A posteriori error estimate

8/6/2019 A posteriori error estimate

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A residual-based a posteriori error estimator for a fully-mixed

formulation of the Stokes-Darcy coupled problemGabriel N. Gatica Ricardo Oyarzua Francisco-Javier Sayas

Abstract

In this paper we develop an a posteriori error analysis of a new fully mixed finite elementmethod for the coupling of fluid flow with porous media flow in 2D. Flows are governed bythe Stokes and Darcy equations, respectively, and the corresponding transmission conditionsare given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman

law. We consider dual-mixed formulations in both media, which yields the pseudostress andthe velocity in the fluid, together with the velocity and the pressure in the porous medium,and the traces of the porous media pressure and the fluid velocity on the interface, as theresulting unknowns. The set of feasible finite element subspaces includes Raviart-Thomaselements of lowest order and piecewise constants for the velocities and pressures, respectively,in both domains, together with continuous piecewise linear elements for the traces. We derivea reliable and efficient residual-based a posteriori error estimator for the coupled problem.The proof of reliability makes use of the global inf-sup condition, Helmholtz decompositionsin both media, and local approximation properties of the Clement interpolant and Raviart-Thomas operator. On the other hand, inverse inequalities, the localization technique basedon triangle-bubble and edge-bubble functions, and known results from previous works, arethe main tools for proving the efficiency of the estimator. Finally, some numerical resultsconfirming the theoretical properties of this estimator, and illustrating the capability of thecorresponding adaptive algorithm to localize the singularities of the solution, are reported.

Key words: a posteriori error analysis, efficiency, reliability, Stokes, Darcy, fully-mixed

Mathematics Subject Classifications (2000): 65N15, 65N30, 74F10, 74S05

1 Introduction

The derivation of new finite element methods for the Stokes-Darcy coupled problem, in whichthe respective interface conditions are given by mass conservation, balance of normal forces,and the Beavers-Joseph-Saffman law, has become a very active research area lately (see, e.g.[5], [10], [13], [14], [20], [22], [23], [24], [29], [30], [35], [38], [40], [41], [42], [43], [47] and the

references therein). The above list includes porous media with cracks, nonlinear problems, andthe incorporation of the Brinkman equation in the model (see [10], [23], and [47]). In addition,

CI2MA and Departamento de Ingeniera Matematica, Universidad de Concepcion, Casilla 160-C, Concepcion,

Chile, email: [email protected] de Ingeniera Matematica, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile, email:

[email protected] de Matematica Aplicada, Centro Politecnico Superior, Universidad de Zaragoza, Mara de

Luna, 3 - 50018 Zaragoza, Spain, e-mail: [email protected] Present address: School of Mathematics, University

of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA.

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most of the formulations employed are based on appropriate combinations of stable elementsfor the free fluid flow and for the porous medium flow, and the first theoretical results in thisdirection go back to [22] and [35]. Indeed, an iterative subdomain method employing the primalvariational formulation and standard finite element subspaces in both domains is proposed in[22], whereas the primal method in the fluid and the dual-mixed method in the porous medium

are applied in [35]. In this way, the approach from [35] yields the velocity and the pressure inboth domains, together with the trace of the porous medium pressure on the interface, as themain unknowns of the coupled problem. This trace unknown is motivated by the fact that one ofthe transmission conditions becomes essential. Then, new mixed finite element discretizationsof the variational formulation from [35] have been introduced and analyzed in [29] and [30].The stability of a specific Galerkin method is the main result in [29], and the resulting mixedfinite element method is the first one that is conforming for the primal/dual-mixed formulationproposed in [35]. The results from [29] are improved in [30] where it is shown that the use ofany pair of stable Stokes and Darcy elements implies the stability of the corresponding Stokes-Darcy Galerkin scheme. The analysis in [30] hinges on the fact that the operator defining thecontinuous variational formulation is given by a compact perturbation of an invertible mapping.

Further techniques utilized in the literature include mortar finite element methods, discontinuousGalerkin (DG) schemes, and stabilized formulations (see, e.g. [5], [13], [14], [20], [21], [24], [38],[40], [41], [42], [43]). In particular, the main motivation for employing stabilized formulationseither in both domains or in one of them, is the possibility of approximating the Stokes andDarcy flows with the same finite element subpaces. Certainly, different finite element subspacesin each flow region may lead to different approximation properties for each subproblem. On thecontrary, using the same spaces guarantees the same accurateness along the entire domain andleads to simpler and more efficient computational codes.

Now, in the recent paper [31] we have developed a new variational approach for the 2DStokes-Darcy coupled problem, which allows, on one hand, the introduction of further unknownsof physical interest, and on the other hand, the utilization of the same family of finite elementsubspaces in both media, without requiring any stabilization term. More precisely, in [31]we consider dual-mixed formulations in both domains, which yields the pseudostress and thevelocity in the fluid, together with the velocity and the pressure in the porous medium, as themain unknowns. The pressure and the gradient of the velocity in the fluid can then be computedas a very simple postprocess of the above unknowns, in which no numerical differentiation isapplied, and hence no further sources of error arise. In addition, since the transmission conditionsbecome essential, we impose them weakly and introduce the traces of the porous media pressureand the fluid velocity, which are also variables of importance from a physical point of view, asthe corresponding Lagrange multipliers. Then, we apply the well known Fredholm and Babuska-Brezzi theories to prove the unique solvability of the resulting continuous formulation and derivesufficient conditions on the finite element subspaces ensuring that the associated Galerkin schemebecomes well posed. Among the several different ways in which the equations and unknownscan be ordered, we choose the one yielding a doubly mixed structure for which the inf-supconditions of the off-diagonal bilinear forms follow straightforwardly. In this way, the argumentsof the continuous analysis can be easily adapted to the discrete case. In particular, a feasiblechoice of subspaces is given by Raviart-Thomas elements of lowest order and piecewise constantsfor the velocities and pressures, respectively, in both domains, together with continuous piecewiselinear elements for the Lagrange multipliers.

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On the other hand, it is well known that in order to guarantee a good convergence behaviourof most finite element solutions, specially under the eventual presence of singularities, one usuallyneeds to apply an adaptive algorithm based on a posteriori error estimates. These are representedby global quantities that are expressed in terms of local indicators T defined on each elementT of a given triangulation T. The estimator is said to be efficient (resp. reliable) if there

exists Ceff > 0 (resp. Crel > 0), independent of the meshsizes, such that

Ceff + h.o.t. error Crel + h.o.t. ,

where h.o.t. is a generic expression denoting one or several terms of higher order. In particular,the a posteriori error analysis of variational formulations with saddle-point structure has alreadybeen widely investigated by many authors (see, e.g. [2], [3], [4], [11], [15], [17], [27], [33], [36], [37],[39], [44], and the references therein). These contributions refer mainly to reliable and efficienta posteriori error estimators based on local and global residuals, local problems, postprocessing,and functional-type error estimates. In addition, the applications include Stokes and Oseenequations, Poisson problem, linear elasticity, and general elliptic partial differential equations

of second order. However, up to our knowledge, the first a posteriori error analysis for theStokes-Darcy coupled problem has been provided recently in [8], where a reliable and efficientresidual-based a posteriori error estimator for the variational formulation analyzed in [29] isderived. Partially following known approaches, the proof of reliability makes use of suitableauxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved,and local approximation properties of the Clement interpolant and Raviart-Thomas operator.Similarly, Helmholtz decomposition, inverse inequalities, and the localization technique basedon triangle-bubble and edge-bubble functions, are the main tools for proving the efficiency ofthe estimator.

Motivated by the discussion in the above paragraphs, our purpose now is to additionallycontribute in the direction of [8] and provide the a posteriori error analysis of the fully-mixed

variational approach introduced in [31]. According to this, the rest of this work is organized asfollows. In Section 2 we recall from [31] the Stokes-Darcy coupled problem and its continuousand discrete fully-mixed variational formulations. The kernel of the present work is given bySection 3, where we develop the a posteriori error analysis. In Section 3.1 we employ theglobal continuous inf-sup condition, Helmholtz decompositions in both domains, and the localapproximation properties of the Clement and Raviart-Thomas operators, to derive a reliableresidual-based a posteriori error estimator. An interesting feature of our proof of reliabilityis the previous transformation of the global continuous inf-sup condition into an equivalentestimate involving global inf-sup conditions for each one of the components of the productspace to which the vector of unknowns belongs. Then, in Section 3.2 we apply again Helmholtzdecompositions, inverse inequalities, and the localization technique based on triangle-bubble and

edge-bubble functions to prove the efficiency of the estimator. This proof benefits partially fromthe fact that some components of the a posteriori error estimator coincide with those obtained in[8] and the related work [15]. Finally, numerical results confirming the reliability and efficiencyof the a posteriori error estimator and showing the good performance of the associated adaptivealgorithm, are presented in Section 4.

We end this section with some notations to be used below. In particular, in what follows weutilize the standard terminology for Sobolev spaces. In addition, if O is a domain, is a closed

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Lipschitz curve, and r R, we define

Hr(O) := [Hr(O)]2 , Hr(O) := [Hr(O)]22 , and Hr() := [Hr()]2 .

However, for r = 0 we usually write L2(O), L2(O), and L2() instead of H0(O), H0(O), and

H

0

(), respectively. The corresponding norms are denoted by r,O (for H

r

(O), H

r

(O), andHr(O)) and r, (for Hr() and Hr()). Also, the Hilbert space

H(div ; O) :=

w L2(O) : div w L2(O)

,

is standard in the realm of mixed problems (see, e.g. [12] or [32]). The space of matrix valuedfunctions whose rows belong to H(div ; O) will be denoted H(div; O). The Hilbert norms ofH(div ; O) and H(div; O) are denoted by div;O and div;O, respectively. On the otherhand, the symbol for the L2() and L2() inner products

, :=

, L2(), , :=

, L2()

will also be employed for their respective extensions as the duality products H1/2() H1/2()and H1/2() H1/2(). Finally, we employ 0 as a generic null vector, and use C and c, withor without subscripts, bars, tildes or hats, to mean generic positive constants independent ofthe discretization parameters, which may take different values at different places.

2 The Stokes-Darcy coupled problem

In this section we follow very closely the presentation from [31] to introduce the model problemand the corresponding continuous and discrete mixed variational formulations.

2.1 The model problem

The Stokes-Darcy coupled problem consists of an incompressible viscous fluid occupying a regionS, which flows back and forth across the common interface into a porous medium living inanother region D and saturated with the same fluid. Physically, we consider a simplified 2Dmodel where D is surrounded by a bounded region S (see Figure 2.1 below). Their commoninterface is supposed to be a Lipschitz curve and we assume that D = . The remainingpart of the boundary of S is also assumed to be a Lipschitz curve S. For practical purposes,we can assume that both S and are polygons. The unit normal vector field on the boundariesn is chosen pointing outwards from S (and therefore inwards to D when seen on ). On we also consider a unit tangent vector field t in any fixed orientation of this closed curve.

The governing equations in S are those of the Stokes problem, which are written in thefollowing non-standard velocity-pressure-pseudostress formulation:

S = pS I + uS in S , divS + fS = 0 in S ,

div uS = 0 in S , uS = 0 on S ,(2.1)

where > 0 is the viscosity of the fluid, uS is the fluid velocity, pS is the pressure, S is thepseudostress tensor, I is the 2 2 identity matrix, and fS L

2(S) are known source terms.

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t

n

nDS

S

Figure 2.1: Geometry of the problem

Here, div is the usual divergence operator acting on vector fields, and div denotes the action

of div along the rows of each tensor. On the other hand, the flow equations in D are those ofthe linearized Darcy model:

uD = K pD in D , div uD = fD in D , (2.2)

where the unknowns are the pressure pD and the flow uD, and the source term, given by fD

L2(D), satisfies

D

fD = 0. The matrix valued function K, describing permeability of D

divided by the viscosity , is symmetric, has L(D) components and is uniformly elliptic.Finally, the transmission conditions on are given by

uS n = uD n on ,

S n + 1 (uS t) t = pD n on , (2.3)

where :=

(K t) t

is the friction coefficient, and is a positive parameter to be determined

experimentally. The first equation in (2.3) corresponds to mass conservation on , whereas thenormal and tangential components of the second one constitute the balance of normal forcesand the Beavers-Joseph-Saffman law, respectively. Throughout the rest of the paper we assume,without loss of generality, that is a positive constant.

We complete the description of our model problem by observing that the equations in theStokes domain (cf. (2.1)) can be rewritten equivalently as

1dS = uS in S , divS + fS = 0 in S ,

pS = 12 trS in S , uS = 0 on S ,

(2.4)

where tr stands for the usual trace of tensors, that is tr := 11 + 22, and

d := 12 (tr ) I

is the deviatoric part of the tensor := (ij)22.

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We end this section by remarking that, though the geometry described by Figure 2.1 waschoosen to simplify the presentation, the case of a fluid flowing only across a part of the boundaryof the porous medium does not yield further complications for the a posteriori error analysisof the problem. We already discussed this issue in [31, Section 2.1], in connection with therespective a priori error analysis, and further details can be found in [24].

2.2 The fully-mixed variational formulation

We first define the global unknows := (S, uD,, ) and u := (uS, pD), where and arethe traces := uS| and := pD|. Then we recall from [31, Lemma 3.5] that the coupledproblem given by (2.2), (2.3), and (2.4) has the one-dimensional kernel defined by

{ ((S, uD,, ), (uS, pD)) : S = c I, uD = 0, = 0, = c, uS = 0, pD = c ; c R} .

Hence, in order to solve this indetermination, we introduce

L20

(D

) := q L2(D) : D q = 0 ,and define the product spaces

X := H(div; S) H(div;D) H1/2() H1/2() , M := L2(S) L

20(D) ,

endowed with the product norms

X := Sdiv,S + vDdiv ;D + 1/2, + 1/2, := (S, vD,, ) X ,

andvM := vS0.S + qD0,D v := (vS, qD) M .

In this way, as explained in [31, Sections 2 and 3]), it suffices to consider from now on thefollowing modified variational formulation of (2.2), (2.3), and (2.4): Find (, u) XM suchthat

A(, ) + B(, u) = F() := (S, vD,, ) X ,B(, v) = G(v) v := (vS, qD) M ,

(2.5)

whereF() := 0, G(v) = G((vS, qD)) := (fS, vS)S (fD, qD)D , (2.6)

and A and B are the bounded bilinear forms defined by

A(, ) := a((S, uD), (S, vD)) + b((S, vD), (, ))

+ b((S, uD), (, )) c((, ), (, )) ,(2.7)

with

a((S, uD), (S, vD)) := 1 (dS,

dS)S + (K

1 uD, vD)D ,

b((S, vD), (, )) := S n, vD n, ,

c((, ), (, )) := 1 t, t + n, n, ,

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and

B(, v) := (div S, vS)S (div vD, qD)D. (2.8)

Hereafter we utilize, for each {S, D}, the following notations

(u, v) := u v, (u, v) := u v, (, ) := : ,for all u, v L2(), u, v L

2(), and , L2(), where : := tr(

t).

We find it important to remark that and constitute the Lagrange multipliers associatedwith the transmission conditions (2.3). In addition, we notice that (2.5) is equivalent to thevariational formulation defined in [31, Section 3.2, eq. (3.2)], in which S is decomposed intoS = + , with H0(div; S) and R, where

H0(div; S) :=

H(div; S) :

S

tr() = 0

.

The following result taken from [31] establishes, in particular, the well-posedness of (2.5).Theorem 2.1 For each pair (F, G) X M there exists a unique (, u) X M solutionto (2.5), and there exists a constant C > 0, independent of the solution, such that

(, u)XM C

FX + GM0

. (2.9)

Proof. See [31, Theorem 3.9].

We end this section with the converse of the derivation of (2.5). More precisely, the followingtheorem establishes that the unique solution of (2.5), with F and G given by (2.6), solves theoriginal transmission problem described in Section 2.1. This result will be used later on inSection 3.2 to prove the efficiency of our a posteriori error estimator. We remark that no extra

regularity assumptions on the data, but only fS L2(S) and fD L2(D), are required here.

Theorem 2.2 Let (, u) H Q be the unique solution of the variational formulation (2.5)with F and G given by (2.6). ThendivS = fS in S,

1dS = uS in S, uS H1(S),

div uD = fD in D, uD = K pD in D, pD H1(D), uD n + n = 0 on ,

S n + n ( t) t = 0 on , = pD on , = uS on , and uS = 0 on S.

Proof. It basically follows by applying integration by parts backwardly in (2.5) and using suitabletest functions. We omit further details.

2.3 The Galerkin formulation

Although the analysis in [31] provides general hypotheses for the well-posedness of a Galerkinscheme of (2.5), it suffices to consider in what follows the particular case described in [31, Section5]. Let TSh and T

Dh be respective triangulations of the domains S and D formed by shape-

regular triangles T of diameter hT, and assume that TSh and T

Dh match in , so that their union

is a triangulation of S D. Then, for each T TSh T

Dh we let RT0(T) be the local

Raviart-Thomas space of order 0, that is

RT0(T) := span

10

,

01

,

x1x2

,

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where x :=

x1x2

is a generic vector ofR2, and for each {S, D} we define the global spaces

Hh() :=

vh H(div;) : vh|T RT0(T) T Th

, (2.10)

andLh() :=

qh : R : qh|T P0(T) T T

h

.

Hereafter, given a non-negative integer k and a subset S ofR2, Pk(S) stands for the space ofpolynomials defined on S of degree k. Next, we let h be the partition of inherited fromTSh (or T

Dh ), and assume, without loss of generality, that the number of edges of h is even. The

case of an odd number of edges is easily reduced to the even case (see [31]). Then, we let 2hbe the partition of arising by joining pairs of adjacent edges of h. Note that because h isinherited from one of the interior triangulations, it is automatically of bounded variation (thatis, the ratio of lengths of adjacent edges is bounded) and, therefore, so is 2h.

Employing the above notations, we now introduce

Hh(S) := { : S R22 : ct Hh(S) c R

2 } ,

Lh(S) := Lh(S) Lh(S) ,

Lh,0(D) := Lh(D) L20(D) ,

h() := { h C() : h|e P1(e) e edge of 2h } ,

h() := h() h() ,

and the product spaces

Xh := Hh(S) Hh(D) h() h() and Mh := Lh(S) Lh,0(D) .

In this way, the Galerkin scheme of (2.5) becomes: Find (h, uh) Xh Mh such that

A(h, ) + B(, uh) = F() := (S, vD,, ) Xh,B(h, v) = G(v) v := (vS, qD) Mh ,

(2.11)

where h = (S,h, uD,h,h, h) and uh := (uS,h, pD,h).

The following theorems, also taken from [31], provide the well-posedness of (2.11), the asso-ciated Cea estimate, and the corresponding theoretical rate of convergence.

Theorem 2.3 Assume that TSh

and TDh

are quasiuniform in a neighborhood of . Then theGalerkin scheme (2.11) has a unique solution (h, uh) Xh Mh. Moreover, there existC1, C2 > 0, independent of h, such that

(h, uh)XM C1

F|XhXh + G|MhM

h

,

and

hX + u uhM C2

inf

hXh hX + inf

vhMhu vhM

.

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Proof. See [31, Theorems 5.3 and 5.4].

Theorem 2.4 Assume the same hypotheses of Theorem 2.3, and let (, u) X M and(h, uh) Xh Mh be the unique solutions of the continuous and discrete formulations (2.5)and (2.11), respectively. Assume that there exists (0, 1] such thatS H

(S), divS

H

(S), uD H

(D), and div uD H

(D). Then, uS H1+

(S), pD H1+

(D), H1/2+(), H1/2+(), and there exists C > 0, independent of h and the continuousand discrete solutions, such that

(, u) (h, uh)XM C h

S,S + divS,S

+ uD,D + div uD,D + uS1+,S + pD1+,D

.

(2.12)

Proof. See [31, Theorem 5.5].

3 A residual-based a posteriori error estimator

We first introduce some notations. For each T TSh TDh we let E(T) be the set of edges of

T, and we denote by Eh the set of all edges of TSh T

Dh , that is

Eh = Eh(S) Eh(S) Eh(D) Eh() ,

where Eh(S) := { e Eh : e S }, Eh() := { e Eh : e } for each {S, D},and Eh() := { e Eh : e }. Note that Eh() is the set of edges defining the partitionh. Analogously, we let E2h() be the set of double edges defining the partition 2h. In whatfollows, he stands for the diameter of a given edge e Eh E2h(). Now, let {D, S} andlet q [L2()]

m, with m {1, 2}, such that q|T [C(T)]m for each T Th . Then, given

e Eh(), we denote by [q] the jump ofq across e, that is [q] := (q|T

)|e(q|T

)|e, where T

andT are the triangles ofTh having e as an edge. Also, we fix a unit normal vector ne := (n1, n2)t

to the edge e, which points either inward T or inward T, and let te := (n2, n1)t be the

corresponding fixed unit tangential vector along e. Hence, given v L2() and L2()

such that v|T [C(T)]2 and |T [C(T)]

22, respectively, for each T Th , we let [v te] and[te] be the tangential jumps of v and , across e, that is [v te] := {(v|T)|e (v|T)|e} teand [te] := {(|T)|e (|T)|e} te, respectively. From now on, when no confusion arises, wesimply write t and n instead of te and ne, respectively. Finally, for suffiently smooth scalar,vector and tensors fields q, v := (v1, v2)

t and := (ij)22, respectively, we let

curl v := v1x2

v1x1

v2x2

v2

x1 , curl q := qx2 , qx1

t

,

rot v :=v2x1

v1x2

, and rot :=

12x1

11x2

,22x1

21x2

t

.

Next, let (, u) XM and (h, uh) := ((S,h, uD,h,h, h), (uS,h, pD,h)) Xh Mh be theunique solutions of (2.5) and (2.11), respectively. Then, we introduce the global a posteriori

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error estimator:

:=

TTSh

2S,T +

TTDh

2D,T

1/2

, (3.1)

where, for each T TSh

:

2S,T := fS + divS,h20,T + h

2T rot

dS,h

20,T + h

2T

dS,h

20,T

+

eE(T)Eh(S)

he [dS,ht]

20,e +

eE(T)Eh(S)

he dS,ht

20,e +

eE(T)Eh()

he uS,h + h20,e

+

eE(T)Eh()

he

S,h n + h n

(h t) t20,e

+ he

1dS,ht + h t20,e

,

and for each T TDh :

2

D,T := fD div uD,h2

0,T + h2

T rot(K1

uD,h)2

0,T + h2

T K1

uD,h2

0,T

+

eE(T)Eh(D)

he[K1uD,h t]20,e +

eE(T)Eh()

he

K1uD,h t + dhdt20,e

+

eE(T)Eh()

he uD,h n + h n

20,e + he pD,h h

20,e

.

3.1 Reliability of the a posteriori error estimator

The main result of this section is stated as follows.

Theorem 3.1 There exists Crel > 0, independent of h, such that

hX + u uhM Crel . (3.2)

We begin the derivation of (3.2) by recalling that the continous dependence result given by(2.9) is equivalent to the global inf-sup condition for the continuous formulation (2.5). Then,applying this estimate to the error ( h, u uh) XM, we obtain

( h, u uh)XM C sup(,v)XM

(,v)=0

|R(, v)|

(, v)XM, (3.3)

where R : XM R is the residual operator defined by

R(, v) := A( h, ) + B(, u uh) + B( h, v), (, v) XM .

More precisely, according to (2.5) and the definitions of A and B (cf. (2.7), (2.8)), we find thatfor any (, v) := ((S, vD,, ), (vS, qD)) XM there holds

R(, v) = R1(S) + R2(vD) + R3() + R4() + R5(vS) + R6(qD) ,

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where

R1(S) := 1

S

dS,h : dS

S

uS,h divS S n,h ,

R2(vD) :=

DK1uD,h vD +

DpD,h div vD + vD n, h ,

R3() := S,h n, n, h +

t,h t ,

R4() := uD,h n, + h n, ,

R5(vS) :=

S

vS (fS + divS,h) ,

and

R6(qD) :=

D

qD (fD div uD,h) .

Hence, the supremum in (3.3) can be bounded in terms of Ri , i {1, ..., 6}, which yields

( h, u uh)XM C

supSH(div;S)S=0

|R1(S)|

Sdiv;S+ sup

vDH(div;D)

vD=0

|R2(vD)|

vDdiv ;D

+ supH1/2()=0

|R3()|

1/2,+ sup

H1/2()

=0

|R4()|

1/2,+ sup

vSL2(S)

vS=0

|R5(vS)|

vS0,S+ sup

qDL20(D)

qD=0

|R6(qD)|

qD0,D

.(3.4)

Throughout the rest of this section we provide suitable upper bounds for each one of the termson the right hand side of (3.4). The following lemma, whose proof follows from straightforwardapplications of the Cauchy-Schwarz inequality, is stated first.

Lemma 3.1 There hold

supvSL

2(S)

vS=0

|R5(vS)|

vS0,S fS + divS,h0,S =

TTSh

fS + divS,h20,T

1/2

, (3.5)

and

supqDL20(D)qD=0

|R6(qD)|

qD0,D fD div uD,h0,D = TTDh

fD div uD,h

2

0,T1/2

. (3.6)

The next lemma estimates the suprema on the spaces defined in the interface .

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Lemma 3.2 There exist C3 , C4 > 0, independent of h, such that

supH1/2()=0

|R3()|

1/2, C3

eEh()

he

S,h n + h n

(h t) t

2

0,e

1/2

, (3.7)

and

supH1/2()

=0

|R4()|

1/2, C4

eEh()

he uD,h n + h n20,e

1/2

. (3.8)

Proof. It is clear from the definition of R3 that

R3() = S,h n + h n

(h t) t, H

1/2() ,

and hence

sup

H1/2()=0

|R3()|

1/2,

= S,h n + h n

(h t) t1/2, . (3.9)In order to estimate

S,h n + h n (h t) t1/2, in terms of local quantities we now applya technical result from [16]. In fact, taking S = 0, vD = 0 and = 0 in the first equation of(2.11), we have

S,h n + h n

(h t) t, = 0 h() ,

which says that S,h n + h n (h t) t is L

2()-orthogonal to h(). Hence, applying [16,Theorem 2], and recalling that h and 2h are of bounded variation, we deduce that

S,h n + h n (h t) t21/2, C

eE2h()

he

S,h n + h n

(h t) t20,e

C

eEh()

he

S,h n + h n

(h t) t20,e

,

which, together with (3.9), yields (3.7).

The proof of (3.8) proceeds analogously. In fact, it is easy to see that

sup

H1/2()=0

|R4()|

1/2,

= uD,h n + h n1/2, ,

and hence, noting also from the first equation of (2.11) that uD,h n +h n is L2()-orthogonal

to h(), another straightforward application of [16, Theorem 2] yields the required estimate.We omit further details here.

Our next goal is to bound the first two suprema on the right hand side of (3.4), for whichwe need several preliminary results. We begin with the following lemma showing the existenceof stable Helmholtz decompositions for H(div;D) and H(div; S).

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Lemma 3.3

a) For each vD H(div; D) there exist w H1(D) and H

1(D), with

S

= 0,

such that there hold vD = w + curl in D, and

w1,D + 1,D CD vDdiv ;D ,

where CD is a positive constant independent of vD.

b) For each S H(div; S) there exist H1(S) and H

1(S) such that there holdS = + curl in S, and

1,S + 1,S CS Sdiv;S ,

where CS is a positive constant independent of S.

Proof. Given vD H(div;D), we let G be a smooth convex domain containing D, and letz H10(G) H

2(G) be the unique solution of

z =

div vD in D

0 in G \D

in G , z = 0 on G .

It follows thatz2,G Cdiv vD0,D CvDdiv ;D ,

and hence, defining w := z in D, we find that

div w = div vD in D and w1,D z2,D z2,G CvDdiv ;D .

In addition, since div (vD w) = 0 and D is connected, there exists H1(D), with

D

= 0, such that vD w = curl in D. In this way, using the generalized Poincare

inequality and the above estimate for w, we deduce that

1,D C||1,D = Ccurl 0,D = CvD w0,D CvDdiv ;D ,

which completes the proof of a).

We now let S H(div; S). Since S is not necessarily connected, we first perform asuitable extension of S to the domain := S D, and then apply a) to each row ofthe resulting tensor. More precisely, let S,i H(div;S) be the i-th row of S, i {1, 2},and let i H

1(D) be the unique solution of the Neumann problem:

i = S,i n, 1|D|

in D , in

= S,i n on , D

i = 0 .

Then we define exti =

S,i in S

i in D, and notice that exti H(div ; ) and

exti div ; S,idiv ;S + idiv ;D

S,idiv ;S + CS,i n1/2, CS,idiv ;S .

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Proceeding as in the proof of a), but now for exti H(div ; ), we deduce the existence of

wi H1() and i H

1(), with

i = 0, such that exti = wi + curl i in , and

wi1, + i1, Cexti div ; CS,idiv ;S .

Hence, the proof of b) follows by defining i-th row of := wi|S and := (1|S , 2|S).

The Raviart-Thomas interpolation operator h : H1() Hh() (cf. (2.10)), {S, D},

which, given v H1(), is characterized by

h(v) Hh() and

e

h(v) n =

e

v n edge e of Th , (3.10)

will also be needed in what follows. Note that as a consequence of (3.10), there holds

div (h(v)) = Ph(div v) , (3.11)

where Ph, {S, D}, is the L2()-orthogonal projector onto the piecewise constant functions

on . A tensor version of

h, say

h : H1

() Hh(), which is defined row-wise by

h,and a vector version of Ph, say Ph, which is the L

2()-orthogonal projector onto the piecewiseconstant vectors on , might also be required. The local approximation properties of

h (and

hence of h) are stated as follows.

Lemma 3.4 For each {S, D} there exist constants c1, c2 > 0, independent of h, such thatfor all v H1() there hold

v h(v)0,T c1 hT v1,T T Th ,

and

v n h(v) n0,e c2 h1/2e v1,Te edge e of T

h ,

where Te is a triangle of T

h containing e on its boundary.Proof. See [12].

We will also utilize the Clement interpolation operators Ih : H1() X,h (cf. [19]),

where

X,h := {v C() : v|T P1(T) T Th } for each {S, D} .

In addition, we will make use of a vector version of Ih, say Ih : H

1() X,h := X,h X,h,which is defined componentwise by Ih. The following lemma establishes the local approximationproperties of Ih (and hence of I

h).

Lemma 3.5 For each {S, D} there exist constants c3, c4 > 0, independent of h, such that

for all v H1() there hold

v Ih(v)0,T c3 hT v1,(T) T Th ,

and

v Ih(v)0,e c4 h1/2e v1,(e) e Eh ,

where

(T) := {T Th : T

T = 0} and (e) := {T Th : T

e = 0} .

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Proof. See [19].

Finally, we require the technical results given by the following two lemmas.

Lemma 3.6 Let H1(S) and H1(S). Then there hold

|R1( Sh())| c1 1 TTSh

hT dS,h0,T 1,T + c2 eEh()

h1/2e uS,h + h0,e 1,Te ,

and

|R1(curl ( ISh()))|

c3 1TTSh

hT rot (dS,h)0,T 1,S(T) + c4

1

eEh(S)

h1/2e [dS,ht]0,e 1,S(e)

+ c4 1

eEh(S)

h1/2e dS,ht0,e 1,S(e) + c4

eEh()

h1/2e

1dS,ht + h t0,e

1,S(e).

Proof. We first let := Sh() and observe, according to (3.10) and (3.11), thate

p n = 0 p [P0(e)]2 , edge e of TSh , and div = div P

Sh(div) .

Then, since dS,h : d = dS,h : and uS,h is a constant vector on each T T

Sh , we deduce from

the definition of R1 and the above identities that

R1() = 1TTSh

TdS,h :

d TTSh

T

uS,h div

eEh()

eh n

= 1 TTSh

TdS,h :

eEh()eh n

= 1TTSh

TdS,h :

eEh()

e

(uS,h + h) n .

On the other hand, we now let := ISh(). Then, using that div(curl ()) = 0, notingthat curl () n = t on , integrating by parts on each T TSh and on , and observingthat h t L

2(), we obtain

R1(curl ()) = 1

SdS,h : curl () curl () n,h

= 1TTSh

T rot (dS,h) +

T

dS,ht

+

eEh()

e (h t)

= TTSh

1T rot (dS,h) +

eEh(S)

1e [dS,h t]

+

eEh(S)

1e dS,h t +

eEh()

e

1dS,h + h

t .

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Hence, straighforward applications of the Cauchy-Schwarz inequality to the above equations,together with the approximation properties provided by Lemmas 3.4 and 3.5, namely

Sh()0,T c1 hT 1,T , n Sh() n0,e c2 h

1/2e 1,T

IS

h()0,T c3 hT 1,S(T) , and IS

h()0,e c4 h1/2

e 1,S(e) ,for each T TSh and for each e E(T), imply the required estimates and finish the proof.

Lemma 3.7 Let w H1(D) and H1(D). Then there hold

|R2(w Dh (w))| c1

TTDh

hT K1 uD,h0,T w1,T + c2

eEh()

h1/2e pD,h h0,e w1,Te ,

and|R2(curl( I

Dh ()))| c3

TTDhhT rot(K

1 uD,h)0,T 1,D(T)

+ c4 eEh(D)

h1/2e [K1 uD,h t]0,e 1,D(e)

+ c4

eEh()

h1/2e

K1 uD,h t + dhdt0,e

1,D(e) .

Proof. Since R1 and R2 have analogue structures, the proof proceeds similarly as for Lemma3.6.

We are now in a position to bound the suprema depending on R1 and R2.

Lemma 3.8 There exists C1 > 0, independent of h, such that

supSH(div;S)

S=0

|R1(S)|

Sdiv;S C1

TTSh

2S,T

1/2

, (3.12)

where, for each T TSh :2S,T := h2T rotdS,h20,T + h2T dS,h20,T+

eE(T)Eh(S)

he [dS,ht]

20,e +

eE(T)Eh(S)

he dS,ht

20,e

+ eE(T)Eh()

he 1dS,ht + h t20,e

+ he uS,h + h20,e

Proof. Given S H(div; S) we know from Lemma 3.3 that there exist H1(S) and

H1(S) such that S = + curl in S and

1,S + 1,S CSdiv;S . (3.13)

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Then, since R1(S,h) = 0 S,h Hh(S), which follows from the first equation of theGalerkin scheme (2.11) taking (vD,, ) = (0, 0, 0), and thanks to the fact that R1 is linear,we obtain

R1(S) = R1(S S,h) S,h Hh(S) . (3.14)

In particular, we let S,h := S

h() + curl (IS

h()), which can be seen as a discrete Helmholtz

decomposition ofS,h, and obtain

R1(S) = R1( Sh()) + R1(curl ( I

Sh()) . (3.15)

Hence, applying Lemma 3.6 and then the discrete Cauchy-Schwarz inequality to the resultingterms, noting that the numbers of triangles in S(T) and S(e) are bounded, and finally usingthe estimate (3.13), we conclude the upper bound (3.12).

Lemma 3.9 There exists C2 > 0, independent of h, such that

supvDH(div;D)

vD=0

|R2(vD)|

vDdiv ;D C2

TTDh

2D,T1/2

, (3.16)

where, for each T TDh :

2D,T := h2T rot(K1uD,h)20,T + h2T K1uD,h20,T + eE(T)Eh(D)

he[K1uD,h t]20,e

+

eE(T)Eh()

he

K1uD,h t + dhdt20,e

+ he pD,h h20,e

.

Proof. It follows basically the same lines of the proof of Lemma 3.8. In fact, given vD

H(div;D) we first apply Lemma 3.3 to deduce the existence of w H

1(D) and H1(D)

such that vD = w + curl and

w1,D + 1,D CvDdiv ;D . (3.17)

Then, since R2(vD,h) = 0 vD,h Hh(D), which corresponds to the first equation of theGalerkin scheme (2.11) with (S,, ) = (0, 0, 0), and thanks to the fact that R2 is linear, weobtain

R2(vD) = R2(vD vD,h) vD,h Hh(D) . (3.18)

Next, we choose vD,h = Dh (w) + curl

IDh (), notice that

R2(vD) = R2(w Dh (w)) + R2

curl( IDh ()) ,and apply Lemma 3.7. Thus, using again the discrete Cauchy-Schwarz inequality, noting thatthe numbers of triangles in D(T) and D(e) are bounded, and employing now the upper bound(3.17), we conclude (3.16).

We end this section by observing that the reliability estimate (3.2) (cf. Theorem 3.1) is adirect consequence of Lemmas 3.1, 3.2, 3.8, and 3.9.

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Lemma 3.12 Let k, l, m N {0} such that l m. Then, there exists c > 0, depending onlyon k, l, m and the shape regularity of the triangulations, such that for each triangle T thereholds

|q|m,T c hlmT |q|l,T, q Pk(T) . (3.23)

Proof. See Theorem 3.2.6 in [18].

In addition, we need to recall a discrete trace inequality, which establishes the existence of apositive constant c, depending only on the shape regularity of the triangulations, such that foreach T TSh T

Dh and e E(T), there holds

v20,e c

h1e v

20,T + he |v|

21,T

v H1(T) . (3.24)

For a proof of inequality (3.24) we refer to Theorem 3.10 in [1] (see also eq. (2.4) in [6]).

The following lemma summarizes known efficiency estimates for ten terms defining 2S,T and

2D,T. In fact, their proofs, which apply the preliminary results described above, are already

available in the literature (see, e.g. [8], [9], [15], [25], [26], [28]). From now on we assume,without loss of generality, that K1 uD,h is polynomial on each T T

Dh . Otherwise, additional

higher order terms, given by the errors arising from suitable p olynomial approximations, shouldappear in the corresponding bounds below, which explains the expression h.o.t. in (3.19).

Lemma 3.13 There exist positive constants Ci , i {1,..., 10}, independent of h, such that

a) h2T rot(K1 uD,h)

20,T C1 uD uD,h

20,T T T

Dh ,

b) h2T rotdS,h

20,T C2 S S,h

20,T T T

Sh ,

c) he |[K1 uD,h t]

20,e C3 uD uD,h

20,we e Eh(D), where the set we is given by

we := T TDh : e E(T),d) he [

dS,ht]

20,e C4 S S,h

20,we e Eh(S), where the set we is given by

we :=

T TSh : e E(T)

,

e) he dS,ht

20,e C5 S S,h

20,T e Eh(S), where T is the triangle of T

Sh having

e as an edge,

f) h2T K1 uD,h

20,T C6

pD pD,h

20,T + h

2T uD uD,h

20,T

T TDh ,

g) h2T dS,h

20,T C7 uS uS,h20,T + h2T S S,h20,T T TSh ,

h) he pD,hh20,e C8

pDpD,h

20,T + h

2T uDuD,h

20,T + he h

20,e

e Eh(),

where T is the triangle of TDh having e as an edge,

i)

eEh()

he

K1 uD,h t + dhdt20,e

C9

eEh()

uD uD,h20,Te + h

21/2,

,where, given e Eh(), Te is the triangle of T

Dh having e as an edge, and

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j)

eEh(S)

he

1dS,ht + ht20,e

C10

eEh(S)

S S,h20,Te + h

21/2,

,where, given e Eh(S), Te is the triangle of T

Sh having e as an edge.

Proof. For a) and b) we refer to [15, Lemma 6.1]. Alternatively, a) and b) follow from straight-forward applications of the technical result provided in [9, Lemma 4.3] (see also [28, Lemma4.9]). Similarly, for c), d), and e) we refer to [15, Lemma 6.2] or apply the technical result givenby [9, Lemma 4.4] (see also [28, Lemma 4.10]). Then, for f) and g) we refer to [15, Lemma6.3] (see also [28, Lemma 4.13] or [25, Lemma 5.5]). On the other hand, the estimate givenby h) corresponds to [8, Lemma 4.12]. In particular, its proof makes use of the discrete traceinequality (3.24). Finally, the proofs of i) and j) follow from very slight modifications of theproof of [25, Lemma 5.7]. Alternatively, an elasticity version of i) and j), which is provided in[26, Lemma 20], can also be adapted to our case.

We find it important to remark that the estimates i) and j) in the previous lemma provide theonly non-local bounds of the present efficiency analysis. However, under additional regularity

assumptions on and , one is able to prove the following local bounds.

Lemma 3.14 Assume that |e H1(e) for each e Eh(), and that |e H

1(e) for eache Eh(S). Then there exist C9, C10 > 0, such that

he

K1 uD,h t + dhdt20,e

C9

uD uD,h

20,Te + he

ddt h20,e

e Eh() ,

and

he

1dS,ht + ht2

0,e C10

S S,h

20,Te + he

d

dt

h

2

0,e

e Eh(S) .

Proof. Similarly as for i) and j) from Lemma 3.13, it follows by adapting the correspondingelasticity version from [26]. We omit details here and refer to [26, Lemma 21].

It remains to provide the efficiency estimates for three residual terms defined on the edges ofthe interface . They have to do with the transmision conditions and with the trace equationuS + = 0 on . More precisely, we have the following lemmas.

Lemma 3.15 There exists C > 0, independent of h, such that for each e Eh(), there holds

he uD,h n + h n20,e C

uD uD,h

20,T + h

2Tdiv(uD uD,h)

20,T + he h

20,e

,

where T is the triangle of TDh having e as an edge.

Proof. We proceed similarly as in [8, Lemma 4.7]. Given e Eh(), we let T be the triangle ofTDh having e as an edge, and define ve := uD,h n + h n on e. Then, applying (3.21), recallingthat e = 0 on T\e, extending e L(ve) by zero in D\T so that the resulting function belongsto H1(D), and using that uD n + n = 0 on , we get

ve20,e c2

1/2e ve

20,e = c2

e

e ve (uD,h n + h n) = c2 uD,h n + h n, e L(ve)

= c2 uD,h n uD n, e L(ve) + c2 h n n, e L(ve) ,(3.25)

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where , stands here for the duality pairing between H1/2() and H1/2(). Next, integra-

ting by parts in D, and noting thath n n

L2(), we find, respectively, that

uD,h n uD n, e L(ve) =

T

e L(ve)

(uD,h uD) +

T

e L(ve)div (uD,h uD) ,

andh n n, e L(ve) =

e

h n n

e ve .

Thus, replacing the above expressions back into (3.25), applying the Cauchy-Schwarz inequalityand the inverse estimate (3.23), and recalling that 0 e 1, we obtain

ve20,e C

h1T uD uD,h0,T + div(uD uD,h)0,T

eL(ve)0,T + c ve0,e h0,e.

But, using again that 0 e 1 and thanks to (3.22), we get

e L(ve)0,T 1/2e L(ve)0,T c

1/23 h

1/2e ve0,e , (3.26)

whence the previous inequality yields

ve0,e C h1/2eh1T uD uD,h0,T + div(uD uD,h)0,T + c h0,e .

Finally, it is easy to see that this estimate and the fact that he hT imply the required upperbound for he ve

20,e, which finishes the proof.


he S,h n + h n

(h t) t

20,e

C

S S,h20,T + h

2Tdiv(S S,h)

20,T + he h

20,e + he h

20,e

,

where T is the triangle of TSh having e as an edge.

Proof. We proceed as in the previous lemma (see also [8, Lemma 4.6]). Indeed, given e Eh(),we let T be the triangle of TSh having e as an edge, and define ve := S,h n + h n

(h t) t

on e. Then, applying (3.21), recalling that e = 0 o n T\e, extending e L(ve) by zero inS\T so that the resulting function belongs to H

1(S), using that S n + n ( t) t = 0

on , and then integrating by parts in S, we arrive at

ve20,e c2

1/2e ve

20,e = c2

e

e ve S,h n + h n

(h t) t

= c2

T

(e L(ve)) : (S,h S) + c2

T

e L(ve)) div(S,h S)

+ c2 e e ve (h ) n

(h t t) t .Next, applying the Cauchy-Schwarz inequality and the inverse estimate (3.23), recalling that0 e 1, and employing the vector version of (3.26), we deduce that

ve0,e C h1/2e

h1T S S,h0.T + div(S S,h)0,T

+ C

h0,e + h0,e

,

which easily yields the required estimate, thus finishing the proof.

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he uS,h + h20,e C

uS uS,h

20,T + h

2T S S,h

20,T + he h

20,e

,

where T is the triangle of TSh having e as an edge.

Proof. Let e Eh() and let T be the triangle of TSh having e as an edge. We follow the proof

of [8, Lemma 4.12] and obtain first an upper bound of h2T |uS uS,h|21,T. Indeed, using that

uS = 1dS in S (cf. Theorem 2.2) and that uS,h is constant in T, adding and substracting

dS,h, and then applying the estimate g) from Lemma 3.13, we deduce that

h2T |uS uS,h|21,T =

h2T2

dS20,T C h

2T

S S,h

20,T +

dS,h

20,T

C

uS uS,h

20,T + h

2T S S,h

20,T

.

(3.27)

Next, since = uS on (cf. Theorem 2.2), we find that

he uS,h + h20,e 2 he

uS uS,h

20,e + h

20,e

,

which, employing the discrete trace inequality (3.24) and the estimate (3.27), yields

he uS,h + h20,e C

uS uS,h

20,T + h

2T |uS uS,h|

21,T + he h

20,e

C

uS uS,h

20,T + h

2T S S,h

20,T + he h

20,e

,

which completes the proof.

We end this section by observing that the efficiency estimate (3.19) follows straightforwardlyfrom Lemmas 3.10, 3.13, 3.15, 3.16, and 3.17. In particular, the terms he h

20,e and

he h20,e, which appear in Lemma 3.13 (item h)), 3.15, 3.16, and 3.17, are bounded as

follows: eEh()

he h20,e h h

20, C h h

21/2, ,

and eEh()

he h20,e h h

20, C h h

21/2, .

4 Numerical resultsIn [31, Section 5] we presented several numerical results illustrating the performance of theGalerkin scheme (2.11) with the subspaces Xh := Hh(S) Hh(D) h() h() andMh := Lh(S) Lh,0(D) defined in Section 2.3. We now provide three examples confirmingthe reliability and efficiency of the respective a posteriori error estimator derived in Section3, and showing the behaviour of the associated adaptive algorithm.

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In what follows, N stands for the number of degrees of freedom defining Xh and Mh, andthe individual and total errors are defined by:

e(S) := S S,hdiv,S , e(uS) := uS uS,hdiv ;S ,

e

(uD) := uD uD,hdiv ;D ,e

(pD) := pD pD,h0,D ,e() := h1/2, , e() := h1/2, ,

and

e(, u) :=

(e(S))2 + (e(uS))

2 + (e(uD))2 + (e(pD))

2 + (e())2 + (e())21/2

,

whereas the effectivity index with respect to is given by

ef f() := e(, u)/ ,

where

(, u) := ((S, uD,, ), (uS, pD)) XM

and(h, uh) := ((S,h, uD,h,h, h), (uS,h, pD,h)) Xh Mh

denote the unique solutions of (2.5) and (2.11), respectively.

Also, we let r(S), r(uS), r(uD), r(pD), r(), r(), and r(, u) be the individual and globalexperimental rates of convergence given by

r(%) :=log(e(%)/e(%))

log(h/h)for each %

S, uS, uD, pD,,

,

andr(, u) :=

log(e(, u)/e(, u))

log(h/h),

where h and h denote two consecutive meshsizes with errors e and e. However, when theadaptive algorithm is applied (see details below), the expression log(h/h) appearing in thecomputation of the above rates is replaced by 12 log(N/N

), where N and N denote thecorresponding degrees of freedom of each triangulation.

The examples to be considered in this section are described next. In all of them we choosefor simplicity = 1, K = I, the identity matrix ofR22, and = 1. Example 1 is employedto confirm the reliability and efficiency of the a posteriori error estimator . Then, Examples 2and 3 are utilized to illustrate the behaviour of the associated adaptive algorithm, which appliesthe following procedure from [46]:

1) Start with a coarse mesh Th := TDh T

Sh .

2) Solve the discrete problem (2.11) for the actual mesh Th.

3) Compute ,T for each triangle T Th , {D, S}.

4) Evaluate stopping criterion and decide to finish or go to next step.

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5) Use blue-greenprocedure to refine each T Th , {D, S}, whose indicator ,T satisfies

T 1

2max

i{D,S}

max

i,T : T T

ih

.

6) Define resulting meshes as actual meshes TDh and T

Sh , and go to step 2.

In Example 1 we consider the regions D := ] 0.5, 0.5[2 and S := ] 1, 1[

2 \ D, whichyields a porous medium completely surrounded by a fluid, and choose the data fS and fD sothat the exact solution is given by the regular functions

uS(x) =

2 sin2(x1) sin(x2) cos(x2)2 sin(x1) sin

2(x2) cos(x1)

x := (x1, x2) S ,pS(x) = x

31 e

x2 x := (x1, x2) S ,

andpD(x) = x

31 sin(x2) x := (x1, x2) D .

In Example 2 we consider D := ] 1, 0[2 and let S be the L-shaped domain given by

] 1, 1[2 \ D, which yields a porous medium partially surrounded by a fluid. Then we choosethe data fS and fD so that the exact solution is given by

uS(x) = curl

0.1

x22 12

sin2(x1)

x := (x1, x2) S ,

pS(x) =1

100(x21 + x22) + 0.1

x := (x1, x2) S ,

andpD(x) =

x1 + 1

10

2sin3(2 (x2 + 0.5)) x := (x1, x2) D .

Note that the fluid pressure pS has high gradients around the origin.

Finally, in Example 3 we take D := ] 1, 1[ ] 2, 1[ and S := ] 1, 1[2 \ [0, 1]2, which

yields a porous medium below a fluid, and choose the data fS and fD so that the exact solutionis given by

uS(r, ) = curl

0.1 r5/3 (r2 cos2() 1)2 (r sin() 1)2 sin2

2

3

(r, ) S ,

pS(x) = 0.1 x1 sin(x2) x := (x1, x2) S ,

andpD(x) = 0.1 (x2 + 2)

2 sin3(x1) x := (x1, x2) D .

Note that uS is defined in polar coordinates and that its derivatives are singular at the origin.

The numerical results shown below were obtained using a MATLAB code. In Table 4.1 wesummarize the convergence history of the mixed finite element method (2.11), as applied toExample 1, for a sequence of quasi-uniform triangulations of the domain. We observe there,

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1

10

100 1000 10000 100000

N

quasi-uniform refinement

3

3

3

3

3

3adaptive refinement

++ +++

+ +

+

++ +

++

+

Figure 4.3: Example 3, e(, u) vs. N for quasi-uniform/adaptive schemes

[4] A. Alonso, Error estimators for a mixed method. Numerische Mathematik, vol. 74, pp.385-395, (1996).

[5] T. Arbogast and D.S. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Computational Geosciences, vol. 11, 3,pp. 207-218, (2007).

[6] D.N. Arnold, An interior penalty finite element method with discontinuous elements.SIAM Journal on Numerical Analysis, vol. 19, 4, pp. 742-760, (1982).

[7] I. Babuska and G.N. Gatica, On the mixed finite element method with Lagrange multi-pliers. Numerical Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).

[8] I. Babuska and G.N. Gatica, A residual-based a posteriori error estimator for theStokes-Darcy coupled problem. SIAM Journal on Numerical Analysis, vol. 48, 2, pp. 498-523,(2010).

[9] T.P. Barrios, G.N. Gatica, M. Gonzalez, and N. Heuer, A residual based a pos-teriori error estimator for an augmented mixed finite element method in linear elasticity.ESAIM: Mathematical Modelling and Numerical Analysis, vol. 40, 5, pp. 843-869, (2006).

[10] C. Bernardi, F. Hecht, and O. Pironneau, Coupling Darcy and Stokes equations forporous media with cracks. ESAIM: Mathematical Modelling and Numerical Analysis, vol.39, 1, pp. 7-35, (2005).

[11] D. Braess and R. Verfurth, A posteriori error estimators for the Raviart-Thomaselement. SIAM Journal on Numerical Analysis, vol. 33, pp. 24312444, (1996).

30


31/34

Figure 4.4: Example 3, adapted meshes with 1863, 3109, 11719, and 60159 degrees of freedom

31


32/34

[12] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag,1991.

[13] E. Burman and P. Hansbo, Stabilized Crouzeix-Raviart elements for the Darcy-Stokesproblem. Numerical Methods for Partial Differential Equations, vol. 21, 5, pp. 986-997,

(2005).[14] E. Burman and P. Hansbo, A unified stabilized method for Stokes and Darcys equations.

Journal of Computational and Applied Mathematics, vol. 198, 1, pp. 35-51, (2007).

[15] C. Carstensen, A posteriori error estimate for the mixed finite element method. Mathe-matics of Computation, vol. 66, 218, pp. 465-476, (1997).

[16] C. Carstensen, An a posteriori error estimate for a first kind integral equation. Mathe-matics of Computation, vol. 66, 217, pp. 139-155, (1997).

[17] C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM inelasticity. Numerische Mathematique, vol. 81, pp. 187-209, (1998).

[18] P. G. Ciarlet, The finite Element Method for Elliptic Problems. North-Holland, Amster-dam, New York, Oxorfd, 1978.

[19] P. Clement, Approximation by finite element functions using local regularisation. RAIROModelisation Mathematique et Analyse Numerique, vol. 9, pp. 77-84, (1975).

[20] M.R. Correa, Stabilized Finite Element Methods for Darcy and Coupled Stokes-DarcyFlows. D.Sc. Thesis, LNCC, Petropolis, Rio de Janeiro, Brasil (in portuguese), (2006).

[21] M.R. Correa and A.F.D. Loula, A unified mixed formulation naturally coupling Stokesand Darcy flows. Computer Methods in Applied Mechanics and Engineering, vol. 198, 33-36,

pp. 2710-2722, (2009).

[22] M. Discacciati, E. Miglio, and A. Quarteroni, Mathematical and numerical modelsfor coupling surface and groundwater flows. Applied Numerical Mathematics, vol. 43, pp.57-74, (2002).

[23] V.J. Ervin, E.W. Jenkins, and S. Sun, Coupled generalized nonlinear Stokes flow withflow through a porous medium. SIAM Journal on Numerical Analysis, vol. 47, 2, pp. 929-952,(2009).

[24] J. Galvis and M. Sarkis, Non-matching mortar discretization analysis for the couplingStokes-Darcy equations. Electronic Transactions on Numerical Analysis, vol. 26, pp. 350-

384, (2007).

[25] G.N. Gatica, A note on the efficiency of residual-based a-posteriori error estimators forsome mixed finite element methods. Electronic Transactions on Numerical Analysos, vol 17,pp. 218-233, (2004).

[26] G.N. Gatica, G.C. Hsiao, and S. Meddahi, A residual-based a posteriori error esti-mator for a two-dimensional fluid-solid interaction problem. Numerische Mathemaik, vol.114, 1, pp. 63-106, (2009).

32


33/34

[27] G.N. Gatica and M. Maischak, A posteriori error estimates for the mixed finite elementmethod with Lagrange multipliers. Numerical Methods for Partial Differential Equations,vol. 21, 3, pp. 421-450, (2005).

[28] G.N. Gatica, A. Marquez, and M.A. Sanchez, Analysis of a velocity-pressure-

pseudostress formulation for the stationary Stokes equations. Computer Methods in AppliedMechanics and Engineering, vol. 199, 17-20, pp. 1064-1079, (2010).

[29] G.N. Gatica, S. Meddahi, and R. Oyarzua, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA Journal of Numerical Analysis,vol. 29, 1, pp. 86-108, (2009).

[30] G.N. Gatica, R. Oyarzua and F-J Sayas, Convergence of a family of Galerkin dis-cretizations for the Stokes-Darcy coupled problem. Numerical Methods for Partial Differen-tial Equations DOI 10.1002/num, to appear.

[31] G.N. Gatica, R. Oyarzua, and F.-J. Sayas, Analysis of fully-mixed finite element

methods for the Stokes-Darcy coupled problem. Preprint 2009-08, Departamento de Inge-niera Matematica, Universidad de Concepcion, Chile, (2009).

[32] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations.Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, 1986.

[33] R.H.W. Hoppe and B.I. Wohlmuth, A comparison of a posteriori error estimators formixed finite element discretizations by Raviart-Thomas elements. Mathematics of Compu-tation, vol. 68, 228, pp. 1347-1378, (1999).

[34] R. Kress, Linear Integral Equations. Springer Verlag, Berlin, 1989.

[35] W.J. Layton, F. Schieweck, and I. Yotov, Coupling fluid flow with porous mediaflow. SIAM Journal on Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).

[36] M. Lonsing and R. Verfurth, A posteriori error estimators for mixed finite elementmethods in linear elasticity. Numerische Mathematik, vol. 97, 4, pp. 757-778, (2004).

[37] C. Lovadina and R. Stenberg, Energy norm a posteriori error estimates for mixed finiteelement methods. Mathematics of Computation, vol. 75, 256, pp. 1659-1674, (2006).

[38] A. Masud, A stabilized mixed finite element method for Darcy-Stokes flow. InternationalJournal for Numerical Methods in Fluids, vol. 54, 6-8, pp. 665-681, (2008).

[39] S. Repin, S. Sauter, and A. Smolianski, Two-sided a posteriori error estimates formixed formulations of elliptic problems. SIAM Journal on Numerical Analysis, vol. 45, 3,pp. 928-945, (2007).

[40] B. Riviere, Analysis of a discontinuous finite element method for coupled Stokes and Darcyproblems. Journal of Scientific Computing, vol. 22-23, pp. 479-500, (2005).

[41] B. Riviere and I. Yotov, Locally conservative coupling of Stokes and Darcy flows. SIAMJournal on Numerical Analysis, vol. 42, 5, pp. 1959-1977, (2005).

33


34/34

[42] H. Rui and R. Zhang, A unified stabilized mixed finite element method for coupling Stokesand Darcy flows. Computer Methods in Applied Mechanics and Engineering, vol. 198, 33-36,pp. 2692-2699, (2009).

[43] J.M. Urquiza, D. NDri, A. Garon, and M.C. Delfour, Coupling Stokes and Darcy

equations. Applied Numerical Mathematics, vol. 58, 5, pp. 525-538, (2008).[44] R. Verfurth, A posteriori error estimators for the Stokes problem. Numerische Mathe-

matik, vol. 55, pp. 309-325, (1989).

[45] R. Verfurth, A posteriori error estimation and adaptive mesh-refinement thecniques.Journal of Computational and Applied Mathematics, vol 50, pp. 67-83, (1994).

[46] R. Verfurth, A Review of A posteriori Error Estimation and Adaptive Mesh-RefinementThecniques. Wiley-Teubner (Chichester), 1996.

[47] X. Xie, J. Xu, and G. Xue, Uniformly stable finite element methods for Darcy-Stokes-Brinkman models. Journal of Computational Mathematics, vol. 26, 3, pp. 437-455, (2008).

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