Mixed Finite Element Methods on Non-Matching Multiblock ...

TICAM REPORT 96-50November, 1996

Mixed Finite Element Methods on Non-MatchingMultiblock Grids

Todd Arbogast, Lawrence C. Cowsar, Mary F.Wheeler and Ivan Yotov

MIXED FINITE ELEMENT METHODSON NON-MATCHING MULTIBLOCK GRIDS*

TODD ARBOGASTt, LAWRENCE C. COWSARt,MARY F. WHEELER§, AND IVAN YOTOVll

Abstract. We consider mixed finite element methods for second order elliptic equations onnon-matching multiblock grids. A mortar finite element space is introduced on the non-matchinginterfaces. We approximate in this mortar space the trace of the solution, and we impose weaklya continuity of flux condition. A standard mixed finite element method is used within the blocks.Optimal order convergence is shown for both the solution and its flux. Moreover, at certain dis-crete points, superconvergence is obtained for the solution, and also for the flux in special cases.Computational results using an efficient parallel domain decomposition algorithm are presented inconfirmation of the theory.

Key words. Mixed finite element, mortar finite element, error estimates, superconvergence,multiblock, non-conforming grids

AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30

1. Introduction. Mixed finite element methods have become popular due totheir local (mass) conservation property and good approximation of the flux variable.In many applications the complexity of the geometry or the behavior of the solutionmay warrant using a multiblock domain structure, wherein the domain is decomposedinto non-overlapping blocks or sub domains with grids defined independently on eachblock. Typical examples include modeling faults and wells in subsurface applications.Faults are natural discontinuities in material properties. Locally refined grids areneeded for accurate approximation of high gradients around wells.

In this work we consider second order linear elliptic equations that in porousmedium applications model single phase Darcy flow. We solve for the pressure p andthe velocity u satisfying

u = -K'1p in n, (1.1)

'1·u=J in n, (1.2)

p=g on an, (1.3)

where neRd, d = 2 or 3, is a multiblock domain and K is a symmetric, uni-formly positive definite tensor with LOO(n) components representing the permeabilitydivided by the viscosity. The Dirichlet boundary conditions are considered merely forsimplicity.

*The first, third, and fourth authors were partially supported by the Department of Energy, andthe third author was partially supported by the National Science Foundation.

tTexas Institute for Computational and Applied Mathematics (TICAM) and Department of Math-ematics, The University of Texas at Austin, Austin, TX 78712; arbogast~ticam. utexas. edu.

t Lucent Technologies Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974;cowsar~research.att.com.

§TICAM, Department of Aerospace Engineering & Engineering Mechanics, and Department ofPetroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712;rnfw~ticam.utexas.edu.

,-rDepartment of Computational and Applied Mathematics, Rice University, 6100 Main Street,Houston, TX 77005-1892, and TICAM, The University of Texas at Austin, Austin, TX 78712;yotov~ticam.utexas.edu.

1

2 ARBOGAST, COWSAR, WHEELER, AND YOTOV

A number of papers deal with the analysis and the implementation of the mixedmethods applied to the above problem on conforming grids (see, e.g., [25, 23, 22, 7,5, 6, 9, 12, 21, 26, 13, 15, 2, 1] and [24, 8]). Mixed methods on nested locally refinedgrids are considered in [14, 16]. These works apply the notion of "slave" or "worker"nodes to force continuity of fluxes across the interfaces. The results rely heavily onthe fact that the grids are nested and cannot be extended to non-matching grids.

In the present work we employ a partially hybridized form [3, 8] of the mixedmethod to obtain accurate approximations on non-matching grids. We assume thatn is a union of non-overlapping polygonal blocks, each covered by a conforming,affine finite element partition. Lagrange multiplier pressures are introduced on theinter block boundaries [3, 8, 17]. Since the grids are different on the two sides ofthe interface, the Lagrange multiplier space can no longer be the normal trace ofthe velocity space. A different boundary space is needed, which we call a mortarfinite element space, using terminology from previous works on Galerkin and spectralmethods (see [4]and references therein). As we show later in the analysis, the methodis optimally convergent if the boundary space has one order higher approximabilitythan the normal trace of the velocity space. Moreover, superconvergence for thepressure and, in the case of rectangular grids, for the velocity is obtained at certaindiscrete points.

We allow the mortar space to consist of either continuous or discontinuous piece-wise polynomials and obtain the same order of convergence in both cases. The methodusing discontinuous mortars provides better local mass conservation across the inter-faces, but numerical observations suggest that this may lead to slightly bigger numer-ical error.

The method presented here has also been considered in [27] in the case of thelowest order Raviart- Thomas spaces [23, 22]. Here we take a somewhat differentapproach in the analysis, which allows us to relax a condition on the mortar gridsneeded to obtain optimal convergence and superconvergence. The relaxed conditionis easily satisfied in practice.

An attractive feature of the scheme is that it can be implemented efficiently inparallel using non-overlapping domain decomposition algorithms. In particular, wemodify the Glowinski-Wheeler algorithm [17,11] to handle non-matching grids. Sincethis algorithm uses Lagrange multipliers on the interface, the only additional cost iscomputing projections of the mortar space onto the normal trace of the local velocityspaces and vice-versa.

The rest of the paper is organized as follows. The mixed finite element methodwith mortar elements is presented in the next section. In Section 3 we construct aprojection operator onto the space of weakly continuous (with respect to the mortars)velocities and analyze its approximation properties. Sections 4 and 5 are devotedto the error analysis of the velocity and the pressure, respectively. In Section 6 themethod is reformulated as an interface problem. A substructuring domain decompo-sition algorithm for the solution of the interface problem is discussed in Section 7.Numerical results confirming the theory are presented in Section 8.

(2.1)

(2.2)

v E H(div; n),w E L2(n),

2. Formulation of the method. A weak solution of (1.1)-(1.3) is a pair u EH( div; n), p E L2(n) such that

(I{-lU, v) = (p, '1. v) - (g, v· v)an,('1. u,w) = (J,w),

It is well known (see, e.g., [8, 24]) that (2.1)-(2.2) has an unique solution.

MIXED METHODS ON NON-MATCHING GRIDS 3

Let n = Ui=l ni be decomposed into non overlapping subdomains ni, and letfi,j = ani n anj, f = Ui,j=lfi,j, and fi = ani n f = ani\an. Let

n

V· = H(div' n·)I , I'

andn

W = E9 Wi = L2(n).i=l

If the solution (u,p) of (2.1)-(2.2) belongs to H(div; n) x H1 (n), it is easy to see thatit satisfies, for 1 ~ i ~n,

(J(-lu, v)ni = (p, '1. v)ni - (p, V· Vi)ri - (g, V· Vi)ani\r,

('1. u,w)ni = (J, W)ni'

v E Vi, (2.3)

wE Wi, (2.4)

where Vi is the outer unit normal to ani.Let Th,i be a conforming, quasi-uniform finite element partition of ni, 1 ~ i ~n,

allowing for the possibility that Th,i and Th,j need not align on f i,j' Let Ih = Ui=l Th,i'Let

Vh . X Wh . C V· x w·,1 ,ttl

be any of the usual mixed finite element spaces, (i.e., the RTN spaces [25, 23, 22];BDM spaces [7]; BDFM spaces [6]; BDDF spaces [5], or CD spaces [9]). We assumethat the order of the spaces is the same on every subdomain. Let

Recall that

n

Vh = E9Vh,i,

i=l

n

Wh = E9Wh,i'i=l

'1. Vh' = Wh',1 , 'l,

and that there exists a projection IIi onto V h,i, satisfying amongst other propertiesthat for any q E (H1/2+e(ni))d n Vi,

('1 . (IIiq - q), w)ni = 0,

((q - IIiq) . Vi, V· Vi)ani = 0,

w E VVh i, (2.5)

(2.6)

Note that, since q E (H1/2+e(ni)), q' vie E He(e) for any element face (edge) e;therefore IIiq is well defined.

Let Th,i,j be a quasi-uniform finite element partition of f i,j. Denote by Ah,i,j CL2 (f i,j) the space of either continuous or discontinuous piecewise polynomials of de-gree k + 1 on Ih,i,j, where k is associated with the degree of the polynomials in V h . v.More precisely, if d = 3, on any boundary element J(, Ah,i,j IK = Pk+l (J(), if J( is atriangle, and Ah,i,j\K = Qk+l(J(), if J( is a rectangle. Let

Ah = E9 Ah,i,j.l~i<j~n


v E V h,i, (2.7)

w E Wh,i, (2.8)

In the following we treat any function µ E Ah as extended by zero on an. Anadditional assumption on the space Ah and hence 7h,i,j will be made below in (2.11)and (3.18). We remark that 7h,i,j need not be conforming if a discontinuous space isused.

In the mixed finite element approximation of (2.1)-(2.2), we seek Uh E V h, Ph EWh, Ah E Ah such that, for 1 ~ i ~n,

(K-1Uh, v)ni = (Ph, '1. v)ni - (Ah, V· vi)ri - (g, V· Vi)ani\r,

('1. Uh, w)ni = (J, w)ni'n

L(Uh . Vi,µ)ri = 0,i=l

µ E Ah. (2.9)

(2.10)

For each sub domain ni, define a projection Qh,i : L2(fd -+ V h,i . vilri such that,for any <P E L2(fd,

(<p - Qh,i<P,V . Vi)ri = 0, v E V h,i.

Let, for <PE L2(f), Qh<P= EB~=l Qh,i<P.LEMMA 2.1. Assume that for any <P E Ah,

Qh,i<P = 0, 1 ~ i ~n, implies that <P= O. (2.11)

Then there exists a unique solution of (2.7)-(2.9).Proof. Since (2.7)-(2.9) is a square system, it is enough to show uniqueness. Let

f = 0, 9 = O. Setting v = Uh, w = Ph, and µ = -Ah, adding (2.7)-(2.9) together,and summing over 1 ~ i ~n, implies that Uh = O. Denote, for 1 ~ i ~n,

Ph,i = I~illiPh dx,

and consider the auxiliary problem

- '1.K'1r.pi = Ph - Ph,i

- K'1r.pi . v = - (Qh,iAh - Qh,iAh)

where Ah = 0 on an n ani. Note that the problem is well posed and regular with r.pidetermined up to a constant. Setting v = -IIiK'1r.pi in (2.7), we have

(Ph,Ph - Ph,i)ni + (Qh,iAh, Qh,iAh - Qh,iAh) ani = 0,

implying

Since now (2.7) is

Ph Ini (1, '1.V)ni - Qh,iAh (1, v . V)ani = 0,

the divergence theorem implies Phlni = Qh,iAh.Since Ah = 0 on an, Phlni = Qh,iAh = 0 for those domains i with ani n an 1= 0.

For any j such that ani n anj = fi,j 1= 0, (2.10) implies that

0= Qh iAhlr· . = Qh J·Ahlr· . = -IfIll Ah ds., I,) , 1,)

i,j ri,j

We conclude that Qh,iAh = 0 for all 1 ~ i ~n; hence, Ph = 0 and Ah = 0 by thehypothesis of the lemma. 0


3. The space of weakly continuous velocities. We first introduce some pro-jection operators needed in the analysis. Let Ph be the L2(r) projection onto Ahsatisfying for any 'l/JE L2(f),

For any r.pE L2(n), let c:P E Wh be its L2(n) projection satisfying

(r.p- C:P,w) = 0, wE Wh.

These operators and the projections II and Qh,i defined earlier have the followingapproximation properties, wherein l is associated with the degree of the polynomialsin Wh:

1I'l/J- Ph'l/JII-s,r ~ CII'l/Jllr,rhr+s,

1Ir.p- C:Pllo~ C 1Ir.pIIrhr,

IIq - IIiqllo,ni ~ Cllqllr,n;hr,

11'1· (q - IIiq)llo,ni ~ CII'1· qllr,nihr,

1I'l/J- Qh,i'l/JII-s,ri ~ CII'l/Jllr,rihr+s,

lI(q - I1iq)· vill-s,ri ~ Cllqllr,rihr+s,

o :::;r ~ k + 2, 0 ~ s ~ k + 2,

o ~ r ~ l + 1,

1 ~ r ~ k + 1,

1 ~ r ~ l + 1,

o ~ r ~ k + 1, 0 ~ s ~ k + 1,

o ~ r ~ k + 1, 0 ~ s ~ k + 1,

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

where 1I'lIr is the Hr-norm and II'II-s is the norm of H-s, the dual of HS. Moreover,

Let

--'1 . IIiq = '1 . q,

(IIiq) . Vi = Qh,i( q . Vi).

Vh,O = {v E Vh : ~(vln; . Vi,µ)r; = 0 V µ E Ah }

(3.7)(3.8)

be the space of weakly continuous velocities, with respect to the mortar space. Thenthe mixed method (2.7)-(2.9) can be rewritten in the following way. Find Uh E V h,O,

Ph E Wh such that

n

(K-1 Uh, v) = 2:)Ph, '1 .V)ni - (g, v . v)an,i=l

n

2::('1. Uh,w)ni = (J,w),i=l

v E Vh,O, (3.9)

(3.10)

Our goal for the rest of this section is to construct a projection operator IIo ontoV h,O with optimal approximation properties such that, for any q E (Hl(n))d,

('1. (IIoq - q),w)n = 0, wE Wh·

By an abuse of notation, define

Vh' V = ((<Pl,<P2) E L2(r) x L2(f): <pllr;,j E Vh,i' Vi and<P2\ri,j EVh,j'//j V l~i<j~n}

(3.11)

6

and

ARBOGAST, COWSAR, WHEELER, AND YOTOV

Vh,O' v = ((<Pl,<P2) E L2(f) x L2(f)::lv E Vh,O such that

<pllri,j = vlni . Vi and <p2Iri,j = vlnj . Vj V 1 ~ i < j ~n}.

Henceforth, for any <PE (L2(f))2, we write <plr;,j = (<Pi, <pj), 1 ~ i < j ~n. Define aprojection Qh,O: (L2(f))Z -+ Vh,o' v such that, for any <PE (L2(f))2,

n

L(<Pi - (Qh,O<P)i,ei)r; = 0, e E Vh,o' V.i=l

(3.12)

LEMMA 3.1. Assume that (2.11) holds. For any <P E (L2 (f))2, there existsAh E Ah such that on f i,j, 1 ~ i < j ~n,

Qh,iAh = Qh,i<Pi - (Qh,O<P )i,Qh,j Ah = Qh,j<Pj - (Qh,O<P)j,(Ah,l)ri,j = ~(<pi + <Ph l)r;,j'

(3.13)(3.14)(3.15)

Proof. Consider the following auxiliary problem. Given <PE (L2(f))2, find 'lj;h EV h . v and Ah E Ah such that

n

L(<Pi - 'lj;h,i - Ah, ei)ri = 0,i=ln

L('lj;h,i,µ)ri = 0,i=l

e E Vh' v, (3.16)

(3.17)

To show existence and uniqueness of a solution of (3.16)-(3.17), take <P= 0, e = 'lj;h,and µ = Ah to conclude that 'lj;h = O. Now (3.16) and (2.11) imply that Ah = O.

With e E V h,O . v in (3.16) we have

n

L(<Pi - 'lj;h,i,ei)r; = o.i=l

Also, from (3.17), 'lj;h E V h,O . v. Therefore'lj;h = Qh,O<P' Equation (3.16) now implies(3.13) and (3.14). Since any constant function is in V h,i . Vi, V h,j . Vj, and Ah,i,j, wehave

2(Ah,1)ri,j = (Qh,iAh, l)ri,j + (Qh,jAh' l)ri,j= (Qh,i<Pi - (Qh,O<P)i, l)r;,j + (Qh,j<Pj - (Qh,O<P)j, l)r;,j= (Qh,i<Pi, l)ri,j + (Qh,j<Pj, l)ri,j

= (<Pi + <Pj, l)ri,j'

and (3.15) follows. 0The next lemma shows that, under a relatively mild assumption on the mortar

space Ah, Qh,O has optimal approximation properties for normal traces:


LEMMA 3.2. Assume that there exists a constant C, independent of h, such that

Then, for any <Psuch that <plr;,j = (<Pi, -<Pi), there exists a constant C, independentof h, such that

(

n ) 1/2 n

~ IIQh,i<Pi - (Qh,o<P)ill=-s,r; ~ ct; II<Pillr,r;hr+s,

o ~ r ~ k + 1, 0 ~ s ~ k + 1. (3.19)

Proof. By Lemma 3.1, there is a Ah E Ah such that

Qh,i/\h = Qh,i<Pi - (Qh,O<P k

Since 2:7=1 (( Qh,O<PL, Ah)ri = 2:7=1 (<Pi, Ah)r; = 0,

n n

L IIQh,iAhI15,ri = L(Qh,iAh, Ah)r;i=l i=l

n

= L(Qh,i<Pi - <Pi,Ah)rii=l

(

n ) 1/2 ( n ) 1/2~ t; IIQh,i<Pi - <PiIl5,ri t; Ph 116,ri

(

n ) 1/2 ( n ) 1/2~ C t; IIQh,i<Pi - <PiIl5,r; t; IIQh,iAhI15,ri '

by (3.18), and (3.19) with s = 0 follows from (3.20) and (3.5).On any interface fi,j take any'P E HS(fi,j), 0 ~ s ~ k + 1, and write

(3.20)

(Qh,iAh' r.p)ri,j = (Ah, Qh,i'P - 'P)r;,j + (Ah, r.p)ri,j~ CPhllo,ri,jhSIl'PlIs,ri,j + (Ah, r.p)ri,j' (3.21)

The last term is

(Ah,r.p)ri,j = (Ah,'P - ~(Qh,ir.p + Qh,jr.p))ri,j+~(Ah, Qh,i'P + Qh,jr.p)ri,j (3.22)

~ CPhllo,r;,jhsllr.plls,r;,j + ~(Ah, Qh,ir.p + Qh,j'P)r;,j'

Using Lemma 3.1, for the last term in (3.22) we have

(Ah, Qh,i'P + Qh,j'P)ri,j= (Qh,i Ah, Qh,i'P)r ;,j + (Qh,j Ah, Qh,jr.p)r i,j

= (<Pi - (Qh,O<P)i, Qh,ir.p)r;,j + (<pj - (Qh,O<P)j, Qh,jr.p)r;,j= (<Pi - (Qh,O<P)i, Qh,ir.p - Phr.p)r;,j + (<pj - (Qh,O<P)j, Qh,jr.p - Phr.p)ri,j

~ Chrll<Pillr,ri,j hS 1Ir.plls,r;,j' 0 ~ r ~ k + 1. (3.23)


Combining (3.21)-(3.23) with (3.18), we obtain (3.19). 0We are now ready to construct our projection. For any q E (Hl/2+e(ni))d n Vi

define

where Jqi solves

Jqi = -'17fi

'1. Jqi = 0

Jqi . Vi = -Qh,iq· Vi + (Qh,oq' V)i

Jqi . Vi = 0 on ani nan,

(3.24)

(3.25)

(3.26)

(3.27)

wherein, on any fi,j, q' vlri,j = (q. Vi,q' Vj). Note that the Neumann problems(3.25)-(3.27) are well posed, since (3.15) and (3.13) imply that

(Qh,iq· Vi - (Qh,oq' V)i, l)ri = o.Also, note that the piece-wise constant Neumann data is in Hl/2-e(and, so Jqi E(Hl-e(ni))d and IIi can be applied to Jqi.

We first notice that by (3.8),n n

L((IIoq)· vi,µ)ri = L((Qh,Oq· v)i,µ)ri = 0, Vµ E Ah;i=l i=l

therefore IIoq E V h,O. Also, by (3.7),

('1. IIoq, W)ni = ('1. IIiq, W)ni + ('1. IIiJq, w)n; = ('1.q, w )n;,

It remains to estimate the approximability of IIo. Since on ni

IIoq - q = IIiq - q + IIiJqi,

with (3.3) we need only bound the correction IIiJqi. By elliptic regularity [18, 19],for any s 2: 0

We now have

IlIIiJqi Ilo,ni ~ IIIIiJ% - Jqi lIo,ni + IIJqi Ilo,ni

~ Chl/21IJqilh/2,n; + IIJqi llo,ni

~ C{IIQh,iq' Vi - (Qh,oq· v)illo,rihl/2

+IIQh,iqi' Vi - (Qh,Oq· v)ill-l/2,rJ,

using an estimate by Mathew [20] for any divergence free vector 'ljJ

(3.28)

(3.29)

Note that the result in [20] is for Raviart- Thomas spaces, but can be trivially extendedto any ofthe mixed spaces under consideration. Together with Lemma 3.2, (3.29) gives

n

IlIIoq - IIqllo ~ C L Ilqllr+l/2,ni hr+l/2, 0 ~ r ~ k + 1,i=l

(3.30)

and, with (3.3),

MIXED METHODS ON NON-MATCHING GRIDS

n

IlIIoq - qllo ~ C L IIqllr,nJ~r, 1 ~ r ~ k + 1.i=l

9

(3.31 )

4. Error estimates for the velocity. We start this section with a lemmaneeded later in the analysis.

LEMMA 4.1. For any function v E V h,i,

IIv, vllo,ani ~ Ch-l/2I1vllo,n;.

Proof. All spaces under consideration admit nodal bases that include the degreesof freedom of the normal traces on the element boundaries. Since for any element Eand any of its faces (edges) e, lei ~ Ch-lIE!, the lemma follows. 0

4.1. Optimal convergence. Subtracting (3.9)-(3.10) from (2.3)-(2.4) gives theerror equations

n

(J{-l(U - Uh), v) =L ((p - Ph, '1. v)n; - (p, V· Vi)r;)i=l

n

L('1'(U-uh),w)ni =0,i=l

We first notice that (4.2) implies that

'1. (IIou - Uh) = \7 . (IIu - Uh) = o.We now take v = IIo U - Uh to get

(J(-l(IIou - uh),IIou - Uh)n

= L(PhP - P, (IIou - Uh) . Vi)r; + (I{-l (IIou - u), IIou - Uh)

i=l

VEVh,o, (4.1)

wE Wh. (4.2)

(4.3)

n~L IIPhP - pllo,ri II(IIou - Uh) . Villo,ri + (1{-l(IIou - u), IIou - Uh)

i=l

~ C (t,IIpllc+l,o;hC+l/2I1llou - uhllo,o;h-1/2 + t, lIullc,o;hclillou - u/,llo) ,

1 ~ r ~ k + 1, (4.4)

where we used (3.1), Lemma 4.1, and (3.31) for the last inequality. With (4.3)-(4.4),(3.4), and (3.31) we have shown the following theorem.

THEOREM 4.2. For the velocity Uh of the mixed method (2.7)-(2.9), if (2.11)holds, then there exists a positive constant C independent of h such that

n

11'1· (u - uh)lIo ~ C L 11'1· ullr,nihr, 1 ~ r ~ l + 1.i=l

Moreover, if (3.18) holds,n

lIu - uhllo ~ C L(llpllr+l,ni + Ilullr,ni )hr, 1 ~ r ~ k + 1.

i=l


4.2. Superconvergence. In this subsection we restrict to the case of diagonaltensor ]( and RTN spaces on rectangular type grids. In this case superconvergenceof the velocity is attained at certain discrete points. To show this we modify the lastinequality in (4.4). In particular, (3.1) gives, for 1 ~ r ~ k + 1,

n n

L IIPhP - pllo,ri II(IIou - Uh) . Villo,ri ~ C L IIPllr+3/2,nihr+lIlIIou - uhllo,nih-l/2,i=l i=l

and (3.30) gives, for 1 ~ r ~ k + 1,n

(](-l (IIou - IIu), IIou - Uh) ~ C L Ilullr+l/2,ni hr+l/2I1IIou - Uh 110,i=l

which, combined with and an estimate by Duran ([13], Theorem 3.1)

(](-l(IIiU - u), IIou - Uh)n; ~ Cliullr+l,nihr+lIlIIou - uhllo,ni, 0 ~ r ~ k + 1,

impliesn

IlIIou - uhllo ~ C L(lIpllr+3/2,ni + lIullr+l/2,nJhr+l/2, 1~ r ~ k + 1. (4.5)i=l

This estimate implies superconvergence along the Gaussian lines. Consider (for d = 3)an element E = [al,bl] x [a2,b2] x [a3,b3]. Denote by gL ... ,g1+l' i = 1,2,3, theGaussian points on [ai, bd, i.e., the roots of the Legendre polynomials of degree k + 1on [ai,bd. As in [15,13], for a vector q = (ql,q2,q3) define

k+l k+l bl

IIIQlllli,E = L A12(b2 - a2) L Aj3(b3 - a3) 1 IQ1(Xl,g]2,g;JI2 dXl,12=1 j3=1 Ul

k+l k+l b2

IIIQ2111~,E= L Ajl (bl - ad L Aj3 (b3 - a3) 1 IQ2(gJl ' X2, g;J12 dX2,h=l j3=1 U2

k+l k+l b3

IIIQ3111~,E= L Ah (bl - ad L Ah(b2 - a2) 1 IQ3(gJl ,g]2' x3)12 dX3,h=l 12=1 U3

where Aji, ji = 1, ... , k + 1 are the coefficients of Gaussian quadrature in [-1,1].Define

3

IIlqlW = L L IIIQilll;,E'i=l EETi.

Note that, for q E V h, IIlqlll is equal to the L2-norm of q.THEOREM 4.3. Assume that the tensor]( is diagonal and the mixed finite element

spaces are RTN on rectangular type grids. For the velocity Uh of the mixed method(2.7)-(2.9), if (3.18) holds, then there exists a positive constant C independent of hsuch that

n

Illu - uhlll ~ C L(lIpllr+3/2,ni + lIullr+l/2,nJhr+l/2, 1~ r ~ k + 1.i=l


Proof. By the triangle inequality,

IIlu - uhlll ~ IIlu - IIulil + IlIIIu - IIoulll + IlIIIou - uhlll·

The theorem follows from (3.30), (4.5), and the bound (see [13])

IIlu - IIulilni ~ Cllullr+l,n;hr+l, 1~ r ~ k + 1.

o

11

5. Error estimates for the pressure. In this section we use a duality argu-ment to derive a superconvergence estimate for i> - Ph. Let r.pbe the solution of

- '1.]('1r.p = -(i> - Ph)

r.p=0

By elliptic regularity,

Take v = IIo]('1r.p in (4.1) to get

n

IIi> - Phll6 = I)i> - Ph, '1. IIo]('1'P)nii=l

in n,on an.

(5.1)

n

= L ((](-l (u - Uh), IIo]('1r.p )ni + (p - PhP, IIo]('1r.p . Vi)r;). (5.2)i=l

The first term on the right can be manipulated as

n

L(I{-l(U - uh),IIo]('1r.p)n;i=l

n

=L ((](-l (u - Uh), IIo]('1r.p - J('1'P )n; + (u - Uh, '1r.p)n;)i=ln

= L ((J(-l(U - uh),IIo]('1r.p - ]('1r.p)nii=l

(5.3)n

~ C L (liu - uhllo,nih + 11'1. (u - uh)llo,nih + lI(u - Uh) . villo,rih3/2)1Ir.p1l2i=l

using (3.31), (3.2), and (3.1) for the last inequality with C = C(maxi 11](lkoo,n;).The last term on the right is

lI(u - Uh) . villo,rih3/2

~ (lI(u - IIiU)' villo,ri + II(IIiu - Uh)' vi\lo,r;)h3/2

~ C(hrllullr,r; + h -1/2I1IIiu - Uh Ilo,ni)h3/2

= C(llullr,rihr+3/2 + IIIIiU - uhllo,nih), 0 ~ r ~ k + 1, (5.4)


o ~ r ~ k + 1,

o ~ r ~ k + 1,

using (3.6) and Lemma 4.1.For the second term on the right in (5.2) we have

(p - PhP, IIoK'1'P' Vi)r;= (p - PhP, (IIoK'1r.p - IIJ('1r.p) . Vi + (IIJ{'1r.p - K'1r.p) . Vi + K'1r.p . vi)ri~ lip - Phpllo,ri(llo(K'1r.p)i . villo,ri + II (IIJ{'1r.p - K'1r.p) . villo,rJ

+lIp - PhPII-l/2,ri IIK\7r.p· vilil/2,ri'

With (3.1), (3.26), (3.19), and (3.6) we have

lip - PhPllo,ri ~ CIlPllr+l/2,rihr+l/2,

lip - PhPII-l/2,ri ~ Cllpllr+l/2,rihr+l,

lIo(K'1'P)i' villo,r; ~ CIlr.p1l2,nih1/2,

II(IIJ('1'P - K'1'P) . villo,ri ~ CIlr.p1l2,nihl/2;

therefore,

A combination of (5.1)-(5.5), Theorem 4.2, and (3.3) gives the following theorem.THEOREM 5.1. For the pressure Ph of the mixed method (2.7)-(2.9), if (3.18)

holds, then there exists a positive constant C, independent of h, such that

n

lip - Ph 110~ C L(lIpllr+l,ni + lIullr,ni + 11'1. ullr,ni )hr+l,i=ln

lip - Philo ~ C L(lIpllr+l,n; + lIullr,ni + 11'1·ullr,nJhr,i=l

where 1 ~ r ~ min(k + 1,l + 1).

6. An interface operator. In this section we introduce a reduced probleminvolving only the mortar pressure. This reduced problem arose naturally in the workof Glowinski and Wheeler [17] on sub structuring domain decomposition methods formixed finite elements and is closely related to the inter-element multiplier formulationof Arnold and Brezzi [3]. The reason to consider the interface operator is twofold.First, we use it to derive a bound on the error in the mortar space. Second, it is thebasis for our parallel domain decomposition implementation.

6.1. The reduced problem. Define a bilinear form dh : L2(r) X L2(r) -+ Rby

n n

dh(A,µ) = Ldh,i(A,µ) = - L(U~(A)' Vi,µ)r;,i=l i=l

where for A E L2(r), (U~(A),p~(A)) E Vh x Wh solve, for 1 ~ i ~n,

(K-l U~(A), v)ni = (p~(A), '1 . V)ni - (A, v . vi)ri'

('1'U~(A),w)ni =0,

VEVh,i,

wE Wh,i.

(6.1)

(6.2)


Define a linear functional gh : L2(f) -+ R by

n n

gh(µ) = Lgh,i(µ) = L(Uh' vi,µ)ri'i=l i=l

13

(J(-lUh, V)ni = (fh, '1. v)ni - (g, V· Vi)ani\r,

('1.Uh, w)n; = (J, w )ni'

v E Vh,i,

wE Wh,i.

(6.3)

(6.4)

It is straightforward to show (see [17]) that the solution (Uh,Ph,Ah) of (2.7)-(2.9)satisfies

(6.5)

with

(6.6)

LEMMA 6.1. The interface bilinear form dh(·,·) is symmetric and positive semi-definite on L2(f). If (2.11) holds, then dh(·,·) is positive definite on Ah'

Proof. With v = uh(µ) in (6.1) for some µ E L2(f), we have

which shows that dh (', .) is symmetric and

(6.8)

For µ E Ah, if (2.11) holds, the argument from Lemma 2.1 shows that dh(µ,µ) = 0implies µ = O. 0

6.2. Error estimates for the mortar pressure. Denote by II . IIdh the semi-norm induced by dh(·,·) on L2(f), i.e.,

THEOREM 6.2. For the mortar pressure Ah of the mixed method (2.7)-(2.9), if(3.18) holds, then there exists a positive constant C, independent of h, such that

n

lip - Ahlldh ~ CL(lIpllr+l,ni + lIullr,n;)hr, 1 ~ r ~ k + 1, (6.9)i=l

n

IIPhP - Ahlldh ~ C L(llpllr+l,ni + lIullr,ni)hr, 1::; r ::; k + 1.i=l

In the case of diagonal tensor J( and RTN spaces on rectangular type grids,

(6.10)

n

lip - Ahlldh ~ C L(lIpllr+3/2,ni + lIullr+l/2,ni )hr+l/2, 1 ~ r ~ k + 1, (6.11)i=l

n

IIPhP - Ah Ildh ~ C I)lIpllr+3/2,ni + lIullr+l/2,ni )hr+l/2, 1 ~ r ~ k + 1.(6.12)i=l


Proof. With (6.8) we have

lip - Ahlldh ~ CIIUh(p) - Uh(Ah)lIo, (6.13)

using that uh (-) depends linearly on its argument. Define, for µ E L2(r),

Uh(µ) = uh(µ) + Uh, Ph(µ) = Ph(µ) + ih,

and note that (Uh(µ),Ph(µ)) E V h X Wh satisfy, for 1 ~ i ~n,

(1(-1 Uh(µ), v)ni = (ph(µ), '1 .v)ni - (µ, v· V)ri

- (g, v . v)an;\r,

('1. Uh(µ),W)n; = (J,w)ni'

We now have

v E Vh,i,

wE Wh,i'

(6.14)

(6.15)

IIUh(p) - uh(Ah)lIo = lIuh(p) - Uh(Ah)\lo= lIuh(p) - Uhllo~ IIUh(p) - ullo + lIu - Uhllo (6.16)

Bound (6.9) now follows from (6.13), (6.16), Theorem 4.2, and the standard mixedmethod estimate for (2.3)-(2.4) and (6.14)-(6.15) [25, 23, 12]

lIuh(p) - ullo,ni :::;C(lIpllr+l,n; + lIullr,ni )hr, 1 ~ r ~ k + 1.

To show (6.11), we modify (6.16) as

Iluh(p) - Uhllo :::; IIUh(p) - IIulio + IIIIu - IIoulio + IlIIou - uhllo. (6.17)

Bound (6.11) now follows from (6.13), (6.17), (3.30), (4.5), and a superconvergenceestimate for the standard mixed method [13] (see also [21, 15])

Iluh(p) - IIiullo,n; ~ Cllullr+l,nihr+l, 1 ~ r ~ k + 1.

To prove (6.10) and (6.12), note that, by (6.1),

(J(-lUh(PhP - p), uh(PhP - p))n; = -(PhP - P, uh(PhP - p) . v)ri~ I\PhP - pllo,ri lIuh(PhP - p) . vllo,ri

~ Cllpllr,r;hrlluh(Php - p)lIo,n;h-l/2, 0 ~ r ~ k + 2,

using (3.1) and Lemma 4.1 for the last inequality. Therefore, with (6.8),n

IIPhP - plldh ~ CL IIpllr+3/2,n;hr+l/2, 0 ~ r ~ k + 1. (6.18)i=l

Bounds (6.10) and (6.12) follow from (6.9) and (6.11), respectively, using the triangleinequality and (6.18). 0

REMARK 6.1. In the case of the lowest order RTN spaces, it is proven in [10]that, for any <P E Ah, dh,i( <P, <p) is equivalent to IIani Qh,i<Ph/2,ani' where Ian; isan interpolation operator onto the space of continuous piece-wise linears on ani.Therefore 1I·lIdh can be characterized as a certain discrete Hl/2-norm on f (see [27]).This is also in accordance with the numerically observed O(h2) convergence for themortars in a discrete L2-norm (see Section 8).


7. A substructuring domain decomposition algorithm. In this section wediscuss the implementation of a parallel domain decomposition algorithm for solvingthe resulting linear system. We apply the substructuring algorithm by Clowinskiand Wheeler [17] to the lowest order RTN discretization on non-matching multiblockrectangular type grids.

The original method for matching grids solves an interface problem in the spaceconsisting of the normal traces of the velocity. In our case we solve an interfaceproblem in the space of the mortar pressures. We use the conjugate gradient methodto solve the interface problem (6.5). Note that Lemma 6.1 guarantees convergence ofthe iterative procedure in Ah.

Every iteration of the conjugate gradient requires an evaluation of the bilinearform dh(·, .), and therefore, solving subdomain problems (6.1)-(6.2) with a givenDirichlet data in the mortar space Ah. Because of the property

the sub domain solves only use projections of the mortar data onto the local spaces.Therefore, no change in the local solvers is needed for the implementation. Moreover,the conjugate gradient is performed in the space

The conjugate gradient residual is the jump in the fluxes across sub domain boundaries.The jump is computed after projecting the local boundary fluxes onto the mortarspace, as indicated by

n n

dh(\µ) = - L(Uh(A)' Vi,µ)ri = - L(PhUh(A)' Vi,µ)r;, A,µ E Ah.i=l i=l

Therefore the only additional computational cost compared to the case of matchinggrids is computing the projections Qh,i : Ah -+ V h,i . Vi and Ph : V h,i . Vi -+ Ah'

8. Numerical results. In this section we present some numerical tests con-firming the theoretical convergence rates. All examples are on the unit square andconsider only the lowest order RTN spaces on rectangles. In the first example we solvea problem with known analytic solution

p(x, y) = x3y4 + x2 + sin(xy)cos(y)

and tensor coefficient

).The boundary conditions are Dirichlet on the left and right edge and Neumann onthe rest of the boundary. The domain is divided into four sub domains with interfacesalong the x = 1/2 and y = 1/2 lines. The initial non-matching grids are shown inFigure 8.1. We test both continuous and discontinuous mortars. The initial mortargrids on all interfaces have 4 elements with 5 degrees of freedom in the continuouscase and 2 elements with 4 degrees of freedom in the discontinuous case, thereforesatisfying the solvability condition (2.11). Convergence rates for the test case aregiven in Table 8.1. Here II . 11M is the discrete L2-norm induced by the midpoint rule


FIG. 8.1. Initial non-matching grids.

Continuous mortars Discontinuous mortarsE(A)

1.36E-023.45E-038.68E-042.19E-045.35E-05O(h2.00)

E(u)4.12E-021.36E-024.55E-031.54E-035.29E-04O(h1.57)

E(A)1.31E-023.31E-038.30E-042.08E-045.39E-05

E(u)2.95E-028.54E-032.42E-036.66E-041.88E-04O(h1.83)

E(p)9.62E-032.41E-036.04E-041.51E-043.91E-05O(h1.99)

l/h8163264128rate

E(p)9.52E-032.40E-036.03E-041.51E-043.75E-05

O(h1.99) I O(h2.00)TABLE 8.1

Discrete norm errors and convergence rates, where E(p) = lip- Ph 11M, E(u) = Iliu - uh III, andE()") = II).. - Qh)..hIIM.

on Ih (or the trace of Th on r). The rates were established by running the test caseand 4 levels of grid refinement (we halve the element diameters for each refinement)and computing a least squares fit to the error. We observe numerically convergencerates corresponding to those predicted by the theory. The computed pressure andvelocity with continuous and discontinuous mortars on the first level of refinementare shown in Figure 8.2. Although both solutions look the same, Table 8.1 indicatesthat they differ. This can also be seen in Figure 8.3, where the magnified numericalerror is shown. The error in the continuous mortar case is concentrated at the cross-points, where the only discontinuities in the mortar space occur. The error in thediscontinuous mortar case is distributed along the interfaces and is somewhat larger.We have to point out however, that the discontinuous mortars provide flux continuityin a more local sense, as indicated by the flux matching condition (2.9).

In the next example we test a problem with a discontinuous coefficient. We chooseJ( = I for 0 ~ x < 1/2 and J( = 10 * I for 1/2 < x ~ 1. The solution

{x2y3 + cos(xy),

p( x, y) = (2~69 ) 2 y3 + cos (2~69 y) ,o ~ X ~ 1/2,1/2 ~ X ~ 1,

is chosen to be continuous and have continuous normal flux at x = 1/2. The domainis divided into two sub domains with an interface along x = 1/2. The initial grids are4 x 8 on the left and 4 x 11 on the right. Continuous mortars on a grid of 7 elementswith 8 degrees of freedom or discontinuous mortars on a grid of 4 elements with 8


.. .,.. ..... .,.. ..... .,.. .,.... .,.. ...~ r' "" ......__ I .. I .. 1..-

". ",.. "".

__ I.. .. ... ". "" ......

". If"." ....... ." If"

" ff'" lit' f!tI'"

" ".. .... ft""

A. Continuous mortars

.. ..... ..... ....,.. ..... .,.. ..... .,.. ..... .. ..... .. .,.... '" ...

B. Discontinuous mortars

FIG. 8.2. Computed pressure (shade) and velocity (arrows).

.. -.... .. ,

.. .r"

.. .. ,':\.:. )' .-: : . : : : : : : : : . : '~'f;;.fl"f.. - ..............-tst~

. ~~~

" .. .' ...• ~: Ifl"JIJM," f%'~ •• J\\#~' '"

I ~"'1 "'<"'1' ..."~~~, }~~' ~-: : ,

.....~az ~i!!ii¥.\.', d_·,;.~... •.. .' y....:..:..JiZ:S. ,jf.\ : ·\i .• •

;·'1\:;[: ....~:\; ..".' l' ,_~% ?··'i~1 •

'. ····1· ..·,··· ,." I:: ,-, .I.j. '11riF l#i0·\*,,·J ".' .:.., te.·~: '--;-:v '".\ r 0'~'

.. , .. ..'<', ... ",.- -'": ?,<:,,:,-., " '-'.;': ::',:"<

A. Continuous mortars

.";;;~.,v •- • • • - • • • • .. .. .. .. .;.~-~ ~:?,- ,~s·•• __ .•••.••• ;;.;;.,'J

· - I.ti

: : : : : : : : : : : : ::. ,.I.EI· '.' ;Y

: : : : : : : : : : : : : ::\0 "i;' F;iH~1· ... : : : : : : : : : : 1.,:1';-- .,. f

i11

.: .• :.,"'" • 1~lh(1• • .. .. .. .. • • .. *·..\:::;::::~~:EI

'"~

If ~ ",.';, .j!#- lI.i "'!, • . • .. , .. I «.," M' <> i«,w, I :.'~" ." ~

_'.'k '~I'"-":'~~::J h;'" "· . . . ..,,"". .. .;.' .. j:: ~ • ..• ' •. !..;-..... .,'.·.\""I ::'~',;,::.:·-.,,,g- ....'."~.

: t~tt+~l"\o;:;:- . ..· •. ,.:~ i£••~ '"1<",, .-;· . ~1j' ".-"C,:' .••• '.

4,; ••

· ..•.':: '".r.' w ••• ,,' .....l~.

B. Discontinuous mortars

FIG. 8.3. Pressure and velocity error.

degrees of freedom are introduced on the interface. Convergence rates for the testcase are given in Table 8.2.

In the last example we would like to compare the mortar element mixed method onlocally refined grids to the more common "slave" or "worker" nodes local refinementtechnique [14, 16]. In the latter, the fine grid interface fluxes within a coarse cell areforced to be equal to the coarse grid flux. We note that this scheme can be recoveredas a special case of the mortar element method with discontinuous mortars, if thetrace of the fine grid is a refinement by two of the interface grid. Indeed, in this casethe flux matching condition (2.9) becomes a local condition over two (four if d = 3)fine grid boundary elements and forces all fine grid fluxes to be equal to the coarsegrid flux. Our theory also recovers the convergence and superconvergence results


E(A)1.22E-033.17E-048.01E-052.01E-055.02E-06O(h1.98)

E(u)3.27E-021.13E-023.93E-031.37E-034.82E-04O(h1.52)

Discontinuous mortarsE(p)

3.38E-048.55E-052.14E-055.34E-061.35E-06

1Lh8163264128

Continuous mortarsE(p) E(u) E(A)

3.20E-04 1.60E-02 3.32E-048.38E-05 4.27E-03 8.18E-052.12E-05 1.18E-03 2.01E-055.35E-06 3.41E-04 4.89E-061.38E-06 1.05E-04 1.15E-06

rate I O(h') O(h') O(h') I O(h1.99)TABLE 8.2

Discrete norm errors and convergence rates for the example with a discontinuous coefficient,where E(p) = lip - phllM, E(u) = IIlu - uhllJ, and E(>") = II>"- Qh>"hIIM'

derived by Ewing and Wang [16]. In the mortar method however, the flux continuitycondition can be relaxed by choosing a coarser mortar space. In this case the fine gridfluxes are not forced to be equal and approximate the solution better. Our numericalexperience shows that this may reduce the flux error on the interface by up to a factorof two.

We solve a problem with a solution

and a coefficient

]{ = ( 10 + 5cos(xy) 0)o l'

on locally refined grids. The domain is divided into four sub domains with interfacesalong the x = 1/2 and y = 1/2 lines. The domains are numbered starting from thelower left corner and first increasing x. The initial grids are 4 x 4 on nl-n3 and16 x 16 on n4' We use discontinuous piece-wise linear mortars on the non-matchinginterface. We report the numerical error on the grid and three levels of refinement fortwo cases. If the coarse grid is n x n, we take a mortar grid with n - 1 elements inthe first case and 2n elements in the second case, which is equivalent to the "slave"nodes method. The results are summarized in Table 8.3. The pressure and velocityerror on the first level of refinement are shown in Figure 8.4.

E(A)5.74E-31.39E-33.41E-48.42E-5O(h2.03)

E(u)1.45E-15.00E-21.74E-26.09E-3O(h1.52)

"Slave" nodesE(A)

3.80E-31.03E-32.72E-46.93E-5

E(u)6.70E-22.48E-29.77E-33.62E-3O(h1.40)

Discontinuous mortarsE(p)

1.12E-32.67E-46.57E-51.64E-5O(h2.03)rate

1Lh8163264

E(p)1.30E-32.90E-46.86E-51.66E-5

O(h1.93) I O(h2.09)

TABLE 8.3Discrete norm errors and convergence rates on locally refined grids, where E(p) = lip - phllM,

E(u) = Illu - uhllJ, and E(>") = II>"- Qh>"hIIM'

Acknowledgments. The authors thank Prof. Raytcho Lazarov for useful com-ments and discussions with the fourth author.


............................._4 ......_.............................................................. ~............................................................-......... - ..- ....................................................... ....................................................................................................................................... -...................................._ .............................'.. E::::::::::::::::::::::::::::....... - - - ........................

~:::::::::::::::::::::::::::- .................................................................................................-....................................................................................................................................:".....:::::::::::::::::::::::::::: :

I,·,,· ii'>: -.-................................................................................ .. .. ... ..~.............................. ...... ..... . ....... . .......... '

A. Discontinuous mortars.

...........................I~ :E=="::::::::::::::::::::::::: :...... - .....................Ii. ;;;;:.-="::::::::::::::::::::::::::Ii. ~-=-~~~~:::::::~~:~~:::~~~~:::..............................Ii.. z::.-;::::::::: :::::: ::::: ::::::...................... - ... - ..~ iEE;::::::::::::::::::::::::: :~ ii-=:~ ~~::::::::::::::~::::::::i-l ii·~~ ~~~~::~:::~:~:~~:~~~:~~::................. - .........

;;J.:':'::::;;;;;:::::::::::::::::::......... ~...............................

.

B. "Slave" nodes.

FIG. 8.4. Pressure and velocity error on locally refined grids.

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