0 v n e P 1 e for BDFM te natural coices of te Lagrange multipliers are given by = { { } P0 e for R 0 λ: e E λ e. P 1 e for BDFM 4. Finite volume metods We sall derive finite volume scemes based on 3.6 by coosing appropriate quadrature procedures first globally in matricial form and ten in terms of te local equations for te degrees of freedom of te scalar variable p. Denote basis functions by double indices for elements or e for edges and j for te degrees of freedom on or e. Let {{ξ j x y}n 1 j=1 } {{η j x y}n j=1 } and {{ζe j x y} N 3 j=1 } e E be bases for Ṽ W and respectively; N 1 = { 4 R0 10 BDFM N = { 1 R0 3 BDFM and N 3 = { 1 R0 BDFM.

8 10 Z. Cai et al. / Higer order finite volume metods for elliptic equations and let Let A = aij = cξ j N 1 N 1 ξ i B = bij = ξ i N 1 N ηj C e = cij e = ξ i N 1 N 3 nζj e N 1 N 1 \ N 1 N A = diaga : N N B = diagb : N N C = C e : e E N NN 1. N 1 N e matrices A and B are block diagonal wit te blocks being te 4 4forteR 0 or for te BDFM and 4 1forteR 0 or 10 3forteBDFM respectively. Let were U = U N 1 P = P N 1 and λ = λ e NN 1 1 U = u j N 1 1 P = p j and λ N 1 e = λ j e N 3 1. Let F = F N 1 be te rigt-and side vector wit F = f η i N 1. en 3.6 can be written in matricial form as AU + BP + Cλ = 0 B U = F 4. C U = 0. Inverting A in te first equation of 4. gives U = A 1 BP A 1 Cλ. 4.3 is is a cell-by-cell calculation and tus inexpensive. Substituting 4.3 into te second and tird equations in 4. yields { B A 1 BP + B A 1 Cλ = F 4.4 C A 1 BP + C A 1 Cλ = 0. e next coice is weter to eliminate P using te first equation of 4.4 or to eliminate λ using te second equation of 4.4. Since eac component of λ is sared by two elements tere are approximately two for R 0 orfourforbdfm components of λ per element and only one for R 0 or tree for BDFM components of P per element. e well-known Fraeijs de Veubeke [317] reduction of te saddle-point problem in 3.6 to a symmetric positive-definite linear system in edge degrees of freedom is to solve for P in terms of λ using te first equation of 4.4 and to substitute tis relation into te second

9 Z. Cai et al. / Higer order finite volume metods for elliptic equations 11 equation of 4.4 to obtain an equation for λ. If instead te second equation of 4.4 is used to solve for λ in terms of P ten we see tat λ = C A 1 C 1 C A 1 B P 4.5 and te following equation in fewer variables tan obtained in te Fraeijs de Veubeke reduction results: { B A 1 C C A 1 C 1 C A 1 B B A 1 B } P = F. 4.6 Wit te block diagonal matrix A te coefficient matrix of te discretization for P in 4.6 is still complicated. Instead let us approximate te integrals cξ j ξ i by a quadrature j cξ ξ i Q j cξ ξ i and employ exact integration for te remaining integrals. We obtain equations of te form ÃŨ + B P + C λ = 0 B Ũ = F C Ũ = were à = à N N wit à = Q cξ j ξ i N 1 N 1. en elimination as above leads to te perturbed equations { B à 1 C C à 1 C 1 C à 1 B B à 1 B } P = F. 4.8 e object is to coose a quadrature rule so tat te matrix à is diagonal instead of block diagonal and suc tat te numerical integration does not decrease te order of convergence of te original mixed finite element metod. Equation 4.8 will be defined as te finite volume metod to be considered in tis paper. Wile in some cases it can be derived in oter ways te point of tis derivation is tat te same procedure can be applied to different finite element spaces to obtain a variety of finite volume procedures. In eac case te coice of te quadrature procedure will be critical. 5. e standard finite volume metod For let its four edges left rigt bottom and top be denoted by α E ={l r b t} respectively. Let n be te unit outward vector normal to sotat n l = l = 1 0 n r = r = 1 0 n b = b = 0 1 n t = t = 0 1.

10 1 Z. Cai et al. / Higer order finite volume metods for elliptic equations For te R 0 space te natural degrees of freedom are v α = v n α α E te constant normal components of te flux on te edges αforṽ ; q tevalueofq on for W and interpreted as its cell-center value; and m α te midpoint values of m on te edges α wit te restriction tat for any two adjacent elements and m β = m α if tey represent te Lagrange multiplier on te common edge of and. e nodal basis functions ξ α on = x i 1x i y j 1 y j are given by ξ l = x i x l ξ r = x x i 1 r ξ b = y j y b and ξ t = y y j 1 t. e basis functions for W and are wit χ B te caracteristic function for B η x y = χ x y and ζ e x y = χ e x y respectively. Let andlete =. e coice µ = ζ e in te tird equation of 3.6 implies tat u e + u e = were u e and u e are te normal components of u and u on te edge e respectively. Recall tat we ave assumed te coefficient Kx to te constant K on eac element sotatc = c on.nowtakev = ξ e and ten ξ e in te first equation of 3.6. Since div ξ e = div ξ e = 1 and ξ e n = ξ e n = 1 c u ξ e p div ξ e + λ e = 0 c u ξ e p div ξ e + λ e = 0 so tat c u ξ e c u ξ e = p p Eac integral in 5.10 involves normal components of u on te edge e and its opposite edge but wit only te two equations 5.9 and 5.10 we cannot solve for te flux on e tereby motivating te introduction of a quadrature rule for tese integrals so tat tey can be approximated by te normal component of u on te edge e alone. Given te vector functions in te R 0 space it suffices to employ te trapezoidal rule: cw v c Q w v = c Wit te above numerical integration 5.10 is replaced by c ũ e c ũ e = p p α E w α v α. 5.11

11 Z. Cai et al. / Higer order finite volume metods for elliptic equations 13 were we denote approximations after application of te quadrature rule by ũ and p. By 5.9 we ten ave ũ e = c + c p p = K K p p. 5.1 K + K Every element as four adjacent elements wic we denote by α α E ; ten denote te restrictions of K and p to tese elements by K = K K α = K α p = p and p α = p α. By taking q = η in te second equation of 3.6 and using te divergence teorem and 5.1 we see tat f η = div ũ η = ũ n ds = α E ũ α = α E K K α K + K α p p α. us we obtain te desired finite volume sceme: α E d α p p α = f η 5.13 were d α denotes te armonic average of K and K α : d α = K αk. K α + K is equation is te classical finite volume metod or cell-centered finite difference equation wit armonically averaged diffusion coefficients. See [1] for a sligtly different derivation of 5.13 and an error analysis. In section 8 we sall derive a finite volume procedure corresponding to te rectangular RN k mixed finite element space for all k tereby finding finite volume metods for all orders of accuracy. 6. A iger order finite volume metod based on BDFM 6.1. Derivation Let us consider an analogous derivation based on te BDFM space. For eac element = x i 1 x i y j 1 y j denote its center and vertices by a = x i 1/ y j 1/ = x i y j a 1 = x i 1 y j 1 a = x i 1 y j a 3 = x i y j 1 a 4 = x i y j

12 14 Z. Cai et al. / Higer order finite volume metods for elliptic equations and te end points of te edges in E ={l r b t} by For te BDFM space b l1 = b b1 = a 1 b l = b t1 = a b r1 = b b = a 3 b r = b t = a 4. V = Span { ξ i αi i= 1 ; ξ α= r l b t i = 1 } W = P 1 = Span{1x x i 1/ y y j 1/ }=Span { } η 0 ηx ηy = { m L : m α P 1 α α E } ; te degrees of freedom for V are v a te value at te center of and v αi = v n α b αi te value of te normal component on te edge α E ={l r b t} at its two end points b α1 and b α.let θ i = x x i 1/ ψ j = y y j 1/. e two nodal basis functions ξ j j = 1 associated wit te center of satisfy ξ 1 1 a = ξ 0 0 a j = and ξ 1 n α = 0 for α E ; 6.14 tus ξ 1 x y = 1 θ i 0 0 ξ x y = 1 ψj xy e nodal basis functions ξ αi i = 1 satisfy ξ li b li = ξ ri b ri = associated wit te edge α E and its end points b αi 1 0 ξ bi b bi = ξ ti b ti = ξ αi a = 0 ξ αi b βj = 0 β α j i For example 1 1 ξ r1 = 4 θ i ψ j 1 + θ i ξ r = 4 θ i + ψ j 1 + θ i en u = u j ξ j + u i α ξ αi j=1 α E i=1 p = p 0 η0 + px ηx + py ηy

13 Z. Cai et al. / Higer order finite volume metods for elliptic equations 15 wit u j and ui α being te values of te jt component at te center and te values of te normal component on te edge α at its end point b αi respectively and p 0 and p x py te value and gradient of p at te center of respectively. e nodal basis function ζα i for associated wit te edge α E and its end point b αi is te linear function on te edge suc tat ζ i α b αj = δ ij for j = 1. Let and coose te test function in te first equation of 3.6 to be ξ k ; ten we see tat c u ξ k = ξ k p Next let be adjacent to and sare an edge e = wit end points b ei i = 1 wit. aking first v = ξ ek and ten ξ ek k = 1 in te first equation of 3.6 gives te relations Since ξ ek c u ξ ek c = ξ ek p ξ ek c u ξ ek u ξ ek = ξ ek p n λ e ξ ek n λ e n = ξ ek n on e differencing of te two equations in 6.19 gives c u ξ ek = ξ ek p ξ ek p k = e finite volume metod will be derived from 6.18 and 6.0. Note tat te Lagrange multipliers ave been eliminated; in fact it was not necessary to introduce tem [1]. e tird equation in 3.6 implies te flux consistency conditions u 1 e + u1 e = 0 and u e + u e = Next we sall introduce a quadrature rule to diagonalize te left-and sides of 6.18 and 6.0. Clearly it is appropriate to coose a five-point rule using te center and four vertices of eac element and te optimum coice is given by 4 gxydx dy Q g = 8ga + ga i 6. 1 since tis is te only rule associated wit tese nodes tat is exact for P ; in fact it is exact on P 3 and will lead to retaining te accuracy of te BDFM mixed metod. Wit tis numerical integration 6.18 and 6.0 are replaced by 3 c ũ k = ξ k p and 6.3 c ũ k e 1 c ũk e = ξ ek p ξ ek p 6.4 for k = 1 were we denote te approximate solution after quadrature by ũ and p. i=1

14 16 Z. Cai et al. / Higer order finite volume metods for elliptic equations It follows easily from 6.15 and 6.3 tat { K ũ k = p x k = 1 K p y k =. 6.5 Next it follows from 6.4 and te analogue of te flux consistency condition 6.1 tat ũ k e = d 6 ξ ek p ξ ek p 6.6 were d = d e = K K K + K is te armonically averaged diffusion coefficient on e =. A straigtforward calculation based on 6.16 or 6.17 sows tat ξ r1 p = p0 + 6 px 1 py ; 6.7 te oter integrals of tis form can be evaluated similarly. It follows tat ũ 1 r = d r 3 p0 p0 r + p x 1 + px y r p py r so tat ũ r = d r r ũ n ds = 3 p0 p0 r + p x 1 + px y r + p py r 6.8a 6.8b ũ j r = d r 3 p 0 p 0 r + p x + p x r. 6.9 j=1 See section 6.3 for a complete set of te ten flux coefficients in terms of te scalar coefficients. Equation 6.8a as te following interpretation. Let z denote te midpoint of te rigt edge of. en to iger order in 3 p0 p0 r + p x + px r p x z and 1 y p p py r x y z so tat te rigt-and side of 6.8a approximates p x z + p z p x y x a 3 and 6.8a is ten a proper approximation of te flux at a 3.

15 Z. Cai et al. / Higer order finite volume metods for elliptic equations 17 e second equation of 3.6 implies tat f η 0 = f 1 = ũ 1 = ũ n ds = Also from te same equation f η x = div ũ η x = ũ η x + A trivial but tedious calculation leads to te equation f x x i 1/ = 3ũ α E j=1 ũ j α ũ n x x i 1/ ds ũi r ũ i l + 1 ũ 1 t ũ 1 t ũ1 b +ũ b. 6.3 e last term above represents an approximation to te second mixed partial derivative of te y-component of te flux at te center of. Similarly f y y j 1/ = 3ũ i=1 ũi t ũ i b i=1 + 1 ũ 1 r +ũ l ũ1 l ũ1 r e BDFM finite volume equations It follows from and te ten equations given by 6.35 tat f 1 = 3 d α p 0 p 0 [ α + dr p x + p x r α d l p x + p x ] [ y l + dt p + py t f x x i 1/ = 3 K p x + d r d b p y + py b] 6.34a α=lr p 0 p0 r d α p x + p x α p 0 d p0 l l d α p x p x α α=bt 6.34b

16 18 Z. Cai et al. / Higer order finite volume metods for elliptic equations f y y j 1/ = 3 K p y + d t α=bt p 0 p0 t d α p y + py α p 0 d p0 b b α=lr d α p y py α. 6.34c e system 6.34 is te finite volume metod tat associate wit te BDFM mixed finite element metod. ese equations can be interpretated as follows. Let te elliptic equation be p = f ; ten dividing 6.34a by and letting 0 leads back to p = f. e second equation 6.34b as as limiting form te equation p x = f x ; analogously 6.34c tends to p y = f y. Note tat we can easily obtain te approximate flux coefficients ũ i α on eac edge of te blocks from 6.8 and teir corresponding relations on te oter edges. e coefficients for te two internal flux components are given by 6.5. ese coefficients are collected below e flux coefficients for BDFM e full set of equations relating te flux variables to te scalar variables are as follows: ũ 1 = K p x ũ = K p y ũ 1 r = d r 3 p0 p0 r ũ r = d r 3 p0 p0 r ũ 1 l = d l 3 p0 p0 l ũ l = d l 3 p0 p0 l + p x 1 + px y r p py r + p x 1 + px y r + p py r p x 1 + px y l p py l p x 1 + px y l + p py l 6.35a 6.35b 6.35c 6.35d 6.35e 6.35f

17 Z. Cai et al. / Higer order finite volume metods for elliptic equations 19 ũ 1 t = d t ũ t = d t ũ 1 b = d b ũ b = d b 3 p0 p0 t 3 p0 p0 t 3 p0 p0 b 3 p0 p0 b + p y 1 + py t p x p x t 6.35g + p y 1 + py t + p x p x t 6.35 p y 1 + py b p x p x b 6.35i p y 1 + py b + p x p x b. 6.35j 7. Error estimates for finite volume metods related to mixed finite element metods 7.1. A general convergence analysis We sall deduce error estimates from classical results for mixed finite element metods for finite volume metods derived from mixed finite element metods by applying an appropriate quadrature rule to te cu v-term. en we sall employ tese estimates to analyze te BDFM -based FVM. e R 0 case as been satisfactorily treated in [1]; te analysis below owever does apply to it. Now let V = V W be an admissible mixed finite element space and let u p V be te solution of te discrete mixed problem 3.1. Let up V be te solution of.4. For BDFM we know tat [8] u u 0 C k p k+1 1/ k = 1 7.1a p p 0 C p 1/ ; 7.1b analogous error estimates are known for all mixed finite element metods considered below. Since te fluxes evaluated by 6.35a from 6.34 satisfy te flux consistency relations we can drop te Lagrange multipliers in te analysis of te convergence of It will be te case tat te flux consistency conditions will be applied explicitly or implicitly in all oter examples considered below. Consequently we can consider 6.34 to be a special case of te perturbed mixed finite element metod of finding ũ p V suc tat Q cũ v v p = 0 v V 7.a ũ q = f q q W 7.b

18 0 Z. Cai et al. / Higer order finite volume metods for elliptic equations were Q g = Q g 7.3 is te quadrature rule associated wit derivation of te FVM related to V. We sall assume tat te quadrature points for Q coincide wit nodal points for a basis for V and tat Q reduces te matrix A see 4.1 to diagonal form; tis constraint seems not to be possible for a number of efficient mixed finite element spaces if te additional constraint tat it cause no reduction in te optimal order of convergence of te resulting FVM but we sall indicate several interesting examples were suc a Q is easily constructed. Let u p denote te solution of 3.1 and set e ũ = u ũ e p = p p. By differencing 7. and 3.1 we see tat cu v Q cũ v ve p = 0 v V 7.4a e ũq = 0 q W. 7.4b e convergence argument below makes serious use of te commuting diagram constructed in [814] and implicitly in [16] and for many oter families of mixed finite element spaces; tus we assume tat tere exists a map : H 1 V along wit te L -projection P : L W suc tat and suc tat div = P div : H 1 W 7.5 v v 0 M v s s 0 s r a q P q M q s s 0 s r 7.6b were r = r in some cases [7915] and r = r + 1 in oters [8190]. We also require te inverse property v s C s s v s 0 s s t v V 7.7 were C is independent of and V P t d wit d being te dimension of and t being independent of ; 7.7 olds for quasi-regular partitions so tat it olds for te uniform partition in particular. It sould be noted tat is usually defined element-by-element in terms of boundary and interior moments so tat it is defined wen u H 1 d or equivalently p H. Consequently te norms on te rigtand sides in 7.6 can be broken norms over te partition. e two inequalities in 7.6 imply tat P r d V P r 1 W.

19 Z. Cai et al. / Higer order finite volume metods for elliptic equations 1 Begin te analysis by taking q = e ũ W in 7.4b to see tat e = 0 so tat e ũ 0. Next let v Ṽ ={v: v V } and assume tat m v = v j ξ j ũ j=1 were {ξ j j= 1...m} is a nodal basis for V over te nodes a j j= 1...m wit te nodes being similarly placed in as in te reference element [ 1 1] d ; i.e. Also assume tat ξ j a k = δ jk Q g = d jk = 1...m. m j=1 ω j ga j. en tere exist positive constants γ 1 and γ independent of suc tat γ1 v 0 Q v m = d j ω j v γ v 0 j=1 since V is a finite-dimensional subset of P t d. us γ 1 v Q v 1/ γ v v Ṽ 7.8 so tat te perturbed L -like norm based on Q is equivalent to te ordinary L norm. Also assume tat Q is exact on P n : Q q = qx dx q P n ; actually we need a sligt generalization of tis relation. If q corresponds to cu v let q i = cu i v i i= 1...d. en let d d q dx = q i dx Q i q i 7.9 and assume tat Q i q i = i=1 i=1 q dx q i P n i = 1...d It is not necessary tat c be constant on eac in te analysis below but it will be assumed tat cx is sufficiently smoot on eac uniformly in and tat it is bounded away from zero and infinity on.

20 Z. Cai et al. / Higer order finite volume metods for elliptic equations Rewrite te error equation 7.4a in te form Q ce ũ v ve p = Q cu v cu v v V Let v = e ũ and recall tat e ũ = 0 so tat c min γ1 e ũ Q c e ũ = Q cu e ũ cu eũ. 7.1 Let g τ = k+l=τ τ g x k y l dx dy. en te Bramble Hilbert lemma [13] implies tat tere exists a constant M independent of suc tat Q cu e ũ cu eũ M cu e ũ n+1 n since te quadrature rule is exact on P n. e quadrature rule is exact on P m for m<n as well; te argument below applies wit n replaced by m<nand is of considerable interest for m = n 1 in te analysis of ep. We must break te proof into two cases. If t n + 1 ten since eac of te factors u and eũ is a polynomial in P t and c as sufficiently many uniformly bounded derivatives on Q cu e ũ cu eũ M u t eũ t n+1 ; ence it follows tat 1/ Q cu e ũ cu eũ M u t 1/ e ũ t n Here we need error estimates analogous to 7.1 for te underlying mixed finite element metod; i.e. assume tat u u 0 M u s s 0 s r a p p 0 M p τ τ 0 τ r. 7.15b Moreover in all practical cases of mixed finite element spaces t = r or t = r + 1. By 7.15a te inverse property 7.7 and a simple approximation property we sall see tat te first term on te rigt-and side of 7.14 is bounded. Given te parti-

21 Z. Cai et al. / Higer order finite volume metods for elliptic equations 3 tion it is a standard polynomial approximation property tat tere exists a piecewisepolynomial χ wit χ = χ P t d suc tat u χ s M u t t s 0 s t wit M independent of since is quasi-regular. en 1/ 1/ + 1/ u t u χ t + u t 1/ χ u t M u χ 0 t + u t M u u 0 + u χ 0 t + u t M u t 7.16 as claimed. e inequality above is valid wit t replaced by s 0 s<tand will be used for suc s below. By te inverse property 7.7 te second term satisfies te inequality 1/ t M e 0 t e ũ were M is also independent of. us it follows tat e ũ 0 C u t n+1 t 7.17 wic implies tat assuming t r + 1 Next assume tat ũ u ũ 0 C u r+1 minr+1n+1 t t n + 1 < t. en only n + 1 derivatives rater tan t can be applied to u e ũ. Consequently cu e ũ n+1 t j=n+1 t u j e n+1 j 7.19 and applying te inequality 7.16 wit j in place of t and te inverse property 7.7 leads to te bound t e 0 M u j j. 7.0 We ave proved te following teorem. ũ j=n+1 t ũ

22 4 Z. Cai et al. / Higer order finite volume metods for elliptic equations eorem 7.1. Let e ũ = u ũ denote te perturbation in te flux caused by te introduction of te quadrature rule Q. en e ũ = 0 C u t n+1 t if n + 1 t e ũ 0 t C u j j if t n + 1 < t j=n+1 t { r + 1 r + 1 u ũ 0 C u r+1 r+1 if t n = r r. 7.1a 7.1b 7.1c If t n equals eiter r + 1 r + 1 or r r we can obtain a collection of oter estimates for e ũ. Corollary 7.1. If t n = r r or r + 1 r + 1 ten e 0 C u s s+1 0 s r. 7. ũ Proof. We sall treat te r r-case first. From te inverse property 7.7 and 7.16 for t replaced by s [0t] 1/ e ũ 0 M 1/ u r e ũ r+1 r 1/ 1/ M u s s r e r r+1 0 M u s e 0 s wic completes te proof for tis case. Wen t n = r + 1 r + 1 [ 1/ e ũ 0 M u r e ũ 1/ + u r+1 e ũ ũ r+1 ũ r 1/ 1/ ] r+ 1/ 1/ M u s e r s+1 r+ 0 ũ M u s e 0 s ũ and te corollary as been establised.

23 Z. Cai et al. / Higer order finite volume metods for elliptic equations 5 It is wort noting tat if t n = r r 7.3 indicates tat te te two approximate solutions u and ũ are closer togeter under a sligtly less restrictive regularity tan eiter can be globally to u for tat regularity. Global error estimates for p p can be derived using a modification of te duality argument given in [14]; ere te eavy use of te projections and P and te bounds given by 7.6 is involved. Since all of our examples are related to t n = r r or t n = r + 1 r + 1 we sall restrict our attention to tese two cases. Write te error equations in te form ce ũ v ve p = fv v V 7.5a e ũw = 0 w W 7.5b were fv = Q cũ v cũ v 7.6 and recall tat e ũ = 0. Let ϕ H H0 1 satisfy te self-adjoint equation a ϕ = e p. 7.7 en by 7.5a and 7.5 e p 0 = e p a ϕ = e p a ϕ + e p a ϕ a ϕ = f a ϕ ce ũ a ϕ + e p P e p e p = f a ϕ ce ũ a ϕ = f a ϕ + ceũ a ϕ a ϕ ce ũ a ϕ = f a ϕ + ce ũ a ϕ a ϕ + e ũ ϕ = f a ϕ + ce ũ a ϕ a ϕ. Now f a ϕ = Q cũ a ϕ cũ a ϕ = Q cũ ϕ cũ ϕ so tat if t n = r r and 0 s r f a ϕ M1 ũ ϕ r+1 r+1 1/ M ũ r ϕ 1/ r+1 r t

24 6 Z. Cai et al. / Higer order finite volume metods for elliptic equations 1/ M 3 ũ s ϕ 1/ s+ 1 t M 4 u s e p 0 s+ were te same justifications ave been used ere as in te analyses of te error in te flux plus te fact tat ϕ 1/ C 1 1 ϕ C e 0. e same result olds for te case tat t n = r + 1 r + 1; see te proof of corollary 7.1. By 7.6a ce ũ a ϕ a ϕ M5 e 0 ϕ M 6 e e 0 0 p. ũ p ũ us we ave sown tat p p 0 = e 0 p M u s s+ + e ũ 0 0 s r M u s s+ M p k k+1 1 k r Again we ave sown tat te approximations p and p are closer togeter tan tey are to p; ere te estimate of teir difference is better wit respect bot to regularity and to te maximum exponent on. We can summarize te error estimates for p in te following teorem. eorem 7.. Assume tat t n = r r or r + 1 r + 1. en p p 0 = e p 0 M p k k+1 1 k r a p p 0 M p k k 0 k r. 7.9b e bound 7.9b is te optimal global rate of convergence for p and requires te minimal regularity for tis rate. A sligtly more careful argument would ave allowed p k to be replaced by te broken norm 1/ ; p k te same remark applies to te bounds for u ũ.

25 Z. Cai et al. / Higer order finite volume metods for elliptic equations Error estimates for te BDFM -based FVM Error bounds for ũ p were p is te solution of te BDFM -based FVM equations 6.34 can be obtained directly from corollary 7.1 and teorem 7.. For tis metod us r = 1 t = n = 3 tn = r + 1 r + 1 r = p p 0 C p s s 1 s 7.31a u ũ 0 C u s s 0 s. 7.31b ese are optimal estimates for te errors in bot variables. 8. Finite volume metods based on RN-rectangular elements e Raviart omas Nedelec rectangular mixed finite elements of index k ave as local bases te tensor product spaces RN k = V k W k were { Pk+1k P kk+1 dim = V k = 8.1a P k+1kk P kk+1k P kkk+1 dim = 3 { Pkk dim = W k = 8.1b P kkk dim = 3. Note tat in ybridizing te mixed finite element equations te space k would consist of a copy of P k for d = andp kk for d = 3 on eac interface between elements; tus te Fraeijs de Veubeke reduction to a positive-definite system in te Lagrange multipliers would lead to a block system wit te unknowns on an interface seeing tose on te oter six faces of te two elements generating te interface if d =. If d = 3 ten eac block of interface unknowns would see tose for te oter ten faces of te adjacent elements. Also note tat tere are two k + 1-blocks for d = or tree k + 1 -blocks for d = 3 associated wit eac element for te Fraeijs de Veubeke reduced equations wile tere would be a single k + 1 -ork block per element in a FVM reduction if feasible. e FVM-block equations would ave te same 5-point or 7-point structures as are associated wit te simplest finite difference equations for te elliptic equation.1 in two or tree space variables. We sall sow tat it is easy to construct a convenient nodal basis for RN k and an associated quadrature relation tat maintains te global accuracy of te underlying mixed metod so tat we can acieve its FVM. Is it desirable to ave te FVM corresponding to RN k? Look first at te treedimensional problem. For k = 0 te finite volume metod associates one parameter for te scalar function to eac element wile tere are tree Lagrange multipliers associated wit an element; moreover te grap structure is simpler for te finite volume metod. Clearly te finite volume route is superior for k = 0. For k = 1 tere are 1 Lagrange

26 8 Z. Cai et al. / Higer order finite volume metods for elliptic equations multipliers per element versus eigt scalar parameters plus a simpler grap structure so tat te finite volume approac is again superior. For k = tere are 7 parameters per element eiter way; ere tere is a sligt advantage to te finite volume tecnique. For k> te number of scalar parameters exceeds tat for Lagrange multipliers and it is doubtful tat te finite volume approac is very elpful. In te two-dimensional case tere are k + 1 Lagrange parameters per element versus k+1 scalar parameters. It appears tat te finite volume reduction is better tan te Lagrange multiplier procedure for R 0 and R 1 but not for k>1. For k = 1 te resulting FVM as four scalar parameters per element i.e. a bilinear function wile te competing BDFM finite volume metod as tree a linear function. Since bot lead to te same error estimates it would appear tat te FVM based on BDFM is somewat more efficient tan te R 1 -based one. As a consequence of tese remarks we sall treat te tree-dimensional case; te two-dimensional case is an obvious specialization of it. oug te iger order finite volume metods based on RN k may be of lesser practical interest te analysis is effectively independent of k and will be given for arbitrary k. Consider te x-component vx of a vector v V. en v P k+1kk. us tere are k + k + 1k + 1 degrees of freedom for v wit k +1 necessarily associated wit eac face of te form x = const. Consider first te reference element ref =[ 1 1] 3. Denote te one-dimensional Lobatto quadrature rule wit k + nodes by L k+ were k+1 L k+ g = gη i ω i 1 = η 0 <η 1 < <η k+1 = 1 i=0 and te k + 1 node Gauss rule by G k+1 were k+1 G k+1 g = gζ j w j 1 <ζ 1 < <ζ k+1 < 1. It is well known tat tus j=1 L k+ q = G k+1 q = 1 1 qxdx q P k+1 [ 1 1] ; k+1 k+1 k+1 Q xref g = L k+ G k+1 G k+1 g = gη i ζ j ζ l ω i w j w l i=0 j=1 l=1 = gx dx ref g P k+1kk. 8. Now let Q x be defined in te obvious way by mapping te nodes η i ζ j ζ l to affinely and multiplying its weigt by 3 and define Q y and Q z analogously. en if u = u 1 u u 3 and v = v 1 v v 3 let

27 Z. Cai et al. / Higer order finite volume metods for elliptic equations 9 Q cu v = Q x cu 1 v 1 + Q y cu v + Q z cu 3 v ake {η i } {ζ j } {ζ l } as nodes for te x-component v 1 P k+1kk of v on te reference element and define te ijl-basis element for tis component by requiring tat { 1 i ξ x ijl η i ζ j ζ l = j l = ijl 0 i j l 8.4 ijl. Similarly take {ζ i } {η j } {ζ l } and {ζ i } {ζ j } {η l } as nodes for te y- and z-components wit corresponding basis elements. Clearly te quadrature rule Q cu v = Q cu v diagonalizes eac matrix A tereby generating a FVM in our sense to be associated wit RN k. Let us consider error estimates for te resulting FVM. Employ te same notation as in section 7. First Q is exact on P k+1k+1k+1. us r = k t = k + 1 n = k + 1 tn = r + 1 r + 1 r = k + 1. So corollary 7.1 and teorem 7. imply te following optimal error bounds: p p 0 C p s s 1 s k a u ũ 0 C u s s+1 0 s k b e estimates 8.5 old in te two-dimensional case as well. 9. e tree-dimensional BDFM -based finite volume metod Here we derive a iger order finite volume sceme based on te treedimensional BDFM space using an appropriate quadrature formula. e error analysis below is applicable for c = cxyz being variable but te quadrature rule 9.1 wic differs in type from tose applied to te oter mixed finite element spaces discussed in tis paper can be gauranteed to diagonalize te cu v-matrix only if c is constant on eac element. Let be te partition of into cubes wit te lengt = N 1 of eac edge: were = N ijk=1 ij k = ij k = [ i 1 i ] [ j 1 j ] [ k 1 k ] =[x i 1 x i ] [y j 1 y j ] [z k 1 z k ].

28 30 Z. Cai et al. / Higer order finite volume metods for elliptic equations Let P k θ for θ = x y or z be te set of all polynomials in te single variable θ of te degree not greater tan k. Set θ i = θ 1...θ i 1 θ i+1...θ 3 and denote by P j omi te omogeneous polynomials of degree j in te variables θ i. For eac element BDFM is given by V = [ P \P om1 ] [ P \P om ] [ P \P om3 ] W = P 1 e = { m L } e: m P 1 e e E were E ={rrflrbt} for rear front left rigt bottom and top faces. Assume te coefficient c to be constant on eac ; for convenience take c = c = 1on. It suffices to discuss one component in te inner product say te x-component. So consider te reference element =[ 1 1] 3 and let were V = V x = V x = Span { 1xyzx xyxz } = Span{ξ j j= 1...7} ξ 1 = 1 x ξ = 1 x1 x ξ 3 = 1 x1 + x ξ 4 = 1 y1 x ξ ξ 6 = 1 y1 + x ξ 5 = 1 z1 x 7 = 1 z1 + x. ese basis functions correspond to te following degrees of freedom: te values at and ±1 0 0 and y and z components of te gradient at ± Basis functions for W can be specified simply as η xyz= 1 0 ηx η y xyz= y y j 1/ xyz= x x i 1/ η z xyz = z z k 1/ so tat p x y = p 0 η0 + px ηx + py ηy + pz ηz. Now wat is needed is a rule assigning approximations for integrals over V V i.e. for products ξ j ξ k wic diagonalizes te matrix a ij = ξ j ξ i. For te oter mixed finite element spaces te quadrature rule was based on te degrees of freedom for te space; unfortunately tere does not exist a quadrature rule based on te seven

29 Z. Cai et al. / Higer order finite volume metods for elliptic equations 31 degrees of freedom for V x tat is exact on P 3. us we are led to a rule involving second derivatives of normal components at facial midpoints given by were Q u v = Q x g 1 + Q y g + Q z g 3 if u v = Q x g = 3 8g x {±100} gx + x {±100} 3 u k v k = k=1 3 g k k=1 g y x + g z x 9.1 and Q y and Qz are defined analogously; Qx is exact on P 3 and vanises for g = ξ j ξ i for i j as desired. Wen scaled for a cube of side lengt Q is replaced by x Q x 3 g = 8ga + gx + g 1 4 y x + g z x. x=a f a rr x=a f a rr e autors are unaware of any previous appearance of te quadrature formula 9.1. Now it is clear tat tere exist positive α 1 and α suc tat α 1 u u Q u α u u u V so tat Q generates a norm on V V tat is equivalent to te ordinary L -norm. In order to make use of te general convergence argument it is necessary to sow tat Q uv u v M u v 4 ; 9. owever tis follows trivially from te exactness of te quadrature rule on P 3. us te same error estimates old for te BDFM -based tree-dimensional procedure as for te two-dimensional one; tat is te errors in te scalar and flux approximations are O. 10. A finite volume metod based on BDFM 3 e BDFM k = V k W k mixed finite element space over rectangles is defined locally by V k = [ P k \ { y k}] [ P k \ { x k}] W k = P k 1. us r = k 1 t = k r = k; in order tat te approximation ũ k to u be of optimal order it is necessary tat te quadrature rule Q k be exact on polynomials of degree n = k 1

30 3 Z. Cai et al. / Higer order finite volume metods for elliptic equations since t = r + 1 so tat r + 1 = k 1. We sall consider te cases k = 3andk = 4 for wic we need quadrature rules exact on P 5 and P 7 respectively. For k = 3 recall [8] tat te projection for BDFM 3 is determined by te degrees of freedom ϕ ϕ ν q e = 0 q P e e ϕ ϕχ = 0 χ P 1. So for te x-component v 1 of v V 3 it would be natural to take te following degrees of freedom wit l and r denoting te left and rigt edges of respectively v 1 x i i = a v 1 x x v 1 0 y x b were x 0 is te center of x i i = is a vertex of andx 5 and x 6 are te midpoints of l and r respectively. It is easy to sow tat tis set of degrees of freedom determine te first component of v V 3. o tese degrees of freedom we sould like to associate a quadrature rule of te form on te reference element [ 1 1] Q g = 6 gx i w i + ω xx g x x 0 + ω xy g y x i=0 Unfortunately te best quadrature rule for tese degrees of freedom is te same rule as we applied for BDFM. It is exact on P 3 but it obviously fails for qx = x 1 x. Clearly tis quadrature rule kills four of te basis functions for eac component of v V k. us it is a bit surprising tat any convergence result can occur wit tis rule but let us sow a suboptimal result. Wit Q defined by 6. n = 3 and following te general argument of section 7 we see tat 3 cu eũ 4 M u j e ũ 3 j 4 M u 0 e ũ from wic it follows tat j=0 e ũ All tat follows from 10.4 is tat 0 M u u ũ 0 M u a distinctly suboptimal result. e autors ave so far not found a set of degrees of freedom for V 3 and a corresponding quadrature rule to improve on tis O error estimate for ũ. Currently BDFM 3 serves as an example of a mixed finite element space for wic our procedure

31 Z. Cai et al. / Higer order finite volume metods for elliptic equations 33 for deriving a satisfactory finite volume procedure fails. In a sense te difficulty lies in te basic efficiency of te BDFM 3 mixed metod; te dimension of V 3 is too small to support a quadrature rule tat is exact for P 5 wereas te less efficient RN metod does. On te oter and tis finite volume metod could be used to find an initial guess for an Uzawa or oter iteration for te solution of te original mixed metod. References [1]. Arbogast C.N. Dawson P.. Keenan M.F. Weeler and I. Yotov Enanced cell-centered finite differences for elliptic equations on general geometry SIAM J. Numer. Anal []. Arbogast M.F. Weeler and I. Yotov Mixed finite elements for elliptic problems wit tensor coefficients as cell-centered finite differences SIAM J. Numer. Anal [3] D.N. Arnold and F. Brezzi Mixed and nonconforming finite element metods: implementation postprocessing and error estimates RAIRO Modél. Mat. Anal. Numér [4] B.R. Baliga and S.V. Patankar A new finite-element formulation for convection diffusion problems Numer. Heat ransfer [5] R.E. Bank and D.J. Rose Some error estimates for te box metod SIAM J. Numer. Anal [6] J. Baranger J.F. Maitre and F. Oudin Connection between finite volume and mixed finite element metods Modél. Mat. Anal. Numér [7] F. Brezzi J. Douglas Jr. R. Durán and M. Fortin Mixed finite elements for second order elliptic problems in tree variables Numer. Mat [8] F. Brezzi J. Douglas Jr. M. Fortin and L.D. Marini Efficient rectangular mixed finite elements in two and tree space variables RAIRO Anal. Numér [9] F. Brezzi J. Douglas Jr. and L.D. Marini wo families of mixed finite elements for second order elliptic problems Numer. Mat [10] F. Brezzi and M. Fortin Mixed and Hybrid Finite Element Metods Springer New York [11] Z. Cai On te finite volume element metod Numer. Mat [1] Z. Cai J. Mandel and S. McCormick e finite volume element metod for diffusion equations on general triangulations SIAM J. Numer. Anal [13] P.G. Ciarlet e Finite Element Metod for Elliptic Problems Nort-Holland Amsterdam [14] J. Douglas Jr. and J.E. Roberts Global estimates for mixed metods for second order elliptic equations Mat. Comp [15] J. Douglas Jr. and J. Wang A new family of mixed finite element spaces over rectangles Mat. Apl. Comput [16] R.S. Falk and J.E. Osborn Error estimates for mixed metods RAIRO Anal. Numér [17] B.X. Fraeijs de Veubeke Stress function approac in: Internat. Congress on te Finite Element Metod in Structural Mecanics Bournemout [18] B. Heinric Finite Difference Metods on Irregular Networks Birkäuser Basel [19] J.C. Nedelec Mixed finite elements in R 3 Numer. Mat [0] P.A. Raviart and J.M. omas A mixed finite element metod for nd order elliptic problems in: Matematical Aspects of Finite Element Metods Lecture Notes in Matematics Vol. 606 eds. I. Galligani and E. Magenes Springer New York 1977 pp [1].F. Russell and M.F. Weeler Finite element and finite difference metods for continuous flows in porous media in: e Matematics of Reservoir Simulation ed. R.E. Ewing SIAM Piladelpia PA 1983.