TICAM REPORT 96-02January 1996

The Application of Mixed Methods to SubsurfaceSimulation

Todd Arbogast, Clint N. Dawson, Philip T.Keenan, Mary F. Wheeler and Ivan Yotov

THE APPLICATION OF MIXED IVlETHODSTO SUBSUR.FACE SIMULATION

Todd Arbogast, Clint N. Dawson, Philip T. Keenan,Mary F. Wheeler, and lvan Yotov

Texas Institute for Computational and Applied MathematicsCenter for Subsurface Modeling, Taylor Hall 2.400

'rhe University of Texas at. Austin, Aust.in, Texas 78712, U.S.A.

SUMMARY

'vVeconsider the application of mixed finite element and finite difference methods togroundwater flow and transport problems. 'vVeare concerned with accurate approxima-tion and efficient implementation, especially when the porous medium may have geometricirregularities, heterogeneities, and either a tensor hydraulic conductivity or a tensor dis-persion. For single-phase How, we develop an expanded mixed finite element methoddefined on a logically rectangular, cmvilinear grid. Special quadratme rules are intro-duced to transform the method into a simple cell-centered finite difference method. Theapproximation is locally conservative and highly accmate. We also show that the highlynonlinear two-phase flow problem is well approximated by mixed methods. The maindifficulty is that the true solution is typically lacking in regularity.

INTRODUCTION

Our primary goal is to develop discretization methods that accurately and efficientlyapproximate the equations governing subsurface multi-phase flow and transport. 'vVecanjudge the accuracy of an approximation by many criteria. Asymptotic convergence resultstell us that we have an accurate solution when the mesh spacing h is small eno'ugh. Often,we cannot use as fine a mesh resolution as we would like, because of the computationaleffort needed to solve the equations. An equally important criterion to consider is theability of the numerical scheme to preserve important qualitative properties of the gov-erning equations so that physically meaningful results are obtained on a relatively coarsediscretization scale. The most important qualitat.ive property in subsurface simulation isconservation of mass. Mass should be conserved locally, that is, element-by-element.

Several additional physical phenomena need to be addressed by our numerical schemes.They should handle tensor permeabilities and dispersivities. Dispersivities are naturallytensors, and tensor permeabilities can arise from the use of homogenization or scale-uptechniques. Subsurface aquifers are irregularly shaped and contain layers with differingmaterial properties. Nonlinear effects are also prevalent especially in multi-phase How.

We present here some of our work on mixed finite element and finite difference meth-ods [9,6, 10, .5,8, 4, 7]. These methods are "mixed" in that they approximate directly bothpressure and velocity (in the flow problem), and they are asymptotically accurate and con-serve mass locally. The standard mixed finite element method was developed by Raviartand Thomas [19,21,12], aud we restrict our attention to their lowest-order method. It was

first used to solve subsurface problems by Douglas. Ewing, and 'vVlleeler [14]. a,lthoughRussell and Wheeler [20] pointed out that the often used cell-centered finite differencemet.hod on rectangular grids [18] for problems with diagona.l permeabilities is the lowest.order Raviart- Thornas mixed finite element method approximated by applying appropri-ate quadrature rules to some of the integrals.

A problem with mixed methods that we address is t.hat they can be difficult t.o im-plement directly, especially if the aquifer domain is 110t.rectangular or the penneabilit.yis a tensor. There has also been very little t.heoret.ica.l basis for concluding tlwt theapproximation of highly nonlinear multi-phase problems is accmate.

Om discretization schemes are based on an expanded mixed rinite element methodthat we define below. An approximation to this expanded mixed method reduces it tocell-centered finite differences; thus, it. is easy t.o implement. and has only one unknownp('r element. 'rhe elements can be deformed rectangles or bricks, alt.hough many of ourresul ts extend to triangles a.nd teLrahed ra [6].

FINITE ELEMENT APPROXIMATION OF SINCLE-PIIJ\SE FLOW

To illustra.te the numerical schemes, we begin by considering incompressible, singlepllase subsurface How on the aquifer domain n end, d = 2 or :3. \Ve solve for t.hepress1ll'e P and the velocity u satisfying

u = -K\7p. x E n.\7 . u = q, x E n,1) = Po, x E r)nD,

u· 1/ = g, x E anN,

(1 )

(2)

en(4 )

where K is the hydraulic conductivity tensor, q is a source term, 1/ is t.he outer unit normalvector to an, po gives a Dirichlet boundary condition on anD, and g gives a Neumanncondition. This is a second order elliptic equation.

Lowest order Raviart- Thomas spaces.Let L2(n) denote the space of square integrable functions, and let H(n; div) denote

t.he space of vector functions that. have a divergence; that is.

L2(n) = {w(x) : J olwl2 d.T < oo},H(n: div) = {v(x) : v E (L2(n))d and \7. v E U(n)}.

'vVesuppose that the domain n is partitioned into a finit.e number of non-overlappingelements or cells E of maximal diameter h. In the lowest order Raviart- Thomas mixedspaces [19, 16, 11], pressures can be approximated over element.s or on element faces (01'edges in 2-D). Element pressures are approximated in

2{;lfh = {tv: tV is constant on each elernent} C L (n),

<wei, on the Neumann pa.rt of the exterior elomain boundary, element face "Lagrangemult.iplier'· pressures [11] are approximated in

Ai: = {fl: fl is constant on each element fa.ce of anN} c [2(dnN).

The nodal degrees of freedom can be considered as the function values at the cent.ers oft.he elements or faces.

The velocity u is approximated in a space of vector valued functions V" such t.1)(\1,

V" c {v E U..2(O))" : v· 1/ is constant on eachelement. face and cont.inuous across elements} C 11(0; div).

On a, 2-D (01' J-D) rectangle E, t.his space of fnnctions is

V" IE = {v : Vi = (Ii + hi:l:i for some constants (Ii and hi, ; = 1, 2(, :3)},

where we use the standard Cartesian decomposition of the vectors x = (;rl,:1:2(,.1::3)) andv = ('111,'1'2(,03)): that is, the ith component of v is lineal' ill the ith coordinate directionand constant in the other direction(s). The importc1.llt fact is that v . 1/ is a constant.:therefore, the nodal degrees of freedom can be considered as t.he values of v . 1/ ;-1.1, t.hecenters of the element faces.

For a relatively general shaped clement E, assume that. there is a map F : E ~ I,,'front a rectangle or brickE (,0 g F'ollowing Thomas [21], we lise t.he Piola tranSfOl'll1 t.odefine V"IE from V"le; this transform preserves normal [Il1xes in an average sense (i.e ..it. is locally mass conservative). Let. the .Jacobian matrix he DF = (uFi/rJ:z:.j). Then

where' J = Idet (DF)I.

v(x) = +DFV(x), (S)

The expanded mixed method.Unlike the standard mixed Illet.hocL we introduce a synunet.ric and positive definite

tensor G and define the "adjusted" pressure gradient u by

Gu = -\7p. (6)

Then the system of equations is

KGli - u = O.

Gu + \7p = 0,\7 . u = q.

Denote inner-products over a set S by

(<p,lp)s = is <p(x) 'z/;(x) dx (or is <p(x) . Ii'(X) dx).

and inner-products over a boundary set as by

(7)

(8 )

(9)

where S is omitted if S' = n. Tbe expanded mixed finite element method is then: Findu E V", II E V", P E TV", and A E Ai: such that

(GKGu, v) - (Gu, v) = 0

(Gu, v) - (p, \7 . v) = -(Po, v . IJ)aon - (A, v . IJ)aoN

(\7. u,w) = (q,w)

(u . v, p)aoN = (g.II,)CioN

for all v E V",for all v E V",

for all 'l/} E Ttl/",

for aU Ii E A".

( J 0)

( Ll)( 12)

(U)

\Ve remark that if G = K-1, then u = u and we recover the standard mixed

met.hod [19, 21, 12]. If G = I, we recover the expanded mixed method considered in[2,1, 15, U, 8, ~)]. Later we will 111akca specia.l choice or G.

The algebraic syst.em of equations that results is a symmetric saddle point linearsystem of the form

(

1\1_AT

oo

-A()

BT

-IJ( 14)

where we represent u by U. u by U, p by p. and /\ by A in the nodal bases {Vi} for VIt,

{Wi} for Wit, and {Pi} for A",. In particular,

( (5)

are symmetric and positive definite. To reduce the size or the linear system, we can solve1'01' the ShIl\' COlli plemcnt by elimina.ting

(I())

( 17)

to obt.ain

(BA-1MA-1BT) ( ~ ) = ( ~~,) + (BA-'M/l-')Fu,

where B = ( ~ ). This system is symmet.ric, positive definite, and relatively small (one

unknown per element plus a few boundary nodes). Unfortunately, a.lthough A is spa.rse,A-I is ill general full. Iterative solution will require the following steps for the application

or the matrix: a matrix vector multiplY;1: = BT ( ~ ); the solution of the system Ay = :r:

and another matrix vector multiply By. Thus, we need inner iterat.ions wit.hin our overallitera.tive solution, which can become somewhat expensive.

CELL-CENTERED FINITE DIFFERENCE APPROXIMATION

Vie now use approximate integration to reduce A in (17) to an easily inverted diagonalmatrix.

The rectangular cell-centered finite difference approximation.Assume in this subsection that the grid is rectangular. Take G = I and use the

tra.pezoidal quadrature rule to approximate the first. three int.egrals (i.e., those involvinga vector-vector product) in (10)-( 1:3). This diagonalizes the matrix A and t.he Shurcomplement system becomes sparse [20,2:3], even when K is a t.ensor [8,9].

It is easy to unravel the procedure in t.erms of the nodal degrees of freedorn of u.11, p, and /\. Consider an element E (not adjacent. t.o t.he outer boundary). Equation(12) requires that the divergence of u be set equal to the source term q. This involvesdifferences of the normal velocities t.hat live on the four edges or six faces of t.he element.Equation (10) relates the velocities to the gradients of pressme. The velocity u on a given

edge OJ' face is related to t.he gradient u that lives on the given edge or face a.nd to thosethat live on the adjacent but perpendicular four edges OJ' eight faces (if K is not diagona,I).Finally, (1 L) relates u living on an edge or face to the difference of the adjacent pressures.Combining this together, we get. a 9 point. stencil for t.he pressure on E if d = 2. and l~)points if d = :L i\/Iore details are given lat.er and also in [9].

The Geometry Mapping.To handle irregular geometry, we assume that there is a smooth mapping F of a

n-'ct.a.ngular, computational domain {) ont.o the domain n. Civen a n'cta,l1glllar grid on {2:F defines a smooth. logically rectangular, curvilil1ear grid on n. (In practice, there aregrid generation codes available for crea.tillg 1" at t.he grid poillts. \Ve use finit.e differencesto a.pproximate DF.)

The Transformed, Computational Problem.III t.he expanded mixed metllod, take

G = .!( DF-I)T DF-'. ( 18)

Transform (10)-( l:q to the cOlllputational domain n. Vector and scalar basis functionst.ransform by t.he Piola and natural transforms, respectively: that. is.

v(x) = .!/X) DF(x)v(x)w(x) = 'II)(X)

(velocit.y),

(pressure).

(Kt.. v) = (u, v) for a.ll v E Vh, (L9)({l. v) - (p. V . v) = -(1)0. v· /i)Dfl

D- (/\, v . /i);3fl

Nfor all v E Vh, (20)

(V . u, 'II)) = (q.1, 'II) ) for all 'II) E I/l;'h. ( 21 )

(A A A) _ (A J) f' , II A \ ( ").))u· /J,p. ariN - g. ",p. ::Jf2N 01 a 1" E 1 h, ~~

where K = .1DF-1 K( DF-l)T. This is the discrete problem in f2:u = -KVp, x E n,

A A

\l .U = (j.1, x E n.p = Po, x E anD:u· /i = g.1,-" x E anN.

(2:q

(2-1)

(25)

(2G)

All computations are perfonned on the rectangular grid of n after preprocessing thecoefficients: K becomes K, q becomes ({./, and g becomes g.h.

The logically rectangular Cell-Centered Finite Difference Approximation.To problem (19)-(22), we use the trapezoidal quadrature rule for approximating t.he

t.hree integrals involving a vector-vector product to obtain am cell-centered finite differ-ence method on the logically rectangular mesh. As an illust ration. consider a. 2- D, uniformgrid, with a constant K. Denote the grid points and cell centers by

Then 1; = - Vp is

:.1.: A

/I . ll'"1+1/2,,j - 1+1,,/,- Pi.jh

(27)

with a sirnilar expression for ttL+I/z' Finally, for each element. E'i,j, \}. tl = (fJ is

[Dii~I/Z.j ~ ttL'/2.,j + Df,j+l/z ~ t'L-l/z] il2 = I. (1.1 d:i:.

h h ]"i)

The solution U 011 n is obtained from P = l) and A = A using (lG).

CONVERGENCE RESULTS

(29)

Let 11·11denote the U-norm, 11'P112= J nl'P(xW d:r, and let III· 111M denot.e the /,2-norm

approximated by the mid poi nt quadrat ure rule. Before stat iug om resul I, t hat I,he schemeis optimally convergent, we lIeed t.he I'ollowing definition.

Definition: An a.symptotic I'amily of grids is said to be generated by a (.'z map if each gridis a.n image by a fixed map of a grid that is uniform in each coordina.te directioll, Eachcomponent of the ma.p mllst be strictly monotone alld in C:2(fl).Theorem 1: There exists a constant C depending on the smoothness of F, K, and t.hesolut.ion, but independent of the maximum grid spacing h, such that. t.he cell-centeredfinite difference approximation satisfies

IIPIrtle - Papprox II S; C h .IIu1rue - uapproxll + IIUtrue - uapPl'Oxl1 S; Ch,II\} . (ut.rue - uapproJl1 S; Ch.

i'doreover,IllPlrue - PapproxlllM S; Ch2

,

IIIu1rue - uapproxlllM + Illutrue - uapproxlllM S; Ch",

where

,~{ :2 if I{ is diagonal and anN = an.:3/2 if J{ is diagonal or the grids are generated by a (.'z map,

1 ot.herwise.

When J{ is diagonal 01' t.he grids are generated by a. C2 map, a ha.lf power of h is lostill the super-convergence result for the velocity. This is due st.rictly to efFects near theboundary of the domain.

Theorem :2 (Interior estimates): Let 0' be compactly contained in O. and suppose t.hateither [{ is diagonal 01' t.he grids are generat.ed by a (:2 map. For any ( > 0, t.here exist.sa constant C, depending all t.he smoothness of F, K, and the solution, but. illdependcnt.of the maximum grid spacing 11, such that

The proofs of these results Ciln be foulld in [6] (see also [2:3, 9]). These results aresha,rp in the sense that t.hey are seen computationally in practical set.t.ings [9, 6].

DISCONTINUOUS MATERIAL PROPERTIES AND THE ENHANCEDCELL-CENTERED FINITE DIFFERENCE METHOD

\i\fe show by example t.hat our cell-centered scheme has difficulties approximating thesolution when the material properties a.re discontinuous. On the unit square, let. the truesolution p a.nd K be

{:l:Y for :1: S 1/2,

p(:z:,y) = :1:11 + (:1: -1/2)(y + 1/2) for.1: > 1/2, (30)

1(i ~)

K(l,y)= (~)~)

for :/; < 1/2,

for:z: > 1/2.

Not.e that t.he eigenvectors for K are at. 45 degrees t.o t.he grid for T < 1/'2. Comput.at.ion-a.lly, we see t.he following convergence rates:

1II/lt.I'1Ie - IlapprnxlllAl S 0.0(; lIu.%.

IIIUt,I'lIe - uapPl'oxlllA1 S 0.2:\ IIY5u.

These arc much worse t.hat predict.ed by Theorem 1. '['he error is concentrated along t.heline :z: = 1/2.

If K or t.he lIlap F is not smooth along an interface, t.hen U = KGli but. not. tl iscont.inuous in the normal direction. However. we have approximated li in tIle same spaceas u, i.e., Vh, which has cont.inuous normal components. \i\fe must. relax t.his continuit.yalong any interface where t.he material properties change discontilluously. To do so for t.heapproximation of li but. not. U would make t.he matrix A in (14) non-square and t.hereforenot fully invertible. We therefore need to relax the coutinuit.y of Vh 1'01' both tl and u.

This idea originates in the hybrid form of the mixed method of A1'llold and Brezzi [ll].Introduce Lagrange multiplier pressures living on the element edges or faces of the dis-continuity interface r (we already have them on fJnN). and add a. condition that specifiest.hat u alone is continuous across r. That is, replace (11) ill (10)-(1:3) and add (:n):

(G'i\, v) - 2)p, V· V)E = -(Po, v· 1/}aoD - 2]'\' v· 1/};JEn(DONUi) for all v E Vh, (:32)E E

2..:(u . 11,II}DEnr = 0 for allfl E Ah. (:n)E

\'Ve call this scheme the enhanced cell-centered finite difl'erence scheme. It ca.n be fonnu-la.ted without explicit reference to Lagrange pressmes by takillg a mesh with zero widthcells on r. The infinitely thin cell's pressure is the Lagrange pressure [22].

Using the enhanced method, the same comput.ational example shows t.he followingconvergence rates:

IIIpIl'lIC - PapPl'oxlllM ::; 0.1811:20:2,IIIUtl'lI" - UapPl'oxlllM ::; 0.10 hI. 19.

The Hybrid Form of the Mixed Method.It should be noted that t.he hybrid form of the mixed method uses Lagrange multiplier

pressures a.long all faces. By eliminating all but the Lagrange pressmes leads to a ShuI'complement system that is sparse, symmetric, and posit.ive definite. Unfortunately, thereare more unknowns than t.he number of elements. In 2-D, if defonned rectangles are used,there are two times as many unknowns. In :3-D, deformed brick elements need three t.imesas many unknowns.

TWO-PHASE FLOW

\Ve consider the accuracy of mixed finite element methods for approximating the highlynonlinear problem of two-pha.se How of incompressible water and air (or oil or a NA.PL).This represents work of two of the authors and Nai- Ying Zhang [10]. The governingequations are

Os .cp-;:- - \l . [/\ /\'u(S)\lfill'] = q",(s).dI

us .-cp 01. - \l . [/\ ).n(~)\lpn] = qn(s).

1),'("") = Pn ~ Pw,

(:H)

(:~!))

( :~())

where 0 ::; .s(x, I.) ::; I is the (norillalized) wet.ting fluid satlll'ation, cp is the porosit.y.\." = 1.:,.,-"(S)/I/(, is the rela.tive mobility of phase n = W,O, and I), is the capillary prcsslll'e.

Define total velocit.yv = -/\'/\w\lpw -/\').n\lp,,·

Hy rearranging (:34)-(:~!)) amI using (:W), we can obta.in the pressure equation

and the saturation equation

(J7)

(:,~8)

The pressme equation is a well behaved eUi ptic equation; howeveL the saturation equationis degenerate parabolic. so concentrate on it.

Kirchhoff Transformation.Dehne the Kirchhoff Transformation

1s (AW'\II ,) IP( s) = - \ .p{ (0') (0'.U All' + A"

'1'1 ( ) /\", All" I 1. • • I I ')(c'n \IP.5 = - \ \ v Pc, allC tIle saturat.lOn equation ta ~es t. Ie lorm/tu+/(l.

us .rP [)t - \I . [0\1P( s) + {:J( fJ( oS))] = ,( fJ(..,)).

(·10 )

(-II )

Regularity (Smoothness) of the Solution.\\le assume that P(s) is strictly monotone increasing, t.hat PI(S) may be zero (degen-

erate parabolic), but there is a constant Co > 0 such t.hat

\Ve a.lso assume that fJ and, are Lipschitz continuous. These assumptions can be just.ifiedon physical grounds [10. :3].

Introduce a new variable

4' = -0\1 P(s) - fJ(P(s)).

It. is known [2, L, :3] that s is continuous and

( ,12)

where H-I is tbe dual of Hel and the outer funct.ion space refers to the time interval ofinterest. and the inner one to O. The low regularit.y is due t.o the degenerate diffusion. :\swe pass from a strictly two-phase region t.o a region with a single phase, t.he solution is1I0t. particularly smooth; thus, it. is very difficult. to approximate the solntion accurately.

]\Iany authors have ('()Ilsiclercd tbe approximation properties of the cont.inuous, piece-wise lineal' hnit.c element Galerkin method. However, cell-centered finite difference met,b-ods are commonly used to solve t.his problem [18]. Since the mixed finit.e eleillent met.hodis st.rongly related to cell-centered finite difl'erence methods, we present a mixed met.hodanalysis. Such an analysis bas not previously appeared. Our approach is t.o consider t.beproblem from t.he point of view of optimal approximat.ion; that is, can we approximate t.hesolution as well as possible, givell t.befinite element. approximatillg space. Since typicallythe int.erfaces between the single and two-phase regions occupy a fairly small part. of t.hedomain, we can then expect to bave very good approximation of the solution.

A NONLINEAR. MIXED METHOD

In a standard mixed variationa.l form for (41),

for all w E He:' (4:~)

Since we can only expect in general that f)s/f)t E U( II-I), t.he trial functiolls must belongto If(lUJ). To avoid this, following Nochetto [17]. we integrate (41) in timc to obtain

1/ 1/~(x, l) + V . 4' dT = ,(P(s)) dT + ·'iO(X),o 0

where So is the initial satlll'ation. Note that

so we can formulate a mixed variational form as

(.s(-,l).w) + (V .114'dT,W) = (11

,(P(S))dT,W) + (so,w) for all w E [2, (~l-!)

(n-1[4' + ,B(P(.s))], v) - (P(.'i), V· v) = -(P(.sD). v· 1/) for all v E H(div). (45)

where .sD is, say, a Dirichlet. boundary condition.Now let. n be partitioned into a conforming finite element mesh with maximal element.

diameter h. Let Wh X Vh C L2 x H(div) be any standard mixed finite element space. Leto.! > 0, tn. = /l.o.!, a.nd 'Pn. = 'PUn).

Our nonlinear mixed method is to find .'in ~ S" E \;Vh and ?jilL ~ WIL E V h sa.tisfying

( n ) (n )(sn.w)+ V·2..:\\1jo.t,w = 2..:,(P(sj))o.t,w + (so,w) forallwEH/h, (46))=1 )=1

(a-1[Wn + ,B(p(sn))], v) - (p(sn), V· v) = -(P(s'DL V· 1/) for all v E Vh. (47)

The first cquation is cquivalent to thc usual ba.ckward Euler forlll

(c,'n _ c,'n- J )'- o.'~ .10 + (V· \lJn,w) = ("y(P(S'")).'w)

(So.w) = (.'io.w) for all wE Wh..

( I~)

(,19 )

Let. Pw" and PI'" denote {2-projection illtO Hlh a,lId Vh, respectively. Let II denotethe usual element based flux-preserving projection operator for mixed methods [19. 12].Such projections are optimal in t.heir approxima.tion properties.

Theorem :J: For the nonlinear mixed fil1it.e element. a.pproxirnatioll,

n II n 1 II?2..:(5') - "'), P(5')) - P(.s))) o.! + 2..: \fJ) ~l - PII" 1"4' <IT -)=1 .1=1

n 1 2 1 2

:S Cj; {IIPwhs' - sil12 + IICPVh - I) 1 J tI'dTII + Ilv. (11- PV,,) 1J tI'dTil

) t 2 1 t 2

+11~ ,(p(Si)) ~t - 1J ,(P(s)) ciT II + II~t l~11/' ciT - 4"jll }o.t,and

liS" - SnIlH-J :S C{hIIPw"s" - S"II + II~WJ o.t - II it" 1!'dTIl

+~ IIP(5'j) - P(,yi)11 o.t + Il,t ,(p(si)) ~1. -11

" ,(P(.s)) dTII2}.

{t } 1/2

Tbe form 1(8 - s, P(S) - P(s)) dT bounds the size or S - s: 1'01' example. it

bounds the norm IIP(S) - P(s)ll. It. is not, however, a norm itself. It ma.)" even rail (,0

he a. metric. Also, " . lin-I = snp ( '. <p)/II<pIIHI. The t.heorem says t.hat the saturation.<pEHJ

ill it weak II-I-norm sense, is honnded by optimal a.pproximation terlllS, time truncatiol1tel'lllS, and a term involving the divergence that is essent.ially optimal ill character.

CONCLUSIONS

\Ve have developed a cell-centered finite difference mixed method as an a.pproximationto an expanded mixed finite element. method. It is suitable for groundwater flow a.ndt.ransport problems on general geometry wit.h tensor penneabilit.ies or dispersivities. It. isboth locally (element- by-element) mass conservat ive al1d highly accu rate.

General geometry can be handled by a mapping bet.ween the computational alld phys-ical domains. The coefficients are t.ransformed in a simple way before commencing thecomputation. On a logically rectangular mesh, special quadratme rules are applied t.ot.he expanded finite element method to transform it into a cell-centered finite differencemethod with a stencil of 9 points in 2-D and 19 points in :3-]), allowing easy and efficientimplementation. Similar results hold 1'01' triangular meshes [4, 5, 6].

The solution of the logically rectangular cell-centered finite difference scheme con-verges t.o t.he true solution at t.he optima.! order as the mesh is refined; moreover, supcr-convergence is attained by the velocity at certain discret.e points of t.he domain on (:2_smoot.h grids away I'rom the boundary. We sa.w that grid geomet.ry st.rongly aJl'ects t.heapproximat.ion error, so it is necessa,ry to define and refine grids in a C'2-smooth manner.

Lagrange multiplier pl'essmes or infinitely thin cells need to be introduced along faceswhere the tensor or geometr.y changes discontinuously. This allows the adjusted pressuregradient i:i to be discontinuous, so that the coefficient. times 11 approximates well t.hecontinuous velocity u.

Mixed methods acc111'atcly approximate two-phase llow. The Kirchoff transrormationInay he useful in computations. The interrace between single and two-phase regions iscaptured ill an approximate (fJ-1-norm) sense.

ACI\NOWLEDCMENTS

This work was supported in part by the U.S. Department or Energy, the U.S. Nat.iollalScience Foundation: and the St.ate of Texas Governor's Energy Office.

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