A Mixed Finite Element Discretization on Non-Matching

TICAM REPORT 97 -17September, 1997

A Mixed Finite Element Discretization onNon-Matching Multiblock Grids for a DegenerateParabolic Equation Arising in Porous Media Flow

Ivan Yotov

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TITLEA Mixed Finite -Element Discretization onNon-Matching MultiblockGrids for aDegenerate Parabolic Equation Arising inPorous Media Flow

AUTHORSIvan Yotov

TYPEdouble-

sided

COPIES~

East- West J. Numer. Math., Vol. 0, No. 0, pp. 1-21 (1997)

© VSP 1997

A mixed finite element discretization on non-matchingmultiblock grids for a degenerate parabolic equationarising in porous media flow

IVAN YOTOV*

Abstract - Mixed finite element methods on multiblock domains are considered for nonlinear degenerateparabolic equations arising in modeling multi phase flow in porous media. The subdomain grids need notmatch on the interfaces, where mortar finite element spaces are introduced to properly impose flux-matchingconditions. The low regularity of the solution is treated through time integration, and the degeneracy of thediffusion is handled analitically via the Kirchhoff transformation. With an appropriate choice of the mortarspaces, the error for both a semidiscrete (continuous time) scheme and a fully discrete (backward Euler)scheme is bounded entirely by approximation error terms of optimal order.

Keywords. Mixed finite element, degenerate parabolic equation, nonlinear, mortar finite element, multi-block, non-matching grids, error estimates, porous media

AMS subject classifications. 65M60, 65M12, 65M15, 35K65, 76S05

Multiblock finite element techniques on non-matching grids have become increasingly popularin recent years. They combine the flexibility of modeling irregularly shaped domains with theconvenience of constructing the grids locally. In porous media applications they also allowaccurate modeling of large scale geological features such as faults, layers, and fractures. Wedefine a multi block domain to be a simply connected domain .0 E Rd, d = 2 or 3, that isa union of non-overlapping sub domains or blocks .oj, i = 1, ... , k. For the purpose of theanalysis we assume that each block .oj is convex.

In the numerical modeling of multiphase flow in porous media the coupled nonlinearsystem of conservation equations is often written as an equation of elliptic or parabolictype for some reference pressure and several saturation equations of advection-diffusion type[7, 13]. The diffusion in the saturation equations is degenerate (is zero at extreme saturationvalues). This causes a very low regularity of the solution and imposes difficulties for thenumerical method.

In this paper we study mixed finite element discretizations for the saturation equationon multiblock domains. The local mass conservation property of the mixed methods isparticularly important in modeling porous media flow. We allow the subdomain grids tobe non-matching along the interfaces. To properly impose flux-matching conditions we

Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin,TX 78712; yotovlQticam. utexas. edu.

2 Ivan Yotov

introduce mortar finite element spaces along the subdomain boundaries. Mortar spaces havebeen a common tool in standard and spectral finite element methods (see e.g. [9]). Recentlymortar [18, 29, 4] and non-mortar [6] techniques have been successfully applied in the contextof mixed finite element methods on multiblock grids.

Due to the degeneracy in the diffusion the solutions can have compact support, thus be-having very differently form the solutions of non-degenerate parabolic problems. In addition,certain error terms cannot be directly bounded in the analysis. Many authors introduce aregularized problem and then approximate it [27, 17,24,15, 14]. The numerical error is thena sum of the discretization error for the regularized problem and the difference between thesolutions to the regularized and the original problems.

Another common technique is to handle the degeneracy analytically via the Kirchhofftransformation [27, 15, 14, 5, 28], which is the approach we take. We combine the useof mortar finite elements along subdomain interfaces with techniques from [5], where mixeddiscretizations for similar degenerate parabolic equations on a single block have been studied.

We consider a continuous time scheme and a fully discrete backward Euler scheme andbound the discretization error in the two versions entirely by approximation error terms ofoptimal order. Critical in the analysis is the choice of mortar spaces along the non-matchinginterfaces. The mortar elements need to provide one order higher approximation than thetraces of the sub domain velocity spaces (see Remark 2.2). At the same time, the mortarspaces should be controled by the subdomain spaces (see hypothesis (HI) and Remark 3.1).

Our analysis improve the single block results from [5] in several ways. We avoid theassumption from [5] on smoothness of an by requiring a minimal smoothness for the flux(see Lemma 2.2.). Also, we weaken the assumption ([5], A2) on the advective and thesource terms and only assume physically reasonable behavior (see (2.7), (2.8)). Finally, bychoosing a different flux test function in (3.4), we avoid a non-standard approximation errorterm involving the divergence of the difference of two discrete projections of the flux, whichappears in the analysis in [5].

The rest of the paper is organized as follows. In the next section we illustrate how adegenerate parabolic saturation equation arises in the fractional flow formulation of twophase incompressible flow. In Section 2 we formulate the mortar mixed method for thesaturation equation. The analysis of the continuous in time and the fully discrete schemesis presented in Sections 3 and 4, respectively.

1. FRACTIONAL FLOW FORMULATION FOR TWO PHASE FLOWTwo phase immiscible flow in porous media is modeled by the system of conservation equa-tions [7, 13]

a( c.pSiPi) + \7 . Pilli = qi

ki(Si)I<lli = - (\7Pi - Pig\7 D)

µi

i = w (wetting), n (non-wetting), coupled with

Sw + Sn = 1,

inn x (O,T],

inn x (O,T],

(1.1)

(1.2)

(1.3)

A multiblock mixed method for a degenerate parabolic equation 3

(1.4)

where Si is the phase saturation, Pi is the phase density, c.p is the porosity, qi is the source term,Ui is the Darcy velocity, Pi is the phase pressure, [{ is the absolute permeability tensor, ki(Si)is the phase relative permeability, µi is the phase viscosity, 9 is the gravitational constant,D is the depth, and T is the final time.

We start by reformulating the problem in a standard way by writing it in a fractionalflow form (pressure and saturation equation). Let

kiAi = -, Z = w, n,{li

denote the phase mobilities, and let

be the total mobility. LetU = Uw + Un

be the total velocity. For simplicity of the presentation, we assume incompressible flow andmedium (constant Pi and c.p) and neglect gravity effects. Multiplying equations (1.1) by l/Piand adding them together, we get

\7. U = q,

where q = qw/ pw + qn/ pn. Let S = Sw, and define the global pressure [13] to be

(1.5)

(Pc(S) (An) (-1 )p=pw+ Jo T Pc (() de.Thus

U = -AI<\7p. (1.6)

Equation (1.5), coupled with (1.6), is referred to as the pressure equation. Since A > a andI< is a symmetric positive definite tensor, this is an elliptic equation. For compressible flowthe pressure equation is parabolic.

To derive the saturation equation, we first observe that

c.p ~; + \7. (,8(S)U + a(s)KV'pe(s)) = qw, (1.7)

where ,8(s) = Aw/A, a(s) = AwAn/A, and qw = qw/Pw. Note that Pe(s) is a strictly monotonedecreasing function; therefore with

Substituting this expression into the water conservation equation (1.1), we get the saturationequation

O'(s) = -a(s) aPeas (1.8)

4 Ivan Yotov

equation (1. 7) takes the advection-diffusion form

c.p~; + \7. (,B(s)u - O'(s)K\7s) = qw. (1.9)

The diffusion term vanishes at s = 0, 1, the minimum and maximum saturation values.This is due to the behavior of the relative permeability and the capillary pressure functions(see, e.g., [8]). This double degeneracy is the main source of difficulties in the numericalapproximation. The solutions of degenerate parabolic equations have very low regularity. Ithas been shown that (see [25, 16, 19, 2, 1, 3])

s E VXl(O, T; £1(.0)),as 2 -1at E £ (0, T; H (.0)).

Since the saturation s satisfies a :::;s(x, t) :::;1, (x, t) E .0 x [0, T], we also have

(1.10)

(1.11)

(1.12)

We handle the degeneracy in the diffusion analytically using the Kirchhoff transformation[27, 15, 14,5]. Let

D(s) = fos O'(() dC.

Then\7D(s) = O'(s)\7s,

and (1.9) becomesasc.p at + \7. (,B(s)u - [-(\7D(s)) = qw. (1.13)

2. A MORTAR MIXED METHOD FOR THE SATURATION EQUATIONIn this section we present a multiblock variational formulation of (1.13) that respects the lowregularity of the solution, and a mixed finite element discretization using mortar elements onthe interfaces. For the rest of the paper we omit the porosity c.p, which is assumed constantand does not effect the error analysis. The saturation s( x, t) satisfies

~; + \7. (,B(s)u - K\7 D(s)) = qw(s)

(,B(s)u - K\7 D(s)) . v = as(x,O) = so(x)

inn x (O,T],

on an x [0, T],

in .0,

(2.1)

(2.2)

(2.3)

where v is the outward unit normal vector to an. For simplicity we assume that no flowboundary conditions are imposed, although more general boundary conditions can also be


treated. We need several assumptions on the coefficients of the above equation. We firstassume that

{

,81IslvI, 0::;s::;a1,0'( s) ~ ,82, a1 ::; s ::;a2,

,8311 - slv2, a2 ::; s ::; 1,where ,8i, 1 ::; i ::;3, are positive constants, and ai and Vi, i = 1,2, satisfy

(2.4)

(2.6)

a < a1 < 1/2 < a2 < 1, a < Vi ::; 2.

Note that (2.4) controls the rate of degeneracy of the diffusion. We also assume that thereexists a positive constant G such that

Here and for the reminder of the paper (', ')5 and (', ')88 denote the L2-inner product (orduality pairing) on S C Rd and as E Rd-1, respectively, and II . 110,5 denotes the L2-normon S. We omit S if S = n. A sufficient condition for (2.5) is

aD0::; as (x,t;s)::; G for (x,t) E .0 x [O,T], 0::; s::; 1.

Bounds (2.4) and (2.6) follow from the physical behavior of the relative permeabilities andthe capillary pressure [3, 15]. Typical relations (see, e.g., [8, 20]) are

Aw(S) "-' S2, Pc(s) "-' S-Ol as s -+ 0,Ao(S) "-' (1- S)2, Pc(s) "-' (1- S)02 as s -+ 1,a < 61 < 1, a < 62 < 1.

Therefore, with (1.8),

O'(s) "-' Sl-01 as s -+ a and O'(s) "-' (1- S)l+02 as s -+ 1,

which implies both (2.4) and (2.6). Finally, we assume that, for 0::; Sl,S2 ::; 1,

1,8(sd - ,8(S2W ::; G(D(sd - D(S2))(Sl - S2),Iqw(sd - qw(s2)12 ::; C(D(sd - D(S2))(Sl - S2)'

Bounds (2.7) and (2.8) are justified by the following lemma, proven in [15].

(2.7)(2.8)

LEMMA 2.1. Suppose 0' satisfies (2.4). If f E G1[0,1] and 1'(0) = 1'(1) = a with f'Lipschitz at a and 1J then there exists a positive constant G such that

If(a) - f(bW ::; G(D(a) - D(b))(a - b) for a ::; a, b::; 1.

Note that the fractional flow function ,8(s) satisfies the conditions of Lemma 2.1.. The wellterm qw( s) satisfies the conditions of Lemma 2.1. at the injection wells. At the productionwells, qw(s) "-' kw(s), so q~(O) = O. Therefore (2.8) holds, if s ::; s* < 1 at the productionwell, which covers all cases of physical interest.

6 Ivan Yotov

REMARK 2.1. The fractional flow function (3( s) and the integrated diffusion functionD( s) are both S -shaped with zero derivatives at the end points. Bound (2.7) relates therates of degeneracy of the derivatives of the two functions and indicates, in a sense, that thediffusion dominates the advection.

In the standard mixed variational formulation, equation (2.1) is multiplied by a testfunction w E £2(.0) and integrated in space. In our case however, because of (1.11), theintegral (~~, w) is not well defined. To avoid this problem we integrate (2.1) in time from ato t [23, 5] to obtain the equivalent equation

s(x, t) + \7 . lot ¢dr = lot qw(s) dr + so(x), inn x [0, T],

where¢ = (3(s)u - f{'V D(s).

Before presenting the variational formulation, we give the following regularity result.

LEMMA 2.2. Assume that there exists some a < c < 1/2 such that, for 1 :::;i :::;k,

Then, for every t E [0, T],

(2.9)

(2.10)

(2.11)

(2.12)

Proof. The argument is similar to one from [5]. The assumption (2.11) is reasonable, sincethe fractional flow function (3(s) has zero deri vati ves at the degeneracy values s = 0, 1. Itfollows from (2.9) and (1.12) that

Using (2.10) we have in ni

-'V. [{\7 lot D(s)dr = 'V. lot ¢dr - \7. lot (3(s)udr.

By (2.11) and elliptic regularity, I~D(s) dr E H1+e(nd, which, along with (2.10) and (2.11)implies that

lot ¢dr E (He(ni))d. 0

Let fi,j = ani nanj, f = Ul~i<j~kfi,j,and fi = ani n f. Let


k

W = EB Wi = £2(.0),i=l

We are now ready to present the multiblock variational formulation of (2.9)-(2.10). Witha = J<-1 and letting / be the trace of D( s) on r, we have, for every time t E [0, T] and1 ~ i ~k,

(a1/;, Vh~i = (D(s), V'. vb; - (" V· /.Ii)r; + (a,8(s)u, vb;, v E Vi,

(s, wb; + (\7 . lot 1/; dT, w) fl.; = (lot iiw(s) dT, w) fl.; + (so, W)fl.;, w E Wi,

t/(t 1/; . /.Ii dT, µ) = 0, µ E A.i=l \Jo ri

(2.13)

(2.14)

(2.15)

Note that, because of (2.12) and the definitions of Vi and A, the boundary integrals in (2.13)and (2.15) are well defined.

Let ~,i be a finite element partition of ni with maximal element diameter h. We allowfor the possibility that ~,i and ~,j need not match on ri,j' Let V h,i X Wh,i be any ofthe usual mixed spaces on ~,i [26, 22, 12, 11, 10]. Let ,;;,) be a finite element partitionof ri,j with maximal element diameter h. Let Ah,i,j C Ai,j be the space of continuous ordiscontinuous piecewise polynomials of degree k + 1 on ,;;,j, where k is associated with thedegree of the polynomials in V h,i . Vi. More precisely, if d = 3, on any boundary elementI<, Ah,i,jIK = Pk+l(I{), if J< is a triangle, and Ah,i,jIK = Qk+l(I{), if J< is a rectangle. Anadditional assumption on Ah,i,j and hence on ,[i,j will be made later. The finite elementspaces on .0 are now defined as

k

Vh = EBVh,i'i=l

k

Wh = EBWh,i'i=l

Ah = EB Ah,i,j'l~i<j~j

REMARK 2.2. We refer to the interface spaces Ah,i,j as ((mortar" finite element spaces)following a terminology for similar techniques used with the standard and spectral finite ele-ment methods [9]. If the order of approximation for Ah is the same as for Vh . v) then themortar mixed methods for elliptic equations lose O(h1/2) from the optimal order of conver-gence [29, 4]. A similar loss is observed for the equations considered here (see the theoremsbelow)) which motivates the above choice of m01'tar spaces.

In the continuous time mixed finite element method for approximating (2.13)-(2.15) weseek, for each t E [O,T], 1/;h(·,t) E Vh, Sh(·,t) E Wh, and /h(·,t) E Ah such that, for1~ i ~k,

(a1/;h, V)fl.i = (D(Sh), V'. V)ni - (,h, V· /.Ii)r; + (a,8(sh)u, v)n;, v E Vh,i, (2.16)

(Sh, w)n; + (\7. lot 1/;hdT, w) fl.; = (lot iiw(Sh) dT, w) fl.; + (SO,h, W)ni' wE Wh,i, (2.17)

~ (lot 1/;h. Vi dT, µ) r; = 0, µ E Ah, (2.18)

8 Ivan Yotov

where SO,h E Wh is an approximation of so.We next consider a backward Euler time discretization. Let {tn}~=o be a monotone

partition of [0, T] with to = a and tN = T, let !:ltn = tn - tn-I, and let fn = f(tn).In the fully discrete mixed method we seek, for any a :::; n :::;N, 'l/Jj;E Vh, sh E Wh, and

,i: E Ah such that, for 1 :::;i :::;k,

(a'l/Jf;, v)n; = (D(sh), \7. v)n; - hi:, v· Vi)r; + (a,B(sh)un, v)n;,

(sh, W)ni + (\7 .t 'l/Ji!:ltj, w))=1 n;

= (tqw(s{)!:ltj,W) + (So,h,W)ni')=1 n;

v E V h,i, (2.19)

w E Wh,i, (2.20)

(2.21)~ (~1f: v;!1ti, µ) r, = 0

As noted in [5], by subtracting equation (2.20) for time levels nand n - 1, it can berewritten in the usual backward Euler form

(

n n-l )Sh - Sh , W + (\7 . 'l/Jf;,W)ni = (qw(sh)' w)n;,

!:ltn ni

(s~,w)ni = (so,h,W)ni'

wE Wh,i, (2.22)

(2.23)

3. ERROR ANALYSIS OF THE SEMIDISCRETE SCHEMEWe start this section with a lemma [29, 4] needed later in the analysis. The proof is includedfor completeness.

LEMMA 3.1. For any function v E Vh,i!

Proof. All spaces under consideration admit nodal bases that include the degrees of freedomof the normal traces on the element boundaries. Since for any element E and any of its faces(edges) e, lei:::; Ch-1IEI, the lemma follows. 0

We need the following projections onto the finite element spaces. The standard mixedprojection operator II : Vi ---+ V h,i satisfies, for q E Vi,

(\7. (q - IIq),w)n; = 0,

((q - IIq)· Vi, V· Vi)an; = 0,wE Wh,i,

v E Vh,i'

(3.1)

(3.2)

It is known [21] that(3.3)


In the analysis we apply II to f~1jJ, which is justified by Lemma 2.2 .. Let, for any c.pE W,cp E Wh be its U-projection, satisfying

(c.p-cp,w) =0, wEWh.

In a similar way we define the U-projections Ph : A -+ Ah, and Qh,i : A -+ V h,i . Vi. Forsmooth enough functions, these operators have optimal order approximation properties. Asin [29, 4], we make explicit the following assumption on the mortar space Ah. There existsa positive constant C independent of h such that, for 1 ~ i < j ~ k,

REMARK 3.1. Hypothesis (HI) imposes a mild condition on the mortar grids and spaces.It implies that the dimension of the mortar space on a given interface, and the distributionof its degrees of freedom, are controlled by the degrees of freedom of the traces of the velocityspaces on the two sides. This condition prevents overconstraining the matching interfaceconditions (2.21) and is easily satisfied in practice. See [29] for details.

We now proceed with the error analysis. Subtracting (2.16)-(2.18) from (2.13)-(2.15),we obtain the error equations

(a(1jJ -1jJh), vb; = (D(s) - D(Sh), \7. Vbi

- b - !h, V· Vi)ri + (a(f3(s) - f3(Sh))U, V)Oi'

(s - Sh, W)Oi + (\7. lot (1jJ -1jJh) dT, w) Oi

= (lot (qw(s) - qw(Sh)) dT, w)0; + (so - SO,h, wb"

t /r (1jJ - 1jJh) . Vi dT, µ) = 0,i=l \Jo ri

vE Vh,i,

wE Wh,i,

(3.4)

(3.5)

(3.6)

To simplify notations, let <1>(t) = IIf~(1jJ -1jJh)dT. We choose SO,h = So and take v = <1>,

w = .6(:;) - D0h)' and µ = Ph! - !h in (3.4)-(3.6). Note that our choice for v differsfrom this in [5], where v is taken to be an L2-projection of the flux error. Consequently, weare able to avoid a non-standard error term involving the divergence of the difference of theII-projection and the U-projection of f~1jJ dT, which appears in [5]. We now have

(s - sh,.6(:;) - D(Sh)) + (a(1jJ -1jJh), <1»

= (fo\qw(s) - qw(Sh)) dT, D0) - D0h))k

+( a(f3( s) - f3( Sh) )u, <1» - Lb - Ph!, <1> • Vi )ri;=1

(3.7)

10 Ivan Yotov

We integrate (3.7) in time form a to t. The first term on the left becomes

(3.8)

whereTI = fat (s - s, D(s) - D(Sh)) dr.

The second term on the left-hand side of (3.7) becomes

where

Combining (3.7)-(3.9), we obtain

(3.9)

where

T3 = fat (fa7' (iiw( s) - qw( Sh)) d~, D(s) - D(.;h)) dr,

T4 = fa\a({3(s) - (3(Sh))U, <I» dr,

k rt

T5 = - 2: in b - Ph" . Vi)ri dr,i=1 0

T6 = t.lot

( (107' 1/J d~ - II107' 1/J d~) . Vi d~, Ph' -'h) rj dr.

We now bound each Tk, k = 1, ... ,6. Note that the terms T3 and T4 are bounded differentlythan in [5], using the weaker and physically reasonable assumptions (2.7) and (2.8), respec-tively. The terms T5 and T6 involve error on the non-matching interfaces and do not appearin the case of a single block. For any 8 > 0, we have

ITII ~ c fat lis - sll~dr + 8 lot IID(s) - D(sh)ll~dr,

IT31 ~ c fat fa7'(s - Sh, D(s) - D(Sh)) d~ dr + 8 fat IID(s) - D(Sh)ll~ dr,

IT41 ~ 8 lot (s - Sh, D(s) - D(Sh)) dr + C lot IIII~dr,

IT51 ~ c {h-I lot 111- Ph,II~,r dr + lot IIII~dr},


using (2.8) for the bound of T3, (2.7) for the bound of T4, and Lemma 3.1. for the bound ofT5. To estimate T2 we integrate by parts in time:

T2 = - lot (loT a(¢ - ¢h)de, :T (n loT ¢de - loT ¢de)) dT

+ (lot a(¢ - ¢h) dT, n lot ¢ dT - lot ¢ dT) ;

therefore,

IT,I s c{lll{(~-~h)d([ d, +l' [I; (IT { ~d( - { ~d() [[ d,

+ lIn lot ¢ dT - lot ¢ dTII:} + J 111o\¢ - ¢h) dTII:

The term T6 is the most difficult to bound. Integration by parts in time gives

T,= - ~ l'(; ({~d( -ll { ~d(). Vi, {(hY -ih)d(),'o d,

+ ~ ( (lot ¢ dT - n lot ¢ dT) . Vi dT, lot (PhI - ,h) dT ) ri .

Therefore,

IT,I s c {1' [I:, ({ ~d( - IT { ~d() '{,r dT

+ II (lot ¢ dT - n lot ¢ dT) . vlI:,r dT} + J lot II loT (PhI - ,h) de[,r dT

+8 II lot (PhI - ,h) dTII:,r .To bound the last two terms, we consider, for 1 ~ i ~n and any fixed t E (0, T], theauxiliary problem

c.p - !:::J.c.p = 0,

\7c.p·Vi = lot Qh,i(Phl - ,h) dT,

(3.11)

(3.12)

Note that the Neumann boundary data is in H1/2-e(ni) for any c > a and by ellipticregularity

1 < m < 2.- - (3.13)

12 Ivan Yotov

We now integrate in time from a to t and take v = II'Vc.p in (3.4) to obtain

II (t Qh,i(Phi -ih) dTI12

Ja a,ani

= - (lot a('l/J - 'l/Jh)dT, II'Vc.p) ni + (lot (D(s) - D(Sh)) dT, c.p) ni

+ ( rta(!3( s) - !3(Sh))U dT, II \7 c.p) + / rt

Qh,i(Phi - i) dT, \7 c.p . Vi)Ja ni \Ja ani= T7 + Ts + Tg + TlO. (3.14)

We bound the terms on the right-hand side of (3.14) as follows. For any 0 > 0,

IT71 ~ c lilt ('l/J - 'l/Jh)dTI12

. + o(II\7c.pII;,ni + 11\7 . \7c.pII~,nJa a,n,

~ ell rt('l/J - 'l/Jh)dTI1

2

. + 0 lilt Qh,i(Phi - ih) dTI12

.'Ja a,n, a a,an,

using (3.3) for the first inequality and (3.13) for the last bound. Note that 0 in the aboveinequalities is an arbitrary small generic constant which may be different each time it appears.Similarly,

ITsl ~ c rtIID(s) - D(Sh)ll~n dT + 0 II rt

Qh,i(Phi-ih) dTI12

,Ja ' 'Ja a,ani

ITgl ~ c t (s - Sh, D(s) - D(Sh))ni dT + 0 lilt Qh,i(Phi-ih) dTI12

.'Ja a a,an,

ITlOl ~ c rtIIPhi -ill~,ani dT + 0 lilt Qh,i(Phi -ih) dTI1

2

.'Ja a a,an,

Combining together (3.14), the bounds on T7-TlO, and (HI), we obtain

II lot (Phi -ih) dT[,r

~c{ll,f'l/JdT-II fot'l/JdTII: +11(t)II~+ fotIID(s)-D(sh)lI~dT

+ fot(s-sh,D(s)-D(sh))dT+ fotIIPhi-ill~,rdT}. (3.15)

Combining (3.10), the bounds on T1-T6, (3.15), and using (2.5) and Gronwall's inequality,we arrive at the following result.

THEOREM 3.1. Assume that (2.4)-(2.8) and (HI) hold. For the semi-discrete mixed fi-nite element approximation (2.16)-(2.18) of problem (2.1)-(2.3), there exists a positive con-stant C such that, for every t E [0, T],

A multiblock mixed method for a degenerate parabolic equation

~ C {fat lis - sll~dT + lin fat ljJdT - fat ljJdT[

+h -1 fat II, - Ph,II~,rdT + II (fat ljJ dT - IIfat ljJ dT) . 1I11:,r2

+ fat :T (faT ljJ d~ - n faT ljJ d~) 0 dT

+ lot II:, ([ ,pd( - II[ ,pd() . {,r d'}.

13

Theorem 3.1. bounds the size of IID(s) - D(Sh)llo by (2.5). It also allows us to derive abound on lis - shll-l (see also [5]), where II . 11-1 is the H-1(n)-norm defined by

11·11-1= sup (.,c.p)rpEH1(n) 1Ic.plh .

THEOREM 3.2. Assume that (2.4)-(2.8) and (HI) hold. For the semi-discrete mixed fi-nite element approximation (2.16)-(2.18) of problem (2.1)-(2.3), there exists a positive con-stant C such that, for every t E [0, T],

IIs(·,t) - sh(·,t)II~1

~ C { h211s - sll~ + fat lis - sll~ dT + IIII fat ljJ dT - fat ljJ dTII:

+h-1 fat liT - Ph,II~,rdT + II (fat ljJ dT - IIfat ljJ dT) . lI[,r

+ fo' II :, ([ ,p d( - II[ ,p d( ) II: dr

+ lot II;, ([ ,pd( -ll [ ,pd() . {,r d'} .

Proof. For any c.pE HJ(!1), we have

(s - Sh,c.p) = (s - Sh,c.p - cp) + (s - Sh,cp) = (s - s,c.p - cp) + (s - Sh,cp).

By (3.5) we have that

(s - Sh,cp) = - ~ (\7. (II lot ljJdT - lot ljJhdT),cp )ni + (fo\qw(s) - qw(sh))dT,cp).

For the first term on the right we write

14 Ivan Yotov

=t (II {t 'ljJdr - (t 'ljJh dr, \7c.p) - t /(II t 'ljJdr - (t 'ljJh dr) . Vi, c.p)i=1 Jo Jo ni i=1 \ Jo Jo C

= ~ (IT J.' ,p dr - 1.',pk dr, V<pt,-1;; ((IT 1.' ,pdr - 1.' ,p dr) Vi, <p) r,

- ~ (lo\ 'ljJ - 'ljJh) . Vi dr, c.p- Phc.p) ri '

using (3.6) for the last equality. Therefore

I(s - sh,c.p)1

~ C { hils - silo + IIII lot 'ljJ dr - lot 'ljJh drllo + II (II lot 'ljJ dr - lot 'ljJdr) . vt.r

+ II lot ('ljJ - 'ljJh) drllo + (lot (s - Sh, D(s) - D(Sh)) dr) 1/2} 11c.plh,

using that11c.p- ~llo ~ Chllc.plll

for the first term on the right, Lemma 3.1. and

1Ic.p- Phc.pllo,c ~ ChI/211c.pIh.ni

for the fourth term, and (2.8) for the last term. An application of Theorem 3.1. completesthe proof. 0

4. DISCRETE TIME ERROR ANALYSISIn this section we present the analysis for the fully discrete scheme (2.19)-(2.21). Thefollowing error equations are obtained by subtracting (2.19)-(2.21) from (2.13)-(2.15) fort = tn.

(a('ljJn - 'ljJ1;), v)ni = (D(sn) - D(sh)' \7. V)ni

- (·'t - II:, v· Vi)C + (a({3(sn) - {3(sh))un, v)ni' V E Vh,i, (4.1)

(s' - sh,w)n, + (V. (t ,pdr - ~,p~L'.tj) ,w) n,

= (1.'" iiw(s) dr - ~ iiw( s~)L'.tj, w) n, + (So - SO,k, w)o" w E Wh,i, (4.2)

k ( tn n )~ lo 'ljJ. Vi dr - [; 'ljJ~ . ViL:1tj, µ ri = 0, µ E Mh. (4.3)

We now take V = ~n = II (I~n'ljJdr - 2::j=1 'ljJ~L:1tj), w = D(;n) - D(;h), and µ = Phln - Ii:,to obtain


We replace n by j in the above equation, multiply by tltj and sum on j. The first term onthe left in (4.4) becomes

n ~ n

2:)sj - s{, D(;J) - D(s{))tltj = 2)sj - s{, D(sj) - D(s{))tltj - T1, (4.5)j=l j=1

wheren

T1 = - 2:(;i - sj, D(sj) - D(s{))tltjj=1

To manipulate the terms involving n we rewrite it as

Note that n is represented as a sum of an approximation error term, a time discretizationerror term, a.nd an error term we are trying to bound. Using (4.6), the second term on theleft in (4.4) becomes

n .

2:( a( ~j - ~i), J )tltjj=1

n J

= 2:(a(~j - ~i), 2:(~1- ~~)tltl)tltj - T2 - T3j=1 1=1

where

For the second equality in (4.7) we used the well known identity, for any sequence {aj},

16 Ivan Yotov

Combining (4.4), (4.5), and (4.7), we arrive atn

I:(sj - s{, D(sj) - D(s{))!ltjj=1

2

+~ Ila1/2 t(ljJj -ljJ~)!ltj + ~t Ila1/2(ljJj -ljJ~)!ltjll: = t Tm, (4.8)

J=1 0 J=1 m=1where

n .

Ts = I:(a(,6(sj) - ,6(s{))uj, cI>J)!ltj,j=1

n k

T6 = - I: I: (,j - Ph,j, cI>j. Vi)r !ltj,j=1i=1 I

n k (( rti tJ) .)T7 = 22 10 ljJ dr - II 10 ljJ dr . Vi, ph,j -,h !ltj.J-1 t-1 ri

We next bound the terms Tm, m = 1, ... , 7. For any J > a we haven n

IT11:::;C I: 11;1- sjll~!ltj +JI: IID(sj) - D(sfJ)II~!ltj,j=1 j=1

n { j ( t' )IT41 :::; Cf; (; L-l qw( s) dr - qw( sI)!ltl

n

+JI: IID(sj) - D(s~)II~!ltj,j=1

: + 1;;(s' - s~, D(s') - D(SU)Ll.t'} M

n n .ITsI :::;oI:(sj - s{, D(sj) - D(s{))!ltj + CI: IIcI>JII~!ltj,

j=1 j=1n

JT61 :::; C I:{h-1Ihj - Ph,jll~,r + IIcI>jlln ~tj,j=1

where we used (2.8) for the bound of T4, (2.7) for the bound of T5, and Lemma 3.1. for thebound of T6. To bound the rest of the terms we need the following discrete integration byparts identity. For sequences {Aj }j=o and {,6j }j=o,

n-1 nI:Aj(,6J+1 - ,6j) = - I:(Aj - Aj-d,6j + An,6n - Ao,6o.j=O j=1

(4.9)

We will apply (4.9) for Aj = 'Li=1 ai, where {aj }j=1 is another sequence. In this case Ao = aand (4.9) leads to

n n-1 j nI: aj,6j = - I: I: al(,6j+1 - ,6j) +I: aj,6n.j=1 j=1 1=1 j=1

(4.10)


. '. J JTo estimate T2, we apply (4.10) with Ctj = a(ljJJ -1/JO~tJ and (3j = IIIo 1/JdT - Io 1/JdT.

n-1 ( j 1 (fJ+1tj+1

))T2 = I: I:a( 1/JI-1/JU~tl, A j+1 II1. 1/JdT -1. 1/JdT ~tj+1j=l 1=1 ut fJ tJ

-(t a(,pj - ,p~)!>tj, II( ,p dr - ( ,p dr)2

o

n-111 1 (tj

+1 tj

+1

) 112+.r; ~ti+1 II1j 1/JdT - L ljJ dT 0 ~tj+1

n-1 j

:::;01I: I:a(1/JI -1/JU~tlll ~tjj=l 1=1

2

+0' lit a( 1/Jj -1/JO~tj + c IIII lotn

1/JdT - lotn

1/JdTI12

,J=l 0 0 0 0

where we assumed that there exists a constant 0 > a such that

(4.11)Similarly,

2

2

o

n-1 II 1 (tj

+1

) 11

2

+f; ~ti+1 1j 1/JdT -1/Jj+1~tj+1 0 ~tj+1

2

IT,I ~ C,~ II~a(,pl_ ,pi)MII !>Ii

n

+0' III:a(1/Jj -1/Ji)~tjj=l o o

and

IT,I ~ {,~ 11t,(pql -'Ii)!>t'2

o,r

n-1 II 1 (tj

+1 tj

+1

) 11

2

+0 f; ~tj+1 L 1/JdT - II1j 1/JdT . v o,r ~tj+1

2

+0' IltCPhl'j -I'i)~tj + ell (ltn

1/JdT - IIltn

1/JdT) . vI12 .J=l 0 rOo o,r,

To further estimate the first and the third terms, we consider, for 1 :::;i :::;k and any fixed1 :::;n :::;N, the auxiliary problem

c.p-~c.p=o, inni, (4.12)n

\7c.p·Vi = I:Qh,i(Phl'j -I'O~tj,j=l

(4.13)

18 Ivan Yotov

Note that the Neumann boundary data is in Hl/2-e(ni) for any c > a and by ellipticregularity

n

11c.pllm-e,ni ::; C II~ Qh,i(Ph,j - ,i)~tjll ' 1::; m ::; 2. (4.14))- m-e-3/2,an;

We now replace n by j in (4.1), multiply by ~tj, sum on 1 ::; j ::;n, and take v = Il\7c.p toobtain

n

L Qh,i(Ph,j - ,i)~tjj=1

2

o,an;

= - (t, a(.p; - .p~)"'1; , II'1'1't + (t,(15(';) - D(;h) )"'1;, 'I't+ (t a(j3(sj) - j3(s~))uj ~tj, Il\7c.p) + (t Qh,i(Ph,j - ,j)~tj, \lc.p. Vi)

)=1 ni )=1 ani= T8 + Tg + TIO+ Tl1. (4.15)

We now have, for any small generic constant 0 > 0,2IT.IS c IIt,W - .pi)Mllo,o; + JlIll'V<pII;,o,2

S c IIt,W - .po"'It 0 + J(II '1'1'11l,o; + 11'1· '1'1'11;,0,, I

2n n

::; C IIL(~j- ~O~tj +0 L Qh,i(Ph,j - ,O~tjj=1 O,ni j=1 1I0,ani

using (3.3) for the second inequality and (4.14) for the third inequality. Similarly,

ITgl ::; C t, I\D(s) - D(Sh)II~,ni~tj + 0 lit, Qh,i(Ph,j - ,i)~tjO,ani

ITIOI ::; ct,(S - Sh, D(s) - D(Sh))ni~tj + 0 lit, Qh,i(Ph,j - ,O~tj

ITl1l ::; C t, IIPh,j - ,jll~,ani~tj + 0 lit, Qh,i(Ph,j - ,i)~tjO,ani

o,an;

Combining (4.15), the bounds on T8-TU, and (HI), we conclude that

2 { 2n n n

L(Ph,j - ,i)~tj ::; C L(~j - ~O~tj + L IID(sj) - D(s{)II~~tj)=1 o,r )=1 0 )=1

A multiblock mixed method for a degenemte parabolic equation 19

(4.16)

Putting together (4.8), the bounds on T1-T7, (4.16), manipulating J in T5 and T6 as in (4.6),applying (2.5) and the discrete Gronwall's inequality, we obtain the following estimate.

THEOREM 4.1. Assume that (2.4)-(2.8) and (HI) hold, and that there exists a constantC1 > a such that

lltj+1 :::; C1lltj, 1:::; j :::;N - 1.

For the fully discrete mixed finite element approximation (2.19)-(2.21) of (2.1)-(2.3), thereexists a positive constant C such that, for any 1 :::;n :::;N,

t,(sj - st D(sj) - D(s~))lltj + 11t,(1/Jj -1/JOlltj2 n

+ L II(1/Jj - 1/JOlltj II~j=1

J ( tt ) 2~ L-l qw(S) dr - qw(sl)lltl 0 + h-11lph,j - ,jll~,r

The following theorem can be shown as in the proof of Theorem 3.2 ..

THEOREM 4.2. Under the assumptions of Theorem 4.1., there exists a positive constantC such that, for any 1 :::;n :::;N,

Iisn - shll~1

:S Ct,{II';! - sj II~++ II;t; (IIt.,p dr - J.:~.,p dr) II:+ II;t; (L ,p dr - ,p;!>t;) Ii:

+ II ;tj (IIL. ,p dr - J.:~.,p dr) .{,r} M + Ch'lIsn - snll~

REMARK 4.1. The estimates from Theorems 3.1.-4.2. bound the discretization error byoptimal order approximation terms in time or space. The term h-1/2111- Ph,llo,r providesapproximation of order h1/2 higher then the other terms, assuming enough regularity of 'Jsince the functions in Ah are piece-wise polynomials of one degree higher than these in Vh . v.

20 Ivan Yotov

Acknowledgments. This work was partially supported by the United States Departmentof Energy. The author thanks Professors Todd Arbogast and Mary F. Wheeler for usefulcomments and discussions.

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A Mixed Finite Element Discretization on Non-Matching