c 2005 Society for Industrial and Applied Mathematics

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1 SIAM J. NUMER. ANAL. Vol. 43, No. 4, pp c 25 Socity for Industrial and Applid Mathmatics SUPERCONVERGENCE OF THE VELOCITY IN MIMETIC FINITE DIFFERENCE METHODS ON QUADRILATERALS M. BERNDT, K. LIPNIKOV, M. SHASHKOV, M. F. WHEELER, AND I. YOTOV Abstract. Suprconvrgnc of th vlocity is stablishd for mimtic finit diffrnc approximations of scond-ordr lliptic problms ovr h 2 -uniform quadrilatral mshs. Th suprconvrgnc rsult holds for a full tnsor cofficint. Th analysis xploits th rlation btwn mimtic finit diffrncs and mixd finit lmnt mthods via a spcial quadratur rul for computing th scalar product in th vlocity spac. Th thortical rsults ar confirmd by numrical xprimnts. Ky words. mixd finit lmnt, mimtic finit diffrnc, tnsor cofficint, suprconvrgnc AMS subjct classifications. 65N6, 65N12, 65N15, 65N22, 65N3 DOI / Introduction. W considr th numrical approximation of a linar scondordr lliptic problm. In porous mdium applications, this quation modls singl phas Darcy flow and is usually writtn as a first-ordr systm for th fluid prssur p and vlocity u: (1.1) u = K grad p in Ω, div u = f in Ω, u n = g on Ω, whr Ω R 2, n is th outward unit normal to Ω, and K R 2 2 is a symmtric uniformly positiv dfinit full tnsor rprsnting th rock prmability dividd by th fluid viscosity. W assum that systm (1.1) satisfis th compatibility condition fdx gds=. Ω Ω In this papr, w analyz th convrgnc of a mimtic finit diffrnc (MFD) mthod on quadrilatral mshs. Th mthod uss discrt oprators that prsrv crtain critical proprtis of th original continuum diffrntial oprators. Consrvation laws, solution symmtris, and th fundamntal idntitis and thorms of vctor Rcivd by th ditors April 15, 24; accptd for publication (in rvisd form) Fbruary 7, 25; publishd lctronically Novmbr 22, 25. This work was prformd by an mploy of th U.S. Govrnmnt or undr U.S. Govrnmnt contract. Th U.S. Govrnmnt rtains a nonxclusiv, royalty-fr licns to publish or rproduc th publishd form of this contribution, or allow othrs to do so, for U.S. Govrnmnt purposs. Copyright is ownd by SIAM to th xtnt not limitd by ths rights. Mathmatical Modling and Analysis Group, Thortical Division, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM (brndt@lanl.gov, lipnikov@lanl.gov, shashkov@ lanl. gov). Th rsarch of ths authors was supportd by th U.S. Dpartmnt of Enrgy undr contract W-745-ENG-36, LA-UR ICES, Dpartmnt of Arospac Enginring and Enginring Mchanics, Dpartmnt of Ptrolum and Gosystms Enginring, and Dpartmnt of Mathmatics, Th Univrsity of Txas at Austin, Austin, TX (mfw@ics.utxas.du). Th work of this author was partially supportd by NSF grant EIA and by NPACI grant UCSD Dpartmnt of Mathmatics, 31 Thackray Hall, Univrsity of Pittsburgh, Pittsburgh, PA 1526 (yotov@math.pitt.du). Th work of this author was partially supportd by NSF grants DMS 17389, DMS , and DMS and by DOE grant DE-FG2-4ER

2 SUPERCONVERGENCE OF THE VELOCITY 1729 and tnsor calculus ar xampls of such proprtis. This mimtic tchniqu has bn applid succssfully to svral applications including diffusion [22, 15, 18], magntic diffusion and lctromagntics [14], continuum mchanics [17], and gas dynamics [8]. For problm (1.1), th mimtic tchniqu uss discrt flux G and divrgnc DIV oprators for th continuum oprators Kgrad and div, rspctivly, which ar adjoint to ach othr, i.., G = DIV. It is straightforward to xtnd th MFD mthod to locally rfind mshs with hanging nods [16], unstructurd thr-dimnsional mshs composd of hxahdra, ttrahdra, and any cll typ having thr facs intrscting at ach vrtx. A connction btwn th MFD mthod and th mixd finit lmnt (MFE) mthod with Raviart Thomas finit lmnts has bn stablishd in [4]. In particular, it was shown that th scalar product in th vlocity spac proposd in [15] for MFD mthods can b viwd as a quadratur rul in th contxt of MFE mthods. Anothr closly rlatd mthod is th control-volum MFE mthod [7, 9]. MFE discrtizations on quadrilatral grids hav bn studid in [25, 26, 2, 13]. Ths mthods ar basd on th Piola transformation [25, 6], which prsrvs continuity of th normal componnt of th vlocity u across msh dgs. Unfortunatly, this rsults in th ncssity to intgrat rational functions ovr quadrilatrals. Th task bcoms vn mor complicatd whn th diffusion tnsor is full and nonconstant. Th rsults in [4] provid an fficint numrical quadratur rul with a minimal numbr of points. Morovr, th connction btwn th two mthods allows for xtnsions of MFE mthods to gnral polygons and polyhdra. Th aformntiond connction provids a suitabl functional fram for rigorous analysis of convrgnc of mimtic discrtizations. In [4], first-ordr convrgnc for th fluid prssur and vlocity was shown. In this papr, w stablish vlocity suprconvrgnc for MFD discrtizations of (1.1) on h 2 -uniform quadrilatral mshs (as dfind in (2.2) (2.3)). Prcis calculation of th fluid vlocity is important for porous mdia and othr applications. Th points or lins whr th numrical solution is suprclos to th xact solution may b usd to improv th accuracy of th ovrall simulation. Various suprconvrgnc rsults for MFE mthods hav bn stablishd for rctangular mshs [21, 19, 27, 1, 11, 12, 3, 1] and gnral quadrilatral mshs [2, 13]. In [13], vlocity suprconvrgnc is stablishd for th MFE discrtization of (1.1) on h 2 -uniform quadrilatral grids. In this papr, w xploit th rlation btwn MFD mthods and MFE mthods with th quadratur rul (3.1) to stablish suprconvrgnc for vlocitis in MFD discrtizations. In particular, w show that th computd normal vlocitis ar suprclos to th tru normal vlocitis at th midpoints of th dgs. In [18], an altrnativ quadratur is introducd, which prsrvs symmtry of th xact solution on polar grids. This symmtry prsrvation is important for problms of radiation transport in th asymptotic diffusion limit. Th analysis of suprconvrgnc for symmtry-prsrving quadraturs is lft for futur invstigation. Th papr outlin is as follows. In sction 2, w dscrib th MFE mthod for (1.1). In sction 3, th MFD mthod is prsntd and rlatd to th MFE mthod with a quadratur rul. Th main suprconvrgnc rsults ar prsntd in sction 4. Suprconvrgnc of th normal vlocitis at th midpoints of th dgs is stablishd in sction 5. In sction 6, numrical xprimnts ar givn that confirm th thortical rsults.

3 173 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV r 4 r 3 r 3 = r 4 r 2 r 1 r 2 = r 1 Fig h 2 -uniform quadrilatral grid. 2. Th MFE mthod. To simplify th xposition, w assum without loss of gnrality that g =, i.., homognous Numann boundary conditions ar imposd on Ω. Throughout this papr, w shall us th notation k,d, div,d, and D for th norms on th Hilbrt spacs H k (D), H(div; D), and L 2 (D), rspctivly, whr D Ω. In addition, k,d will dnot th sminorm on H k (D). To simplify notation, w shall omit th subscript D whn D = Ω. Finally, w dnot by (, ) th L 2 -innr product on Ω of ithr scalar or vctor functions. Lt V = {v H(div; Ω) : v n =onω} and W = { w L 2 (Ω) : Ω } wdx =. Th variational formulation of (1.1) is as follows: find a pair (u, p) V W such that (2.1) (K 1 u, v) (p, div v) =, (div u, w)=(f, w) (v, w) V W. For th discrtization of (2.1), dnot by T h a shap-rgular partition (s [5, Rmark 2.2, p. 113]) of Ω into convx quadrilatral lmnts of diamtr not gratr than h. For two xampls of shap-rgular grids, s Figur 6.1. W assum that th grid is h 2 -uniform. Following [13], th quadrilatral partition T h is calld h 2 -uniform if ach lmnt is an h 2 -paralllogram, i.., (2.2) (r 2 r 1 ) (r 3 r 4 ) Ch 2, and any two adjacnt quadrilatrals form an h 2 -paralllogram, i.., (2.3) (r 2 r 1 ) (r 2 r 1) Ch 2, whr r 1, r 2, r 3, and r 4 ar th vrtics of th adjacnt lmnt (s Figur 2.1). For any convx quadrilatral, thr xists a bijction mapping F :ê, whr ê is th rfrnc unit squar with vrtics ˆr 1 =(, ) T, ˆr 2 =(1, ) T, ˆr 3 =(1, 1) T, and ˆr 4 =(, 1) T. Dnot by r i =(x i,y i ) T, i =1, 2, 3, 4, th four corrsponding vrtics of lmnt as shown in Figur 2.2. Thn, F is th bilinar mapping givn by (2.4) F (ˆr) =r 1 (1 ˆx)(1 ŷ)r 2 ˆx(1 ŷ)r 3 ˆxŷ r 4 (1 ˆx)ŷ.

4 ˆn 4 ˆr 4 ˆl 4 n 2 SUPERCONVERGENCE OF THE VELOCITY 1731 ˆr 3 ˆn 3 n 3 r 3 ˆl 3 r 4 l 3 l 2 ê ˆl2 ˆn 2 F l 4 n 4 l 1 r 2 ˆr 1 ˆl 1 ˆr 2 r 1 n 1 ˆn 1 Fig Bilinar mapping and orintation of normal vctors. Not that th Jacobi matrix DF and its Jacobian J ar linar functions of ˆx and ŷ. Indd, straightforward computations yild (2.5) DF = [(1 ŷ) r 21 ŷ r 34, (1 ˆx) r 41 ˆx r 32 ] and (2.6) J =2 T 124 2( T 123 T 124 )ˆx 2( T 134 T 124 )ŷ, whr r ij = r i r j and T ijk is th ara of th triangl with vrtics r i, r j, and r k. Sinc is convx, th Jacobian J is always positiv, i.., J >. Lt l i and ˆl i, i =1, 2, 3, 4, b th dgs of and ê, rspctivly. Lt n i and ˆn i b th unit outward normal vctors to l i and ˆl i, rspctivly (s Figur 2.2). Similarly, lt τ i and ˆτ i b th unit tangntial vctors to l i and ˆl i, rspctivly. It is asy to s from (2.5) that for any dg l i, (2.7) n i = 1 l i J DF T ˆn i and τ i = 1 l i DF ˆτ i. Th radr is rfrrd to [6] for suitabl choics for th pair of finit lmnt spacs V h V and W h W. In this papr, w considr th lowst-ordr Raviart Thomas finit lmnt spacs RT [25, 2] dfind on th rfrnc lmnt ê as ˆV(ê) =P 1, (ê) P,1 (ê), Ŵ (ê) =P (ê), whr P 1, (or P,1 ) dnots th spac of polynomials linar in th ˆx (or ŷ) variabl and constant in th othr variabl, and P dnots th spac of constant functions. Th vlocity spac on any convx quadrilatral is dfind through th Piola transformation [6] 1 DF : L 2 (ê) L 2 (ê) L 2 () L 2 () T h. J Th RT spacs on T h ar givn by V h = {v V : v = J 1 DF ˆv F 1, ˆv ˆV(ê) T h }, (2.8) W h = {w W : w =ŵ F 1, ŵ Ŵ (ê) T h}. Two proprtis of Piola s transformation will b important in our analysis. For any ˆv ˆV(ê) and th rlatd v = J 1 DF ˆv F 1, (2.9) J div v = div ˆv and l i v n i = ˆv ˆn i.

5 1732 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV Not that sinc V h H(div; Ω), any vctor in V h has continuous normal componnts on th dgs. A function in W h is uniquly dtrmind by its valus at th cll-cntrs and a vctor in V h is uniquly dtrmind by its normal componnts on th dgs. Thrfor, dimw h = N p and dimv h = N, whr N p is th numbr of lmnts and N is th numbr of intrior dgs. Lt {ψi h}, i =1,N p, b a basis for W h such that { ψi h 1, i = j, (c j )=δ ij, i j, whr c j is th cntr of lmnt j, j =1,N p. Similarly, lt φ h i, i =1,N, b a basis for V h such that φ h i n j = δ ij, whr n j is a fixd unit normal vctor on dg l j, j =1,N. In ordr to simplify notation, w us th sam way for global and local indxing of msh dgs and corrsponding normal vctors. Givn th finit lmnt spacs V h and W h, w dfin th discrt problm: find (u h,p h ) V h W h such that (2.1) (K 1 u h, v h ) h (p h, div v h )=, (div u h,w h )=(f, w h ) (v h,w h ) V h W h, whr (, ) h is a continuous bilinar form corrsponding to th application of a numrical quadratur rul for computing (, ). A dtaild discussion of this quadratur rul is givn in sction MFD discrtizations. In this sction, w driv an MFD discrtization of (1.1) and show its connction with th MFE mthod (2.1). Th first stp in th mimtic tchniqu is to spcify discrt dgrs of frdom for prssur and vlocity. Th discrt prssur unknowns ar dfind at th cntrs of th quadrilatrals, on unknown pr msh cll. Th discrt vlocitis ar dfind at th midpoints of msh dgs as normal componnts. In othr words, an dg-basd unknown is a scalar and rprsnts th orthogonal projction of a vlocity vctor onto th unit vctor n i normal to th msh dg l i. Th scond stp in th mimtic tchniqu is to quip th spacs of discrt prssurs and vlocitis with scalar products. W dnot th vctor spac of cll-cntrd prssurs by Q d. Th dimnsion of Q d quals th numbr of msh clls N p. Th scalar product on th vctor spac Q d is givn by (3.1) N p [p d,q d ] Q d = i p d i qi d p d,q d Q d, i=1 whr i dnots th ara of cll i and p d i, qd i ar cll-cntrd prssur componnts. It is asy to s that th vctor spac Q d is isomtric to th MFE spac W h in (2.8). Indd, for any p h W h, thr xists a uniqu p d =(p d 1,p d 2,...,p d N p ) T Q d such that p h = N p i=1 pd i ψh i and (p h,q h )=[p d,q d ] Q d. Not that th discrt MFD prssur variabl, p d i, corrsponds to th valu of th MFE prssur function at th cll-cntr, p h (c i ).

6 SUPERCONVERGENCE OF THE VELOCITY 1733 W dnot th vctor spac of dg-basd vlocitis by X d. Th dimnsion of X d quals th numbr of intrior msh dgs N. Th scalar product on X d is givn by (3.2) [u d, v d ] X d = [u d, v d ] X,, d T h whr [u d, v d ] X d, is a scalar product ovr cll involving only th normal vlocity componnts on cll dgs. Rcall that a vlocity vctor can b rcovrd from two orthogonal projctions on any two noncollinar vctors. Sinc th msh cll is convx, any pair of normal vctors to dgs with a common point satisfis th abov rquirmnt. Th orthogonal projctions ar xactly th dgrs of frdom associatd with cll dgs. As shown in Figur 3.1, four rcovrd vlocity vctors can b associatd with th four vrtics of th quadrilatral. For xampl, vlocity v 1 is rcovrd from its projctions onto th normal vctors n 1 and n 2. For a gnral quadrilatral, w dnot by v d (r j ) th vlocity rcovrd at jth vrtx r j, j =1, 2, 3, 4. Thn, th cll-basd scalar product is givn by (3.3) [u d, v d ] X d, = 1 4 T j K 1 (r j )u d (r j ) v d (r j ), 2 j=1 whr T j is th ara of th triangl with vrtics r j 1, r j, and r j1 (s Figurs 2.2 and 3.1). For xampl, triangls T 1 and T 4 ar th shadd triangls in Figur 3.1. Not that (3.3) is indd an innr product, sinc K is a symmtric and positiv dfinit tnsor and (3.4) [v d, v d ] X d C v d 2, whr is th Euclidan vctor norm. n 4 v 4 T 4 T 1 n 1 n 1 n 2 v 1 Fig Rcovrd vctors v 1, v 4 and triangls T 1, T 4. Th vctor spac X d is isomorphic to th MFE spac V h in (2.8), sinc both spacs hav th sam dfinitions of dgrs of frdom. In particular, for any v h V h, thr xists a uniqu v d =(v1,v d 2,...,v d N d )T X d such that v h = N i=1 vd i φh i. Not that th discrt MFD vlocity variabl, vi d, corrsponds to th MFE normal vlocity componnt, v h n i, on dg l i. Th third stp in th mimtic tchniqu is to driv a discrt approximation to th divrgnc oprator, DIV : X d Q d, which w shall rfr to as th prim

7 1734 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV oprator. For a cll, th Gauss divrgnc thorm givs DIV u d = 1 ( u d (3.5) 1 l 1 u d 2 l 2 u d 3 l 3 u d 4 l 4 ), whr u d 1,...,u d 4 ar th normal vlocity componnts on lmnt and th normal vctors ar orintd as shown in Figur 2.2. Th fourth stp in th mimtic tchniqu is to driv a discrt flux oprator G (for th continuous oprator Kgrad ) adjoint to th discrt divrgnc oprator DIV with rspct to scalar products (3.1) and (3.2), i.., [DIV u d,p d ] Q d [u d, Gp d ] X d u d X d p d Q d. To driv th xplicit formula for G, w considr an auxiliary scalar product, and rlat it to scalar products (3.1) and (3.2). Dnot by, th standard vctor dot product. Thn [p d,q d ] Q d = Dp d,q d and [u d, v d ] X d = Mu d, v d, whr D is a diagonal matrix, D = diag{ 1,..., Np }, and M is a spars symmtric mass matrix with a 5-point stncil. Rstrictd to a cll, this stncil conncts dg-basd unknowns if and only if th corrsponding dgs hav a common point. Combining th last two formula, w gt [u d, DIV p d ] X d = u d, MDIV p d =[DIV u d,p d ] Q d = u d, DIV t D p d u d X d p d Q d, whr DIV t Thrfor, (3.6) is th adjoint of DIV with rspct to th auxiliary scalar product. G = M 1 DIV t D. Th MFD mthod approximating first-ordr systm (1.1) may b summarizd as follows: (3.7) u d = G p d, DIV u d = f d, whr f d =(f d 1,...,f d N p ) t, and ntry f d i is th intgral avrag of f ovr cll i. Th basic tool for th rror analysis of th discrt solution (u d,p d ) X d Q d is basd on th following transformation. Multiplying th first quation in (3.7) by Mv d and th scond on by Dq d,wgt (3.8) [u d, v d ] X d [p d, DIV v d ] Q d =, [q d, DIV u d ] Q d =[f d,q d ] Q d (v d,q d ) X d Q d. Using th isomorphism btwn th finit lmnt spac V h W h and th vctor spac X d Q d, w dfin finit lmnt functions p h, q h, f h, u h, and v h corrsponding to vctors p d, q d, f d, u d, and v d, rspctivly. Thn [p d, DIV v d ] Q d =(p h, div v h ) and [q d, DIV u d ] Q d =(q h, div u h ). Th dfinition of f d implis that [f d,q d ] Q d =(f h,q h )=(f, q h ).

8 SUPERCONVERGENCE OF THE VELOCITY 1735 Finally, by introducing th quadratur rul (3.9) (K 1 u h, v h ) h [u d, v d ] X d, w rduc problm (3.7) to th finit lmnt problm (2.1). Th scalar product in th spac of vlocitis givn by (3.3) is obviously not uniqu. In th contxt of MFE mthods, it is a quadratur rul for numrical intgration of (K 1 u h, v h ): (K 1 u h, v h ) h, = 1 4 (3.1) T j K 1 (r j )u h (r j ) v h (r j ), 2 j=1 whr u h (r j ) is th rcovrd vlocity at vrtx r j. In th contxt of MFE mthods, w shall rfr to (3.1) as th MFD quadratur rul. Th global scalar product is obtaind by summing ovr quadrilatrals, i.., (3.11) (K 1 u h, v h ) h = (K 1 u h, v h ) h,. T h Not that (3.4) implis that thr xists a constant C > such that (3.12) (K 1 v h, v h ) h C v h 2 v h V h. It was shown in [4] that th lmnt quadratur rul (3.1) is xact for any constant vctor u h, constant tnsor K, and v h V h. 4. Suprconvrgnc stimats for th vlocity. W bgin by rcalling th mixd projction oprator Π : H 1 (Ω) H 1 (Ω) V h satisfying (4.1) (div (Π v v), w)= w W h. Th oprator Π is dfind locally on ach lmnt by Πv = ˆΠˆv, whr ˆΠ :H 1 (ê) H 1 (ê) ˆV(ê) is th rfrnc lmnt projction oprator satisfying (4.2) (ˆΠˆv ˆv) ˆn i =, ˆl i i =1, 2, 3, 4. Th approximation proprtis of Π hav bn stablishd in [25, 26]: (4.3) Πv div C v 1, (4.4) Πv v Ch v 1, (4.5) div (Πv v) Ch v 2. Th following lmma givs svral approximation proprtis of ˆΠ which will b usd in th analysis. Lmma 4.1. Th oprator ˆΠ dfind in (4.2) satisfis, for any ˆv =(ˆv 1, ˆv 2 ) in H 1 (ê) H 1 (ê), th following: ê ˆx (ˆΠˆv (4.6) ˆv) 1 dˆxdŷ =, ê ŷ (ˆΠˆv ˆv) 2 dˆxdŷ =, ˆx 1 ê C ˆx 1 ê, ŷ 2 ê C ŷ (4.7) 2 ê, (4.8) ˆΠˆv 1,ê C ˆv 1,ê.

9 1736 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV Proof. Th idntitis in (4.6) follow asily from dfinition (4.2). In particular, writing (4.2) for th two vrtical dgs givs 1 (ˆΠˆv ˆv) 1 (, ŷ) dŷ =, 1 (ˆΠˆv ˆv) 1 (1, ŷ) dŷ =. Subtracting th abov quations and applying th fundamntal thorm of calculus implis th first idntity in (4.6). Th proof of th scond idntity is similar. Not that (4.6) mans that ˆx (ˆΠˆv) 1 and ŷ (ˆΠˆv) 2 ar th L 2 -orthogonal projctions of ˆx ˆv 1 and ŷ ˆv 2, rspctivly, onto th spac of constants, which implis (4.7). Finally, it is asy to s that (4.2) implis ˆΠˆv ê C ˆv 1,ê, which, combind with (4.7), givs (4.8). W also mak us of th L 2 -projction oprator P h : W W h such that for p W, (4.9) (P h p p, w) = w W h. Dnot th quadratur rror by (4.1) σ(q, v) (q, v) (q, v) h. Th variational formulation (2.1) and th discrt problm (2.1) giv ris to th rror quations (4.11) (K 1 (Πu u h ), v h ) h =(P h p p h, div v h ) (K 1 (Πu u), v h ) σ(k 1 Πu, v h ), (div (Πu u h ),w h )=, whr w usd (4.9) and (4.1) in th first and scond quations, rspctivly. W not that, using (2.9), th scond quation in (4.11) givs = (div (Πu u h ),w h ) =( div (ˆΠû û h ), ŵ h )ê w h W h. Sinc div ˆV h = Ŵh, taking ŵ h = div (ˆΠû û h ) implis that div (ˆΠû û h )=and thrfor, by (2.9), (4.12) div (Πu u h )=. Taking v h =Πu u h V h and w h = P h p p h in (4.11) givs (4.13) (K 1 (Πu u h ), Πu u h ) h =(K 1 (Πu u), Πu u h ) σ(k 1 Πu, Πu u h ). Th stimat for th first trm on th right-hand sid of (4.13) follows from Thorm 5.1 in [13] and (4.12): (4.14) (K 1 (Πu u), Πu u h ) Ch 2 ( u 2 Πu u h u 1 div (Πu u h ) ) = Ch 2 u 2 Πu u h.

10 SUPERCONVERGENCE OF THE VELOCITY 1737 Th scond trm on th right-hand sid of (4.13) can b boundd using Lmma 4.3: (4.15) σ(k 1 Πu, Πu u h ) Ch 2 u 2 Πu u h. Combining (4.14), (4.15), and (3.12), w obtain th following suprconvrgnc rsult. Thorm 4.2. Lt K 1 W 2, (Ω). For th vlocity u h of th MFE mthod (2.1), on h 2 -uniform quadrilatral grids, thr xists a positiv constant C indpndnt of h such that (4.16) Πu u h Ch 2 u 2. W now procd to prov stimat (4.15). Lmma 4.3. Lt v V h, and lt K 1 W 2, (Ω). Thr xists a positiv constant C indpndnt of h such that (4.17) σ(k 1 Πu, v) Ch 2 ( u 2 v u 1 div v ). Proof. For an lmnt T h, w dfin th rror (4.18) σ (K 1 Πu, v) = K 1 Πu v dx (K 1 Πu, v) h,. With (3.1), th scond trm on th right-hand sid of (4.18) can b writtn as (4.19) (K 1 Πu, v) h, = 1 2 = 1 2 = 1 2 = T j K 1 (r j )Πu(r j ) v(r j ) j=1 4 j=1 j=1 ( ) ( ) T j ˆK (ˆr j ) DF ˆΠû (ˆr j ) DF ˆv (ˆr j ) J J 4 T j 1 J (ˆr j ) J (ˆr j ) DFT (ˆr j ) ˆK 1 (ˆr j )DF (ˆr j ) ˆΠû(ˆr j ) ˆv(ˆr j ) 4 B (ˆr j ) ˆΠû(ˆr j ) ˆv(ˆr j ) j=1 (B ˆΠû, ˆv)T, whr th subscript T dnots th trapzoidal rul on lmnt ê and w dfin B = 1 J DF T ˆK 1 T DF. Hr w usd (2.6) to conclud that j J = 1 (ˆr j) 2. Considring th first trm on th right-hand sid of (4.18), w obtain K 1 Πu v dx = ˆK DF ˆΠû DF ˆvJ d ˆx ê J J 1 (4.2) = DF T ê J ˆK 1 DF ˆΠû ˆv d ˆx = B ˆΠû ˆv d ˆx. ê Substituting (4.19) and (4.2) into (4.18), w obtain σ (K 1 ( Πu, v) = B ˆΠû ˆv d ˆx B ˆΠû, ˆv )T σ ) (4.21) ê( B ˆΠû, ˆv. ê

11 1738 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV Hraftr w shall omit th subscripts and ê. Lt E(f) f(ˆx, ŷ)dˆxdŷ (f) T ê b th rror of th trapzoidal rul for intgrating a function f(ˆx, ŷ) onê. Thn, (4.22) σ(b ˆΠû, ˆv) =E ( (B ˆΠû) 1ˆv 1 ) E ( (B ˆΠû)2ˆv 2 ). W nxt bound th first trm on th right-hand sid in (4.22). Th argumnt for th bound on th scond trm is similar. Using th trapzoidal rul rror rprsntation from Lmma A.1 basd on th Pano krnl thorm (s [23, Thorm 5.2-3, p. 142]), w writ (4.23) E((B ˆΠû) 1ˆv 1 )= φ(ˆx) 2 ( ) (B ˆΠû)1ˆv ˆx 2 1 (ˆx, ) dˆxdŷ φ(ŷ) 2 ŷ 2 (B ˆΠû) 1 (, ŷ)ˆv 1 (, ŷ) dˆxdŷ ψ(ˆx, ŷ) 2 ˆxŷ (I)(II)(III), ( (B ˆΠû)1ˆv 1 ) (ˆx, ŷ) dˆxdŷ whr φ(t) =t(t 1)/2 and ψ(s, t) =(1 s)(1 t) 1/4. Dnot by B 11, B 12, B 21, and B 22 th componnts of th tnsor B. Sinc ˆv 1 (, ŷ) is constant in ŷ, th scond trm in (4.23) is (4.24) (II)= 2 φ(ŷ) 2 ŷ 2 B 11(, ŷ)(ˆπû) 1 (, ŷ)ˆv 1 (, ŷ) dˆxdŷ φ(ŷ) 2 ŷ 2 B 12(, ŷ)(ˆπû) 2 (, ŷ)ˆv 1 (, ŷ) dˆxdŷ (II) 1 (II) 2 (II) 3. φ(ŷ) ŷ B 12(, ŷ) ŷ (ˆΠû) 2 (, ŷ)ˆv 1 (, ŷ) dˆxdŷ Using (4.8), for th first two trms on th right-hand sid, w hav (II) 1 (II) 2 C B 2,,ê û 1,ê ˆv 1 ê. Sinc ŷ (ˆΠû) 2 is a constant, w rwrit th last trm in (4.24) as (II) 3 =2 φ(ŷ) ŷ B 12(, ŷ) ŷ (ˆΠû) 2 (ˆx, ŷ)ˆv 1 (, ŷ) dˆxdŷ C B 1,,ê ŷ (ˆΠû) 2 ê ˆv 1 ê C B 1,,ê û 1,ê ˆv 1 ê, using (4.7). A combination of th last two bounds implis that (4.25) (II) C ( B 2,,ê û 1,ê B 1,,ê û 1,ê ) ˆv1 ê.

12 (4.26) SUPERCONVERGENCE OF THE VELOCITY 1739 For th last trm in (4.23), sinc ˆv 1 (ˆx, ŷ) is constant in ŷ, whav (III)= Using (4.7), ψ(ˆx, ŷ) 2 ˆxŷ (B ˆΠû) 1 (ˆx, ŷ)ˆv 1 (ˆx, ŷ) dˆxdŷ ψ(ˆx, ŷ) ŷ (B ˆΠû) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ (III) 1 (III) 2. (4.27) (III) 1 C( B 2,,ê û 1,ê B 1,,ê û 1,ê ) ˆv 1 ê. To bound (III) 2 w not that ˆx ˆv 1(ˆx, ŷ) is a constant and 1 ψ(ˆx, ŷ) dˆxdŷ =. Thrfor, by th Brambl Hilbrt lmma [5], and using (4.7), (III) 2 C ŷ (B ˆΠû) (4.28) 1 ˆv 1 ê C( B 2,,ê û 1,ê B 1,,ê û 1,ê ) ˆv 1 ê. 1,ê (4.29) Th first trm in th rror rprsntation (4.23) is (I) = 2 φ(ˆx) 2 ˆx 2 (B ˆΠû) 1 (ˆx, ) ˆv 1 (ˆx, ) dˆxdŷ φ(ˆx) ˆx (B ˆΠû) 1 (ˆx, ) ˆx ˆv 1(ˆx, ) dˆxdŷ =(I) 1 (I) 2. Th first trm on th right-hand sid can b boundd in a way similar to (II): 1 (4.3) (I) 1 C( B 2,,ê û 1,ê B 1,,ê û 1,ê ) ˆv 1 ê. W rwrit th scond trm on th right-hand sid in (4.29) as (4.31) 1 2 (I) 2 = ( φ(ˆx) ˆx (B ˆΠû) 1 (ˆx, ) ˆx (B ˆΠû) 1 (ˆx, ŷ) ) ˆx ˆv 1(ˆx, ) dˆxdŷ φ(ˆx) ˆx (B ˆΠû) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ) dˆxdŷ (I) 2,1 (I) 2,2. To stimat th first trm in (4.31), w writ ˆx (B ˆΠû) 1 (ˆx, ŷ) ŷ ˆx (B ˆΠû) 2 1 (ˆx, ) = ˆxŷ (B ˆΠû) 1 (ˆx, ˆt) dˆt. This allows us to bound th first trm in (4.31) in a way similar to bounds (4.25) and (4.3): (4.32) (I) 2,1 C( B 2,,ê û 1,ê B 1,,ê û 1,ê ) ˆv 1 ê. Th scond trm on th right-hand sid in (4.31) can b rwrittn as (4.33) (I) 2,2 = φ(ˆx) ˆx (B(ˆΠû û)) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ φ(ˆx) ˆx (Bû) 1(ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ (I) 2,2,1 (I) 2,2,2,

13 174 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV whr w usd that ˆx ˆv 1(ˆx, ) = ˆx ˆv 1(ˆx, ŷ) on, sinc ˆv 1 is a constant in ŷ. To stimat th scond trm in (4.33), w us th idntity (4.34) ˆx ˆv 1 = ŷ ˆv 2 div ˆv. W rwrit (I) 2,2,2 as (4.35) (I) 2,2,2 = = ˆl 1 φ(ˆx) ˆx (Bû) 1 ŷ ˆv 2 dˆxdŷ φ(ˆx) ˆl 3 ˆx (Bû) 1 ˆv 2 dˆx φ(ˆx) ˆx (Bû) 1 div ˆv dˆxdŷ. Clarly, th last two trms can b boundd by φ(ˆx) φ(ˆx) ˆx (Bû) 1 div ˆv dˆxdŷ 2 ˆxŷ (Bû) 1 ˆv 2 dˆxdŷ (4.36) C( (Bû) 1 2,ê ˆv 2 ê (Bû) 1 1,ê div ˆv ê). W postpon th stimat of th dg intgrals in (4.35) for latr. To bound th first trm on th right-hand sid in (4.33), w hav (4.37) (I) 2,2,1 = φ(ˆx) ˆx B 11(ˆx, ŷ)(ˆπû û) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ φ(ˆx)b 11 (ˆx, ŷ) ˆx (ˆΠû û) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ φ(ˆx) ˆx B 12(ˆx, ŷ)(ˆπû û) 2 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ φ(ˆx)b 12 (ˆx, ŷ) ˆx (ˆΠû û) 2 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ (I) 2,2,1,1 (I) 2,2,1,2 (I) 2,2,1,3 (I) 2,2,1,4. Sinc ˆΠû is xact for constants, using th Brambl Hilbrt lmma and th invrs inquality, w can bound th first and th third trms in (4.37) as (4.38) (I) 2,2,1,1 (I) 2,2,1,3 C B 1,,ê û 1,ê ˆv 1 ê. For th scond trm in (4.37), a Taylor xpansion of B 11 (ˆx, ŷ ) ê givs about any fixd point (4.39) whr (4.4) (I) 2,2,1,2 = φ(ˆx)b 11 (ˆx, ŷ ) ˆx (ˆΠû û) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ R, R C B 1,,ê û 1,ê ˆv 1 ê, using (4.7) for th last inquality. To bound th first trm on th right-hand sid in (4.39), w not that (φ 2 ) (ˆx) =6φ(ˆx) 1 2, (φ2 ) ()=(φ 2 ) (1)=.

14 SUPERCONVERGENCE OF THE VELOCITY 1741 Thrfor, using (4.6), w hav φ(ˆx)b 11 (ˆx, ŷ ) ˆx (ˆΠû û) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ = ˆx 2 (φ2 )(ˆx)B 11 (ˆx, ŷ ) ˆx (ˆΠû û) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ (4.41) = 1 6 ˆx (φ2 )(ˆx)B 11 (ˆx, ŷ ) 2 ˆx 2 (ˆΠû û) 1 (ˆx, ŷ) ˆx ˆv 1(ˆx, ŷ) dˆxdŷ C B,ê û 2,ê ˆv 1 ê. A combination of (4.39) (4.41) givs (4.42) (I) 2,2,1,2 C( B 1,,ê û 1,ê B,ê û 2,ê ) ˆv 1 ê. To complt th stimat of (I) 2,2,1, it rmains to bound (I) 2,2,1,4. Using that ˆx (ˆΠû) 2 = and (4.34), w hav (4.43) (I) 2,2,1,4 = = φ(ˆx)b 12 ˆxû2 ŷ ˆv 2 dˆxdŷ 1 φ(ˆx) B 12 ˆv 2 dˆx ˆl ˆxû2 1 1 φ(ˆx)b div 12 ˆv dˆxdŷ. ˆxû2 ˆl 3 1 Th last two trms abov ar boundd by (4.44) 1 φ(ˆx)b 12 φ(ˆx) ŷ div ˆv dˆxdŷ ( ˆxû2 ) B 12 ˆv 2 dˆxdŷ ˆxû2 C[( B 1,,ê û 1,ê B,ê û 2,ê ) ˆv 2 ê B,ê û 1,ê div ˆv ê]. Combining (4.23) (4.44), w obtain (4.45) whr (4.46) (4.47) E ( (B ˆΠû) 1ˆv 1 ) = T1 T 2 T 3, T 1 C[( B 2,,ê û 1,ê B 1,,ê û 1,ê B,ê û 2,ê ) ˆv ê ( B 1,,ê û ê B,ê û 1,ê ) div ˆv ê], T 2 = φ(ˆx) ˆl 1 ˆl 3 ˆx (Bû) 1(ˆx, ŷ)ˆv 2 (ˆx, ŷ) dˆx, and T 3 = φ(ˆx) B 12 (ˆx, ŷ) ŷ)ˆv 2 (ˆx, ŷ) dˆx. ˆl 3 ˆl 1 ˆxû2(ˆx, Using Lmma 4.4 blow, T 1 can b boundd as follows: (4.48) T 1 C [( h 2 K 1 2,, u 1, h K 1 1,, h u 1, K 1, h 2 ) u 2, ˆv ê ( h K 1 1,, u K 1 ) ], h u 1, h div v Ch 2 ( K 1 2,, u 2, v K 1 1,, u 1, div v ),

15 1742 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV using th fact that û j,ê Ch j u j, and div ˆv ê Ch div v (s [13, Lmma 5.5]). Th trm T 2 is tratd in Lmma 4.5 blow. Finally, trm T 3 in (4.45) can b rwrittn as T 3 = φ(ŝ) B 12 (ŝ, ŷ) ˆl 3 ˆl 1 ŝ (û ˆn k)ˆv ˆn k dŝ. A similar trm appars in th proof of Thorm 5.1 in [13]. Following th argumnt thr, it can b shown that (4.49) T 3 C h 2 u 2, v. A combination of stimats (4.45), (4.48), (4.51), and (4.49) implis that E ( (B ˆΠû) ) 1ˆv 1 Ch 2 ( u 2 v u 1 div v ). Th argumnt for E((B ˆΠû) 2ˆv 2 ) is analogous. This complts th proof of th lmma. W nxt giv th proofs of th two auxiliary lmmas usd in th abov argumnt. Lmma 4.4. If K 1 W 2, (Ω), thn for all T h thr xists a positiv constant C indpndnt of h such that B s,,ê Ch s K 1 s,,, s =, 1, 2. Proof. First, for a quasi-uniform msh, w hav c 1 h DF,ê c 2 h, c 3 h 2 J,ê c 4 h 2 with som positiv constants c 1 c 4. This implis that (4.5) B,ê C ˆK 1,ê. Scond, for an h 2 -uniform msh, w hav additional stimats. Lt α =(α 1,α 2 ), α i, b a doubl indx, and lt α = α 1 α 2. In th cas α = 1, th dfinition of th bilinar mapping (2.4) (2.6) and (2.2) imply that ˆ α DF,ê Ch 2 and ˆ α 1 J DF,ê C. In th cas α = 2, w hav th stimats ˆ α DF,ê = and ˆ α 1 J DF,ê Ch. As a rsult, w gt ˆ α B,ê C (h ˆK 1,ê ˆ α ˆK 1,ê ) for α = 1 and ˆ α B,ê C (h 2 ˆK 1,ê h ˆ α 1 ˆK 1,ê ˆ α ˆK 1,ê )

16 SUPERCONVERGENCE OF THE VELOCITY 1743 for α = 2. Sinc ˆK 1 = K 1 F, using th chain rul and ˆ α F,ê Ch α for α 2, w obtain which implis ˆ α ˆK 1,ê Ch α K 1 α,,, α =, 1, 2, ˆ α B,ê Ch α K 1 α,,, α =, 1, 2, complting th proof. Lmma 4.5. If K 1 W 2, (Ω), thn (4.51) T 2 =, whr T 2 is dfind in (4.47). Proof. Summing ovr all lmnts in (4.47), w hav T 2 = (4.52) φ(ŝ) ˆl k ŝ ((Bû) ˆτ k)ˆv ˆn k dŝ. k=1,3 Using (2.7), w hav that for any dg l, (Bû) ˆτ = 1 J DFT ˆK 1 DFû l DF 1 τ = l (K 1 u) τ. Thrfor, using (2.9), th sum in (4.52) bcoms T 2 = l k 2 φ(s) ( (K 1 ) (4.53) u) τ k v nk ds. l k s k=1,3 Sinc v V h, v n = on xtrior dgs and v n is continuous across intrior dgs. Th assumd rgularity for K and u implis that K 1 u and s (K 1 u) ar continuous across intrior dgs. Not that ach intrior dg l appars twic in th sum in (4.53), which now can b rwrittn as a sum of intrior dg intgrals T 2 = l 2 φ(s) l l s ((K 1 u) τ )[v n] ds =, whr [v n] dnots th jump in th normal componnt of v. 5. Suprconvrgnc to th avrag dg fluxs and at th dg midpoints. W now discuss how th suprconvrgnc rsult from sction 4 can b applid to obtain suprconvrgnc for th computd vlocity to th avrag dg fluxs and at th midpoints of th dgs. Dfin, for any v (H 1 (Ω)) 2, (5.1) (5.2) 4 ( ) 2 T h, v 2 = v n k ds, k=1 l k v 2 = v 2. T h

17 1744 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV Using th wll-known proprty of th Piola transformation [6], (5.3) v n ds = ˆv ˆn dŝ v (H 1 (Ω)) 2, l ˆl and transforming to th rfrnc lmnt and back, it is asy to s that is a norm on V h and thr xist constants c 1 and c 2 indpndnt of h such that c 1 v v c 2 v v V h. It is clar from (4.2) and (5.3) that Πv v = for any v (H 1 (Ω)) 2. Thrfor, (5.4) u u h Πu u h c 2 Πu u h Ch 2 u 2, using Thorm 4.2. This implis dgwis suprconvrgnc of th computd vlocity u h n to 1 l l u n ds in a discrt L2 -sns. Rmark 5.1. Th suprconvrgnc rsult (5.4) implis similar suprconvrgnc for u u h M with v 2 M = T h k=1 4 l k 2 (v n k ) 2 (m k ), whr m k is th midpoint of l k. Our choic of rporting th rsults in is motivatd by th fact that avrag fluxs ar asir to masur than pointwis valus and thrfor ar of gratr practical intrst. 6. Numrical xprimnts. In this sction, w prsnt th dtails of th numrical implmntation. Instad of solving saddl point problm (2.1), w rduc it to an quivalnt systm with a symmtric positiv dfinit matrix using th standard hybridization tchniqu. Lt V h b th rstriction of V h to quadrilatral and Λ h l b th spac of constant functions ovr dg l. Dfin Ṽ h = V h and Λ h = l Λ h l. Not that th normal componnt of v h V h is continuous across intrior msh dgs and v h n = on xtrior dgs. Thrfor, { } V h = ṽ h Ṽh : (μ h, ṽ h n ) = μ h Λ h, whr n is th outward normal vctor for quadrilatral. It has bn shown by many authors (s,.g., [6]) that th original formulation (2.1) is quivalnt to th mixd-hybrid formulation: find (ũ h,p h,λ h ) Ṽh W h Λ h such that (6.1) (K 1 ũ h, ṽ h ) h, (p h, div ṽ h ) (λ h, ṽ h n ) = ṽ h Ṽh, (div ũ h,w h ) =(f, w h ) w h W h, (μ h, ũ h n ) = μ h Λ h.

18 SUPERCONVERGENCE OF THE VELOCITY 1745 Systm (6.1) can b writtn in th matrix form as M B T C T u (6.2) B p = C λ f, whr ( M B T D = B ) is a block-diagonal matrix (aftr a prmutation of columns and rows) with as many blocks as msh lmnts. Each block is a 5 5 matrix. Thrfor, vctors u and p can b xplicitly liminatd from (6.2) rsulting in a systm (6.3) Sλ= b, whr S is a spars symmtric positiv dfinit matrix. For logically rctangular mshs, S has at most svn nonzro lmnts in ach row and column. Its nonzro ntris rprsnt connctions btwn dg-basd unknowns blonging to th sam cll. Problm (6.3) was solvd with th prconditiond conjugat gradint (PCG) mthod. In th numrical xprimnts, w usd on V-cycl of th algbraic multigrid mthod [24] as a prconditionr. Th stopping critrion for th PCG mthod was th rlativ dcras in th norm of th rsidual by a factor of W solvd th boundary problm (1.1) with a known analytic solution p(x, y) =x 3 y 2 x cos(xy) sin(x) and tnsor cofficint K(x, y) = ( ) (x 1) 2 y 2 xy xy (x 1) 2. It is prtinnt to not hr that th suprconvrgnc rsult stablishd in th prvious sction for th homognous Numann boundary condition can b xtndd to th cas of gnral Numann boundary valu problm. In xampl 1, th computational domain Ω is th unit squar. Th computational grid is constructd from a uniform rctangular grid via th mapping x(ξ,η) = ξ.6 sin(2πη) sin(2πξ), y(ξ,η) = η.6 sin(2πη) sin(2πξ), whr <η,ξ<1, and subsqunt random distortion of msh nod positions (s Figur 6.1). Th maximum valu of th distortion is proportional to th squar of th local msh siz; i.., th rsulting grid satisfis assumptions (2.2) and (2.3). W tst both Numann and Dirichlt boundary conditions. Th rsults for th Numann problm ar shown in Tabl 6.1. Th convrgnc rats wr computd using th linar rgrssion for th data in th rows for 1/h =32, 64, 128, 256. In addition to norm (5.2), w show th convrgnc rat in th discrt L -norm: u u h = max 1 l k u n k ds u h n k l k, l k

19 1746 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV Fig Exampls of mshs usd in numrical xprimnts. Tabl 6.1 Convrgnc rats for xampl 1: Numann boundary conditions. 1/h u u h u u h p p h p p h Rat whr th maximum is takn ovr all msh dgs. Th convrgnc rats for th prssur variabl ar shown in th following discrt norms: p p h 2 = p(c i ) p h (c i ) 2 i i T h and p p h = max i T h p(c i ) p h (c i ), whr c i is th gomtric cntr of lmnt i. Th us of th gomtric cntr instad of th mass cntr is du to th following proprty of th MFD mthod. Th mthod is xact for linar solutions whn th prssur variabl, p(c i ), is valuatd at th gomtric cntr c i [15]. Th scond-ordr convrgnc rat is obsrvd for both th prssur and vlocity variabls in th discrt L 2 - and L -norms. In th cas of Dirichlt boundary conditions, a loss of on half ordr in th convrgnc rat for th vlocity in th L 2 -norm is xpctd (s,.g., [12, 3]). Th convrgnc rats ar shown in Tabl 6.2. Not that th vlocity convrgnc rat in th L 2 -norm is largr than th thortical bound of O(h 1.5 ). Howvr, th convrgnc rat in th L -norm is only O(h). In xampl 2, th computational domain Ω consists of thr quadrilatrals (s Figur 6.1). A squnc of grids is obtaind by uniform rfinmnt of ths quadrilatrals. Th lft bottom cornr of th domain is locatd at th point (1, ). Th rsults

20 SUPERCONVERGENCE OF THE VELOCITY 1747 Tabl 6.2 Convrgnc rats for xampl 1: Dirichlt boundary conditions. 1/h u u h u u h p p h p p h Rat Tabl 6.3 Convrgnc rats for xampl 2: Numann boundary conditions. 1/h u u h u u h p p h p p h Rat of our numrical xprimnts ar shown in Tabl 6.3. W raliz that th grid is only locally h 2 -uniform. Howvr, th scond-ordr convrgnc rat for th vlocity variabl in th L 2 norm is attaind. 7. Conclusion. W hav provd th suprconvrgnc stimat for th vlocity variabl on h 2 -uniform quadrilatral grids whn th xact intgration of vlocitis is rplacd by a novl 4-point quadratur rul. Th thortical rsults for th full diffusion tnsor hav bn confirmd with numrical xprimnts. Appndix. Rprsntation of th trapzoidal rul rror. Lmma A.1. Lt f(x, y) b a function dfind on a rctangular domain [a, b] [c, d]. Th trapzoidal rul rror can b rprsntd as b E(f) =(d c) (b a) b d E(f) a d Proof. Dfin a function a c b d a c f(x, y) dx dy (f) T 2 (x a)(x b) f(x, c) dx 2 x2 c (y c)(y d) 2 ( (x b)(y d) g k (x, s) (x s) k 2 f(a, y) dy y2 ) 2 (b a)(d c) 4 { (x s) k, x s,, x<s, f(x, y) dx dy. xy

21 1748 BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV whr k. Th Pano krnl thorm (s [23, Thorm 5.2-3, p. 142]) stats that th rror of th trapzoidal rul is givn by (A.1) E(f) = b a d A 2, (s)f (2,) (s, c) ds c b d a c A,2 (t)f (,2) (a, t) dt A 1,1 (s, t)f (1,1) (s, t) ds dt, whr f (i,j) (x, y) = ij x i y f(x, y) for i, j and j A 2, (s) =E(g 1 (x, s)), A,2 (t) =E(g 1 (y, t)), A 1,1 (s, t) =E(g (x, s)g (y, t)). Straightforward calculations giv (A.2) Similarly, w gt (A.3) A 2, (s) = A,2 (t) =(b a) b d a c =(d c) =(d c) (t c)(t d) 2 g 1 (s, x) dx dy ( b s (x s) dx b a 2 (s a)(s b). 2 (b a)(d c) 4 4 g(x j,s) j=1 (g(a, s) g(b, s)) and A 1,1 (s, t) =(s b)(t d) A substitution of (A.2) and (A.3) into (A.1) complts th proof. ) (b a)(d c). 4 REFERENCES [1] T. Arbogast, L. C. Cowsar, M. F. Whlr, and I. Yotov, Mixd finit lmnt mthods on nonmatching multiblock grids, SIAM J. Numr. Anal., 37 (2), pp [2] T. Arbogast, C. N. Dawson, P. T. Knan, M. F. Whlr, and I. Yotov, Enhancd cll-cntrd finit diffrncs for lliptic quations on gnral gomtry, SIAM J. Sci. Comput., 19 (1998), pp [3] T. Arbogast, M. F. Whlr, and I. Yotov, Mixd finit lmnts for lliptic problms with tnsor cofficints as cll-cntrd finit diffrncs, SIAM J. Numr. Anal., 34 (1997), pp [4] M. Brndt, K. Lipnikov, J. D. Moulton, and M. Shashkov, Convrgnc of mimtic finit diffrnc discrtizations of th diffusion quation, J. Numr. Math., 9 (21), pp [5] D. Brass, Finit Elmnts: Thory, Fast Solvrs, and Applications in Solid Mchanics, Cambridg Univrsity Prss, Cambridg, UK, [6] F. Brzzi and M. Fortin, Mixd and Hybrid Finit Elmnt Mthods, Springr Sr. Comput. Math. 15, Springr-Vrlag, Nw York, [7] Z. Cai, J. E. Jons, S. F. McCormick, and T. F. Russll, Control-volum mixd finit lmnt mthods, Comput. Gosci., 1 (1997), pp [8] J. Campbll and M. Shashkov, A tnsor artificial viscosity using a mimtic finit diffrnc algorithm, J. Comput. Phys., 172 (21), pp [9] S.-H. Chou, D. Y. Kwak, and K. Y. Kim, A gnral framwork for constructing and analyzing mixd finit volum mthods on quadrilatral grids: Th ovrlapping covolum cas, SIAM J. Numr. Anal., 39 (21), pp

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