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Mod´elisation Math´ematique et Analyse Num´erique M2AN, Vol. 36, No2, 2002, pp. 241–272 DOI: 10.1051/m2an:2002011

A-POSTERIORI ERROR ESTIMATES FOR LINEAR EXTERIOR PROBLEMS VIA MIXED-FEM AND DTN MAPPINGS

Mauricio A. Barrientos

1

, Gabriel N. Gatica

1

and Matthias Maischak

2

Abstract. In this paper we combine the dual-mixed finite element method with a Dirichlet-to- Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we de- velop two differenta-posteriorierror analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions.

Mathematics Subject Classification. 65N15, 65N30, 65N38, 65N50.

Received: June 5, 2001. Revised: January 8, 2002.

1. Introduction

The coupling of dual-mixed finite element methods (FEM) and boundary integral equation methods (BEM) has been frequently applied during the last few years to numerically solve a wide class of linear and nonlinear boundary value problems appearing in physics and engineering sciences (see,e.g.[6, 9, 18, 22, 24, 25, 32], and the references therein). As it is well known in mechanics, the utilization of dual-mixed FEM allows to compute stresses more accurately than displacements, and the use of BEM is more appropriate for linear homogeneous materials in bounded and unbounded regions. Analogously, according to the terminology in heat conduction problems, the above method combines the advantage of BEM for treating homogeneous domains and that of dual-mixed FEM for getting better approximations of the flux variable in heterogeneous media.

An alternative procedure, when dealing with exterior problems, consists of employing Dirichlet-to-Neumann (DtN) mappings. The combination of this approach with the usual FEM has been applied to several elliptic operators, including the Laplacian and the Lam´e system in elasticity (see, e.g.[20, 26, 28–30]). In these works

Keywords and phrases.Dirichlet-to-Neumann mapping, mixed finite elements, Raviart-Thomas spaces, residual based estimates, Bank-Weiser approach.

This research was partially supported by Fondecyt-Chile through the research project 2000124 and the FONDAP Program in Applied Mathematics, by the German Academic Exchange Service (DAAD) through the project 412/HP-hys-rsch, and by the Direcci´on de Investigaci´on of the Universidad de Concepci´on through the Advanced Research Groups Program.

1 GI2MA, Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.

e-mail:mbarrien@ing-mat.udec.cl, ggatica@ing-mat.udec.cl

2 Institut f¨ur Angewandte Mathematik, Universit¨at Hannover, Welfengarten 1, 30167 Hannover, Germany.

e-mail:maischak@ifam.uni-hannover.de

c EDP Sciences, SMAI 2002

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the corresponding DtN mapping either depends on a boundary integral operator or is expressed in terms of a Fourier-type series expansion. Now, in [16] we utilized the DtN mapping from [29] together with our dual-mixed finite element method from [25] to analyze an exterior transmission problem in hyperelasticity. Then, in [22]

we combined a modified dual-mixed FEM with the DtN mapping from [20] and [30] to study the solvability of a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. This modified dual-mixed method, which is based on the Hu-Washizu principle from elasticity, leads to two-fold saddle point operator equations, which are also called dual-dual mixed formulations (see [17, 18]).

On the other hand, in order to guarantee a good rate of convergence of the discrete solutions, one usually applies a mesh-refinement algorithm based on a suitablea-posteriorierror analysis. To this respect, concerning the combination of the usual FEM with BEM, we may refer to [10, 13, 14], where mainly reliable a-posteriori error estimates are provided. More recently, this kind of result has been extended to the coupling of dual-mixed FEM and BEM for linear and nonlinear problems (see [5, 6, 12, 19, 21, 23]). Here, the estimates for the linear problems are of explicit residual type, and those for the nonlinear ones are based on the classical Bank-Weiser implicit approach. Up to the authors’s knowledge, there is no further contributions in this direction for the combination of dual-mixed FEM with either BEM or DtN mappings.

The main purpose of the present work is to derive explicit and implicit reliablea-posteriorierror estimates for linear exterior problems in the plane, whose variational formulations are obtained by the combination of dual-mixed FEM with DtN mappings. As a model, we consider the exterior transmission problem from potential theory studied in [32] (see also [12, 21, 24]). In addition, we use the DtN mapping from [20, 30], which is given in terms of the hypersingular boundary integral operator for the Laplacian. The rest of the paper is organized as follows. In Section 2 we introduce the model problem, derive the associated mixed variational formulation, and prove the corresponding solvability and continuous dependence results. Actually, this is done through an equivalent formulation arising from a direct sum decomposition of one of the unknowns. In Section 3 we use Raviart-Thomas spaces to define the discrete scheme, show that it is stable and uniquely solvable, obtain the Cea error estimate, and state the associated rate of convergence. Then, a reliablea-posteriorierror estimate of explicit residual type is derived in Section 4. Our analysis here follows very closely the techniques from [12, 21].

In Section 5 we apply a Bank-Weiser typea-posteriorierror analysis and provide a reliable estimate that depends on the solution of local problems. An explicit estimate, based on bounds of these local solutions and a suitable averaging technique, is also deduced in this section. Finally, several numerical experiments illustrating the efficiency of these estimators for the adaptive computation of the discrete solutions are given in Section 6.

In what follows, the symbolsC, ˜C, and ¯Care used to denote generic positive constants with different values at different places.

2. The model problem

Let Ω0 be a bounded and simply connected domain in R2 with Lipschitz-continuous boundary Γ0. Also, let Ω1 be the annular domain bounded by Γ0and another Lipschitz-continuous closed curve Γ1 whose interior region contains Ω0. Then, givenf1 L2(Ω1), g H1/20) and a matrix valued function κ1 C(Ω1), we consider the exterior transmission problem: Find u1∈H1(Ω1)andu2∈Hloc1 (R201)such that

u1=g on Γ0, div (κ1∇u1) = f1 in Ω1, u1=u2 and (κ1∇u1)·n = ∂u2

∂n on Γ1,

∆u2= 0 in R201, u2(x) =O(1) as ||x|| →+∞,

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wheren:= (n1, n2)T denotes the unit outward normal to Γ1.

We assume thatκ1 induces a strongly elliptic differential operator, that is there existsα1>0 such that α1||ξ||2 1ξ)·ξ ξR2. (2)

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We now introduce a sufficiently large circle Γ with center at the origin such that its interior region contains Ω01. Then we let Ω2 be the annular region bounded by Γ1 and Γ, put Ω := Ω1Γ12, and define the global unknownu:=

u1 in Ω1

u2 in Ω2 , the data f :=

f1 in Ω1

0 in Ω2 , and the flux variableσ :=κ∇uin Ω, whereκ:=

κ1 in Ω1

I in Ω2 , andIdenotes the identity matrix.

Next, we apply the boundary integral equation method in the region exterior to the circle Γ, and obtain the following Dirichlet-to-Neumann mapping (see [20, 30])

σ·ν = 2W(λ) on Γ, (3)

whereνis the unit outward normal to∂Ω := Γ0Γ,λ:=u|Γis a further unknown, andWis the hypersingular boundary integral operator.

We remark that if Γ is choosen as a polygonal boundary instead of a circle, then we would need all the boundary integral operators to expressσ·ν in terms ofλ. The advantage of using a circle in this case lies on the simplicity of the resulting Dirichlet-to-Neumann mapping (3).

We recall here thatW is the linear operator defined by Wµ(x) :=

∂ν(x) Z

Γ

∂ν(y)E(x, y)µ(y) dsy ∀x∈Γ, ∀µ∈H1/2(Γ),

where ν(z) stands for the unit outward normal at z Γ, and E(x, y) :=−1 log||x−y|| is the fundamen- tal solution of the two-dimensional Laplacian. It is well known that W maps continuously H1/2+δ(Γ) into H1/2+δ(Γ) for allδ∈[1/2,1/2], and that there existsα2>0 such that

hW(µ), µiΓ α2||µ||2H1/2(Γ) ∀µ∈H01/2(Γ), (4) where

H01/2(Γ) :={µ∈H1/2(Γ) : h1, µiΓ = 0} ·

In addition,W(1) = 0 andWis symmetric in the sense thathW(µ), ρiΓ=hW(ρ), µiΓ for allµ,ρ∈H1/2(Γ).

Hereafter,h·,·iΓ (resp. h·,·iΓ0) denotes the duality pairing ofH1/2(Γ) andH1/2(Γ) (resp. H1/20) and H1/20)) with respect to theL2(Γ) (resp. L20)) inner product.

In this way, the model problem (1) is reformulated as a boundary value problem in Ω with the nonlocal boundary condition (3). Hence, by performing the usual integration by parts procedure in Ω, we find that the corresponding mixed variational formulation reads: Find((σ, λ), u)∈H×Qsuch that

A((σ, λ),, µ)) +B((τ, µ), u) = hτ·ν, giΓ0, B((σ, λ), v) = Z

f vdx, (5)

for all ((τ, µ), v)∈H×Q, whereH :=H(div; Ω)×H1/2(Γ),Q:=L2(Ω), and the bilinear formsA:H×H R andB :H×Q→Rare defined as follows:

A((σ, λ),, µ)) :=

Z

1σ)·τdx+ 2hWλ, µiΓ − hτ·ν, λiΓ + hσ·ν, µiΓ, (6) B((τ, µ), v) :=

Z

vdivτdx, (7)

for all (σ, λ), (τ, µ)∈H, for allv∈Q.

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At this point we recall thatH(div; Ω) is the space of functionsτ [L2(Ω)]2such that divτ ∈L2(Ω), which, provided with the inner product

hσ,τiH(div;Ω) :=

Z

divσdivτdx + Z

σ·τdx,

becomes a Hilbert space. In addition, for allτ ∈H(div; Ω),τ·ν|Γ∈H1/2(Γ),τ·ν|Γ0 ∈H1/20), and both

||τ ·ν||H−1/2(Γ) and||τ·ν||H−1/20) are bounded above by||τ||H(div;Ω).

On the other hand, eachµ∈H1/2(Γ) can be uniquely decomposed asµ:= ˜µ+q, with ˜µ:=

µ−|Γ1|

R

Γµds H01/2(Γ) andq := |Γ1|R

Γµds R, which states that H1/2(Γ) = H01/2(Γ)R. Further, it is easy to see that

||µ||2H1/2(Γ)=||µ˜||2H1/2(Γ)+|Γ| |q|2, and hence||µ||H1/2(Γ)and||µ, q)||H1/2(Γ)×R:=||µ˜||H1/2(Γ)+|q|are equivalent.

Then we write λ = ˜λ+p, with ˜λ H01/2(Γ), p R, and consider the alternative formulation: Find ((σ,λ),˜ (u, p))∈H˜×Q˜ such that

A((σ,˜λ),(τ,µ)) + ˜˜ B((τ,µ),˜ (u, p)) = hτ·ν, giΓ0, B˜((σ,˜λ),(v, q)) = Z

f vdx, (8)

for all ((τ,µ),˜ (v, q)) H˜ ×Q, where ˜˜ H := H(div; Ω)×H01/2(Γ), ˜Q := L2(Ω)×R, and the bilinear form B˜: ˜H×Q˜Ris defined as

B((τ˜ ,µ),˜ (v, q)) :=

Z

vdivτdx qhτ ·ν,1iΓ. (9) Then we have the following result.

Theorem 2.1. Problems (5)and(8)are equivalent. More precisely:

1. If ((σ, λ), u) H ×Q is a solution of (5), where λ := ˜λ+p, with ˜λ H01/2(Γ) and p R, then ((σ,λ),˜ (u, p))∈H˜×Q˜ is a solution of(8).

2. If((σ,λ),˜ (u, p))∈H˜×Q˜is a solution of(8), then((σ, λ), u)∈H×Qis a solution of(5)withλ:= ˜λ+p.

Proof. Let ((σ, λ), u)∈H×Qbe a solution of (5), whereλ:= ˜λ+p, with ˜λ∈H01/2(Γ) andp∈R, and consider ((τ,µ),˜ (v, q))∈H˜×Q. Since˜ W(p) = 0, it follows that

A((σ,λ),˜ (τ,µ)) + ˜˜ B((τ,µ),˜ (u, p)) = A((σ, λ),,µ)) +˜ B((τ,µ), u) =˜ hτ ·ν, giΓ0. (10) Now, taking µ = 1 and τ = 0 in the first equation of (5), and using the symmetry of W and the fact that W(1) = 0, we find thathσ·ν,1iΓ= 0, and hence

B((σ,˜ ˜λ),(v, q)) =B((σ,λ), v) =˜ B((σ, λ), v) = Z

f vdx.

This equation and (10) prove that ((σ,˜λ),(u, p))∈H˜×Q˜is a solution of (8).

Conversely, let ((σ,λ),˜ (u, p))∈H˜ ×Q˜ be a solution of (8), and defineλ:= ˜λ+p. Takingv= 0 andq= 1 in the second equation of (8), we deduce thathσ·ν,1iΓ = 0. Then we consider ((τ, µ), v)∈H×Q, such that µ:= ˜µ+q, with ˜µ∈H01/2(Γ) andq∈R, and observe that

A((σ, λ),, µ)) + B((τ, µ), u) = A((σ,˜λ),,µ)) + ˜˜ B((τ,µ),˜ (u, p)) = hτ·ν, giΓ0. (11)

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Also, according to the second equation in (8), we find that

B((σ, λ), v) = ˜B((σ,λ),˜ (v,0)) = Z

f vdx, which, together with (11), shows that ((σ, λ), u)∈H×Qis a solution of (5).

In virtue of Theorem 2.1, from now on we concentrate on the equivalent problem (8). The corresponding continuous and discrete analyses are based on the classical Babuˇska-Brezzi theory.

At this point we remark, which is easy to prove, that the bilinear formsA,B, and ˜B are all bounded.

We end this section with the following theorem providing the unique solvability and the continuous depen- dence result for the mixed variational formulation (8) (and hence also for (5)).

Theorem 2.2. There exists a unique ((σ,λ),˜ (u, p)) H˜×Q˜ solution of (8). Moreover, there exists C >0, independent of the solution, such that

||((σ,λ),˜ (u, p))||H˜×Q˜ C

||f||L2(Ω) + ||g||H1/20) ·

Proof. We first prove the continuous inf-sup condition for ˜B. Thus, given (v, q) Q˜ := L2(Ω)×R, we let z∈H1(Ω) be the weak solution of the mixed boundary value problem:

∆z=v in Ω, z= 0 on Γ0, ∂z

∂ν =q on Γ,

for which one can easily show that ||z||H1(Ω) C{||v||L2(Ω)+|q|}. Then we set τ0 :=−∇z and observe that divτ0=vin Ω, τ0·ν=−qon Γ, and ||τ0||H(div;Ω) C˜{||v||L2(Ω)+|q|}. It follows that

sup

(τµ)∈H˜ (τµ)6=0

B˜((τ,µ),˜ (v, q))

||,µ)˜ ||H˜

B((τ˜ 0,0),(v, q))

||τ0||H(div;Ω) = ||v||2L2(Ω)+|Γ| |q|2

||τ0||H(div;Ω) β||(v, q)||Q˜, whereβ depends on|Γ|and ˜C.

We now let ˜V be the kernel of the operator induced by the bilinear form ˜B, that is V˜ :={,µ)˜ ∈H˜ : B((τ,µ),˜ (v, q)) = 0 (v, q)∈H˜}, which yields

V˜ ={,µ)˜ ∈H(div; Ω)×H01/2(Γ) : divτ = 0 in Ω and hτ·ν,1iΓ= 0} · It follows, using (6), (2), and (4), thatAis strongly coercive on ˜V, that is, for all (τ,µ)˜ ∈V˜ it holds

A((τ,µ),˜ (τ,µ)) =˜ Z

1τ)·τdx+ 2hW(˜µ),µ˜iΓ α||,µ)˜ ||2H(div;Ω)×H1/2(Γ), whereαdepends onα1 andα2.

Finally, a straightforward application of the abstract Theorem 1.1 in Chapter II of [8] completes the proof.

3. The discrete scheme

Hereafter we assume, for simplicity, that Γ0 and Γ1 are polygonal boundaries. In order to discretize the circle Γ, we proceed similarly as in [22]. This means that givenn∈N, we let 0 =t0< t1<· · ·< tn = 2πbe a

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uniform partition of [0,2π] with tj+1−tj = ˜h= n forj∈ {0,1, ..., n1}. In addition, we letz: [0,2π]Γ be the parametrization of the circle Γ given byz(t) :=r(cos(t),sin(t))T for allt∈[0,2π]. We denote by Ω˜hthe annular domain bounded by Γ0 and the polygonal line Γ˜h whose vertices are{z(t1),z(t2), ...,z(tn)

Then we let T˜h be a regular triangulation of Ω˜h by triangles T of diameterhT such thath:= supT∈T˜

h hT. We assume that for eachT ∈ Th˜, either T 1 or T 2. Then, we replace each triangleT ∈ T˜h with one side along Γ˜h, by the corresponding curved triangle with one side along Γ. In this way, we obtain fromT˜h a triangulationTh of Ω made up of straight and curved triangles.

Next, we consider the canonical triangle with vertices ˆP1 = (0,0)T, ˆP2 = (1,0)T and ˆP3 = (0,1)T as a reference triangle ˆT, and introduce a family of bijective mappings {FT}T∈Th, such that FT( ˆT) = T. In particular, ifT is a straight triangle ofTh, thenFT is the affine mapping defined by FTx) =BTxˆ+bT, where BT, a square matrix of order 2, andbT R2depend on the vertices ofT.

On the other hand, if T is a curved triangle with vertices P1, P2 andP3, such thatP2 =z(tj1) Γ and P3=z(tj)Γ, thenFTx) =BTˆx+bT+GTx) for all ˆx:= (ˆx1,xˆ2)∈Tˆ, where

GTx) = xˆ1

1−xˆ2 {z(tj1+ ˆx2(tj−tj1))[z(tj1) + ˆx2(z(tj)z(tj1))]} · (12) We now let J(FT) and D(FT) denote, respectively, the Jacobian and the Frˆechet differential of the mapping FT. Then we summarize their main properties in the following lemma.

Lemma 3.1. There exists h0>0such that for all h∈(0, h0) FT is a diffeomorphism of classC that maps one-to-oneTˆonto the curved triangleT in such a way thatFT( ˆPi) =Pi for all i∈ {1,2,3}. In addition,J(FT) does not vanish in a neighborhood of T, and there exist positive constantsˆ Ci, i∈ {1, ...,5}, independent of T andh, such that for allT ∈ Th there hold

C1h2T ≤ |J(FT)| ≤ C2h2T, |J(FT)k|W1,∞( ˆT) C3h1+2kT ∀k∈ {−1,1}, and

|(DFT)|Wk,( ˆT) C4hk+1T , |(DFT)1|Wk,( ˆT) C5hkT1 ∀k∈ {0,1} · Proof. See Theorem 22.4 in [36].

Herafter, given s 0,k · kWs,∞( ˆT) and | · |Ws,∞( ˆT) (resp. k · k[Ws,∞( ˆT)]2×2 and | · |[Ws,∞( ˆT)]2×2) denote the norm and semi-norm of the usual Sobolev spaceWs,( ˆT) (resp. [Ws,( ˆT)]2×2). In addition,| · |[H1( ˆT)]2 is the semi-norm of [H1( ˆT)]2, and given a non-negative integerkand a subsetS ofRorR2,Pk(S) denotes the space of polynomials defined onS of degree≤k.

We now introduce the lowest order Raviart-Thomas spaces. For this purpose, we first let RT0( ˆT) := span 1

0

, 0

1

, xˆ1

ˆ x2

, (13)

and for each triangleT ∈ Th, we put

RT0(T) :={τ : τ =J(FT)1(DFT) ˆτ◦FT1, τˆ∈ RT0( ˆT)} · (14) Then, we define the finite element subspaces for the unknownsσ,λ, andu, as follows:

Hσ

h :={τh∈H(div; Ω) : τh|T ∈ RT0(T) ∀T ∈ Th}, (15) Hhλ:=h: ΓR, µh= ˆµhz1, µˆh∈Hhλ(0,2π)}, (16)

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with

Hhλ(0,2π) :=

n ˆ

µh: [0,2π]R, µˆh is continuous and periodic of period 2π, ˆ

µh|[tj−1,tj]P1(tj1, tj) ∀j ∈ {1, ..., n}o ,

and

Qh := {vh∈L2(Ω) : vh|T P0(T) ∀T ∈ Th} · (17) Thus, we setHh :=Hσ

h ×Hhλ and state the Galerkin scheme associated with the continuous problem (5) as:

Find ((σh, λh), uh)∈Hh×Qh such that

A((σh, λh),(τh, µh)) +B((τh, µh), uh) = hτh·ν, giΓ0, B((σh, λh), vh) = Z

f vhdx, (18)

for all ((τh, µh), vh)∈Hh×Qh.

Next, similarly as for the continuous problem, we introduce an alternative formulation, which is the discrete analogue of (8). To this end, we define

Hh,0λ := Hhλ H01/2(Γ), H˜h := Hσ

h × Hh,0λ , Q˜h:=Qh×R, (19) and consider the Galerkin scheme: Find ((σh˜h),(uh, ph))∈H˜h×Q˜h such that

A((σh˜h),(τh˜h)) + ˜B((τh˜h),(uh, ph)) = hτh·ν, giΓ0, B((σ˜ h,˜λh),(vh, qh)) = Z

f vhdx, (20)

for all ((τh˜h),(vh, qh))∈H˜h×Q˜h. Then we have the following result.

Theorem 3.2. Problems (18)and(20)are equivalent. More precisely:

1. If ((σh, λh), uh)∈Hh×Qh is a solution of(18), whereλh:= ˜λh+ph, withλ˜h∈Hh,0λ andphR, then ((σh˜h),(uh, ph))∈H˜h×Q˜h is a solution of(20).

2. If ((σh˜h),(uh, ph))∈H˜h×Q˜h is a solution of(20), then((σh, λh), uh)∈Hh×Qh is a solution of(18) with λh:= ˜λh+ph.

Proof. It is similar to the proof of Theorem 2.1 since it is based on the decomposition Hhλ :=Hh,0λ R. We omit further details.

Our next goal is to show that the Galerkin scheme (20) is stable and uniquely solvable. To this end, we consider first the equilibrium interpolation operator Eh : [H1(Ω)]2 Hσ

h , which, according to the Piola transformation used in (14), is given by (see,e.g.[8, 34])

Eh(τ)|T := J(FT)1(DFT) ˆEτ)◦FT1 ∀T ∈ Th,

where ˆτ :=J(FT) (DFT)1τ ◦FT and ˆE : [H1( ˆT)]2 → RT0( ˆT) is the local equilibrium interpolation operator on the reference triangle ˆT.

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Lemma 3.3. There existsC >0, independent ofh, such that

||τ− Eh(τ)||[L2(Ω)]2 C h||τ||[H1(Ω)]2 (21) and

kdiv (Eh(τ))kL2(Ω) CkdivτkL2(Ω) (22) for allτ [H1(Ω)]2.

Proof. Using the change of variablex=FTx), we find that kτ− Eh(τ)k2[L2(T)]2=

Z

T

kτ(x)J(FT)1(DFT) ˆEτ)(FT1(x))k22dx

= Z

Tˆ

|J(FT)|◦FT)(ˆx)−J(FT)1(DFT) ˆEτ)(ˆx)22x

= Z

Tˆ

|J(FT)|J(FT)1(DFT) h

ˆ

τ(ˆx)−Eˆ(ˆτ)(ˆx)i22x

Z

Tˆ |J(FT)|1k(DFT)k22τˆ(ˆx)−Eˆ(ˆτ)(ˆx)22x, (23) wherek · k2 is the usual euclidean norm for both vectors and matrices inR2 andR2×2, respectively.

Now, since|J(FT)1| =O(hT2) andk(DFT)k2 ≤C4hT (see Lem. 3.1), and because of the approximation property of ˆE, we deduce from (23) that

kτ− Eh(τ)k2[L2(T)]2 ≤Cˆkτˆ−Eˆ(ˆτ)k2[L2( ˆT)]2 Cˆ|τˆ|2[H1( ˆT)]2 = ˆC|J(FT) (DFT)1◦FT)|2[H1( ˆT)]2

≤Cˆ

n|J(FT)|W1,∞( ˆT)k(DFT)1)k[W0,∞( ˆT)]2×2kτ◦FTk[L2( ˆT)]2

+kJ(FT)kW0,∞( ˆT)|(DFT)1|[W1,∞( ˆT)]2×2kτ◦FTk[L2( ˆT)]2

+kJ(FT)kW0,∞( ˆT)k(DFT)1k[W0,∞( ˆT)]2×2|τ◦FT|[H1( ˆT)]2

o2

, (24)

with a constant ˆC >0, depending only on ˆT.

Next, applying the corresponding norm estimates for J(FT) and (DFT)1 (see again Lem. 3.1), changing back the variable ˆxbyFT1(x), and using chain rule in the term |τ◦FT|[H1( ˆT)]2, we conclude from (24) that

kτ− Eh(τ)k2[L2(T)]2 C hˆ 2kτk2[H1(T)]2 ∀T ∈ Th. (25) On the other hand, we know from the conmuting diagram property on the reference triangle ˆT that

kdiv ˆEτ)kL2( ˆT)≤ kdiv ˆτkL2( ˆT).

Then we use the above inequality, identity (1.49) (cf. Lem. 1.5) in Chapter III of [8], and Cauchy-Schwarz’s inequality, to find that

kdivEh(τ)k2L2(T):=

Z

T

divEh(τ) divEh(τ) dx = Z

Tˆ

div\Eh(τ) div ˆEτ) dˆx

≤ kdiv\Eh(τ)kL2( ˆT)kdiv ˆEτ)kL2( ˆT) ≤ kdiv\Eh(τ)kL2( ˆT)kdiv ˆτkL2( ˆT), (26)

(9)

wherediv\Eh(τ) stands for divEh(τ)◦FT.

Then, applying the inequalities (1.40) (cf. Lem. 1.4) and (1.54) (cf. Lem. 1.6) in Chapter III of [8], and the estimate forJ(FT) given in Lemma 3.1, we deduce that

kdiv\Eh(τ)kL2( ˆT) C hT1kdivEh(τ)kL2(T) and kdiv ˆτkL2( ˆT) C hTkdivτkL2(T), which replaced back into (26) yields

kdiv (Eh(τ))kL2(T) CkdivτkL2(T) ∀T∈ Th. (27) Hence, summing up over all the trianglesT ∈ Thin (25) and (27), we conclude, respectively, (21) and (22).

We are now in a position to prove the discrete inf-sup condition for the bilinear form ˜B.

Lemma 3.4. There existsβ>0, independent ofh, such that for all(vh, qh)∈Q˜h it holds sup

(τh,µh˜ )Hh˜ (τh ,µh˜ )6=0

B˜((τh˜h),(vh, qh))

||h˜h)||H β||(vh, qh)||Q˜.

Proof. Given (vh, qh)∈Q˜h, we note that sup

(τh ,µh˜ )∈Hh˜ (τh,µh˜ )6=0

B((τ˜ h˜h),(vh, qh))

||h˜h)||H sup τh∈Hσ

τh6=0h

B((τ˜ h,0),(vh, qh))

||τh||H(div;Ω) ·

Then, we define ˜vh :=



vh in Ω

1

|0| Z

vhdx+qh|Γ|

in Ω¯0

, put ˜Ω := ΩΩ¯0, and let z H1( ˜Ω) be the weak solution of

∆z= ˜vh in Ω,˜ ∂z

∂ν =qh on Γ, Z

˜

zdx= 0.

Since ˜Ω, being the interior region of the circle Γ, is clearly convex, the usual regularity result (see, e.g. [27]) implies thatz∈H2( ˜Ω) and

||z||H2( ˜Ω) C{ ||vh||L2(Ω)+|qh| } ·

Thus we define ˜τ :=−∇z|[H1(Ω)]2, and observe that div ˜τ =vh in Ω, ˜τ·ν=−qhon Γ, and

||τ˜||[H1(Ω)]2 = ||∇z||[H1(Ω)]2 ≤ ||z||H2( ˜Ω) C{ ||vh||L2(Ω)+|qh| } · (28) Further, it is easy to see that

||τ˜||H(div;Ω) C

||vh||L2(Ω) +|qh| · (29) Then, using the approximation property (21) and the estimate (22) (cf. Lem. 3.3), we find that

||Ehτ)||2H(div;Ω)=||Ehτ)||2[L2(Ω)]2 + ||div (Ehτ))||2L2(Ω)

≤C

nkτ˜− Ehτ)k2[L2(Ω)]2 +kτ˜k2[L2(Ω)]2 + kdiv ˜τk2L2(Ω)

o

≤C n

h2kτ˜k2[H1(Ω)]2 +kτ˜k2H(div;Ω)

o ,

(10)

which, using (28) and (29), implies

||Ehτ)||H(div;Ω) C

||vh||L2(Ω) + |qh| · (30) We now let Ph be the orthogonal projection from L2(Ω) onto the finite element subspace Qh. Then, using the identity (1.49) (cf. Lem. 1.5) in Chapter III of [8] and the conmuting diagram property on the refer- ence triangle ˆT, similarly as we did in the proof of Lemma 3.3, we deduce that in this case there also holds Ph(divEhτ)) =Ph(div ˜τ), which yields

Z

vhdivEhτ) dx = Z

vhdiv ˜τdx = kvhk2L2(Ω). Next, sinceR

eEhτ)·νeds=R

eτ˜·νedsfor all the edgeseofTh, withνe being the unit outward normal toe, and since ˜τ·ν=−qh on Γ, we deduce thathEhτ)·ν,1iΓ= −qh|Γ|.

According to the above analysis we can write sup

τh∈Hσ τh6=0h

B((τ˜ h,0),(vh, qh))

||τh||H(div;Ω) ≥B((˜ Ehτ),0),(vh, qh))

||Ehτ)||H(div;Ω) = ||vh||2L2(Ω) +|Γ|q2h

||Ehτ)||H(div;Ω)

≥β||(vh, qh)||Q˜, where the last inequality follows from (30). This finishes the proof.

We are now in a position to provide the stability and unique solvability of the Galerkin scheme (20), and the corresponding Cea estimate.

Theorem 3.5. There exists a unique ((σh˜h),(uh, ph))∈H˜h×Q˜h solution of(20). In addition, there exists C >0, independent of h, such that

||((σh˜h),(uh, ph))||H˜×Q˜ C

||f||L2(Ω) +||g||H1/20) , and

||((σ,˜λ),(u, p))((σh˜h),(uh, ph))||H˜×Q˜≤C min

((τhµh),vh)H˜h×Qh ||((σ,λ), u)˜ ((τh˜h), vh)||H˜×Q. Proof. Let ˜Vhbe the discrete kernel of the operator induced by the bilinear form ˜B. It is easy to show, according to the definition of ˜B (cf. (9)) and Lemma 5.7 in [22], that

V˜h :={h˜h)∈H˜h: hτh·ν,1iΓ= 0 and divτh= 0 in Ω}, and hence the bilinear formAis uniformly strongly coercive on ˜Vh.

In this way, Lemma 3.4 and direct applications of the abstract Theorems 1.1 and 2.1 in Chapter II of [8]

complete the proof.

We end this section with a result on the rate of convergence of the Galerkin scheme (20). For this purpose, we recall the following approximation properties of the subspacesHσ

h ,Hh,0λ , and Qh, respectively (see,e.g.[2, 8, 31, 34]):

1. (APσ

h): For allτ∈[H1(Ω)]2with divτ ∈H1(Ω), it holds

||τ− Eh(τ)||H(div;Ω) C h

||τ||[H1(Ω)]2 + ||divτ||H1(Ω) ·

(11)

2. (APλh,0): For all ˜µ∈H3/2(Γ)∩H01/2(Γ), there exists ˜µh∈Hh,0λ such that

||µ˜ −µ˜h||H1/2(Γ) C h||µ˜||H3/2(Γ). 3. (APh): For allv∈H1(Ω) it holds

||v − Ph(v)||L2(Ω) C h||v||H1(Ω), wherePh is the orthogonal projection fromL2(Ω) ontoQh.

Then we can establish the following theorem.

Theorem 3.6. Let((σ,λ),˜ (u, p))and((σh˜h),(uh, ph))be the unique solutions of the continuous and discrete mixed formulations(8) and(20), respectively. Assume thatσ [Hs(Ω)]2,divσ ∈Hs(Ω),λ˜∈Hs+1/2(Γ)and u∈Hs(Ω), for somes∈(0,1]. Then there exists C >0, independent ofh, such that

||((σ,˜λ),(u, p))((σh˜h),(uh, ph))||H˜×Q˜≤C hsn

||σ||[Hs(Ω)]2+||divσ||Hs(Ω)+||˜λ||Hs+1/2(Γ)+||u||Hs(Ω)

o·

Proof. It follows from the Cea estimate in Theorem 3.5, the above approximation properties, and suitable interpolation theorems in the Sobolev spaces.

4. An explicit residual

A-POSTERIORI

estimate

Let us first introduce some notations. We letE(T) be the set of edges ofT ∈ Th, and letEhbe the set of all edges of the triangulationTh. Then we writeEh=Eh(Ω)∪Eh0)∪Eh(Γ), whereEh(Ω) :={e∈Eh: e⊆}, Eh(Γ) :={e∈Eh : e⊆Γ}, and similarly forEh0). In what follows, hT and he stand for the diameters of the triangleT∈ Thand edgee∈Eh, respectively. Also, given a vector-valued functionτ := (τ1, τ2)T defined in Ω, an edgee∈E(T)∩Eh(Ω), and the unit tangential vectortT alonge, we letτT be the restriction ofτ toT, and letJ·tT] be the corresponding jump acrosse, that isJ·tT] := (τTτT0)|e·tT, whereT0 is the other triangle ofTh having eas edge. Here, the tangential vectortT is given by (−ν2, ν1)T whereνT := (ν1, ν2)T is the unit outward normal to∂T. Finally, we let curl (τ) be the scalar ∂x∂τ2

1 ∂τ∂x12· Next, we define the finite element space

Xh := {vh∈C(Ω) : vh|T = ˆv◦FT1, ˆv∈P1( ˆT), ∀T ∈ Th},

and letIh:H1(Ω)−→Xhbe the usual Cl´ement interpolation operator (see [7,15]). The following lemma states the local approximation properties ofIh.

Lemma 4.1. There exist positive constants C1 and C2, independent of h, such that for all ϕ ∈H1(Ω) there holds

||ϕ−Ih(ϕ)||L2(T) C1hT||ϕ||H1(∆(T)) ∀T ∈ Th, and

||ϕ−Ih(ϕ)||L2(e) C2h1/2e ||ϕ||H1(∆(e)) ∀e∈ Eh, where∆(T) :=∪{T0∈ Th: T0∩T 6=∅}, and ∆(e) :=∪{T0 ∈ Th: T0∩e6=∅}·

Proof. See Theorem 4.1 in [7].

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