Adaptive finite element methods for elliptic problems : abstract framework and applications

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DOI:10.1051/m2an/2010010 www.esaim-m2an.org

ADAPTIVE FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS:

ABSTRACT FRAMEWORK AND APPLICATIONS

Serge Nicaise

¹

and Sarah Cochez-Dhondt

¹

Abstract. We consider a general abstract framework of a continuous elliptic problem set on a Hilbert spaceV that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included intoV. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convection- reaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.

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

Received June 26, 2008.

Published online February 4, 2010.

1. Introduction

The convergence of adaptive algorithms for elliptic boundary value problems approximated by a conforming FEM started with the papers of Babuˇska and Vogelius [5] in 1d and of Dörfler [12] in 2d. Since this time some improvements have been proved in order to take into account the data oscillations [23–25] or to prove optimal arithmetic works [8]. On the other hand, the discontinuous Galerkin method becomes recently very popular and is a very efficient tool for the numerical approximation of reaction-convection-diffusion problems for instance. Some a posteriori error analysis were performed recently, let us quote [1,2,7,10,14,16,18,20,21,26] for pure diffusion (or diffusion-dominated) problems and [9,13,16] for singularly perturbed problems (i.e., dominant advection or reaction); for Maxwell system see for instance [17]. In all these papers, no convergence results are proved and to our knowledge, only the recent paper of Karakashian and Pascal [19] provides a convergence result for a purely diffusion problem.

Reading carefully the papers [12,19,23–25] we can remark some similarities in the convergence proof. Hence the goal of the present paper is threefold:

– Give an abstract framework as large as possible in order to contain the setting of the previous papers.

In particular since DG methods use a stability parameterγ >0, we assume that our variational form depends on such a parameter.

Keywords and phrases. A posterioriestimator, adaptive FEM, discontinuous Galerkin FEM.

1 Universit´e de Valenciennes et du Hainaut Cambr´esis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr;

Sarah.Cochez@univ-valenciennes.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2010

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– In this setting, give a series of realistic conditions on an error estimator (inspired from the above mentioned references) that allows to prove that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. In this way the convergence proof becomes quite easy to understand since it is not hidden between some technicalities.

– Apply this framework to some new examples, like the ones from [9,10,16], in order to deduce the convergence of the associated adaptive algorithm. According to the two points before, the convergence result is reduced to check the above mentioned conditions.

We further hope that this framework will be applied to other a posteriori error estimators, like the ones from [13,16] where robust methods were analyzed for problems with dominant advection or reaction.

The schedule of the paper is as follows: we state in Section2the abstract framework, and prove the convergence of the adaptive algorithm under some realistic conditions. In the remainder of the paper we check the conditions from this section and deduce the convergence of the associated adaptive algorithm of bulk type for different estimators built for some finite element approximations of some specific boundary value problems. In Section 3, we apply our theory to diffusion problems approximated by a discontinuous Galerkin method and an error estimator based on equilibrated fluxes; some numerical tests are also presented that confirm the theoretical results. Finally Section 4 performs the same analysis for convection-diffusion-reaction problems approximated by a discontinuous Galerkin method and an error estimator of residual type.

2. An abstract framework

LetV be a real Hilbert space with norm denoted by · V.Letabe a bilinear continuous form onV, which is coercive in the usual sense, namely

∃α >0 :a(u, u)≥αu²_V ∀u∈V. (2.1) Consider the standard variational problem: givenf ∈V, letu∈V be the unique solution of

a(u, v) =f, v, ∀v∈V. (2.2)

This problem is approximated by a discrete family of problems set on a ﬁnite dimensional spaceV_h, h >0, which we suppose to be nested:

V_H ⊂V_h ifH > h,

and that approachV ashgoes to zero. Note that we do not assume thatV_h is a subspace ofV, this allows us to consider non conforming approximation like discontinuous Galerkin methods for instance.

For allh >0 and a family of parametersγ >0, we assume given a norm · _h,γ well deﬁned onV +V_h and a family of bilinear forma_h,γ also well deﬁned onV +V_h such that a_h,γ is coercive on V_h, namely we assume that there existγ₀>0 andα₀>0 independent ofhandγ such that

a_h,γ(v_h, v_h)≥α₀v_h²_h,γ ∀v_h∈V_h, ∀γ≥γ₀. (2.3) With these assumptions, for γ≥γ₀, we can consideru_h ∈V_h solution of (for shortness, we do not specify the dependence ofu_hwith respect to γ)

a_h,γ(u_h, v_h) =f, v_h, ∀v_h∈V_h, (2.4) assuming that f ∈V_h.

Ifa_h,γ would be coercive on V +V_h, thena_h,γ(u−u_h, u−u_h) would dominate the error u−u_h²_h,γ, since we do not assume this coerciveness, we would lose this property. Since it plays a key rule in our analysis, we assume that it holds: there existsα₀>0 independent ofh,γ,uandu_hsuch that

a_h,γ(u−u_h, u−u_h)≥α₀u−u_h²_h,γ ∀γ≥γ₀. (2.5)

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Let us emphasize on the fact that we require (2.5) only foru∈V the exact solution of (2.2) and its approximation u_h∈V_h solution of (2.4).

Our goal is to show that under some basic assumptions (satisﬁed by a quite large family of approximated problems, see below) then reﬁnement strategy based on the bulk criterion [12,23,24] described below leads to the convergence of the algorithm.

To describe this reﬁnement strategy in our framework, for allh >0 we suppose given a ﬁnite familyTh of elements, calledT. Formally this family Th allows to built the spaceV_h by using polynomial functions on the elementsT ofTh for instance. Now for allT ∈ Th we assume that we have at our disposal an estimatorη_T(u_h) (that can be computed with the help ofu_h) that measures the local error onT (henceη_T(u_h) is a nonnegative real number) and for which we have the following upper bound: there exist two positive constants C₁, c₁ >0 independent ofhand γsuch that

a_h,γ(u−u_h, u−u_h)≤C₁η_h²+c₁osc²_h, (2.6) where the ﬁrst term of this right-hand side is the global error estimator which is the sum of local contributions, while the second term osc²_h is the so-called oscillation term that is also the sum of local contributions

η²_h=

T∈Th

η_T(u_h)², osc²_h=

T∈Th

osc_h(T)².

Now the abstract reﬁnement strategy (of bulk type, see [12,23,24]) can be expressed as follows:

Definition 2.1(marking strategy). Given two parameters 0< θ₁, θ₂<1, the new familyTh (allowing to build the new spaceV_h) is designed with the help of a subset ˆTH ofTH constructed such that

T∈TˆH

η_H(T)²≥θ₁²η_H², (2.7)

T∈TˆH

osc_H(T)²≥θ₂²osc²_H. (2.8)

Note that these two conditions yield

T∈TˆH

(η_H(T)²+ osc_H(T)²)≥θ²(η²_H+ osc²_H), (2.9)

withθ= min{θ₁, θ₂}.

We further assume that the next error reduction holds: there exist a positive constantC₂and a non negative constantC₃ such that for two consecutive parametersH > h

T∈TˆH

η_T(u_H)²≤C₂(u_h−u_H²_h,γ+ osc²_H) +C₃

γ u−u_h²_h,γ ∀γ≥γ₀. (2.10) Note that such estimate is usually proved locally, the local version leading to the estimate (2.10) by super- position. For the proof of the convergence of the algorithm the global version is only necessary, moreover in some applications we have in mind (see Sect. 3.2below), only the global version is available. Hence we have restricted ourselves to the global estimate.

For the oscillation terms, we make the assumption that it reduces from one step to another one with a factor

<1 up to the consecutive error, namely we assume that there exist constants 0< ρ₁<1 andρ₂>0 independent ofhandγsuch that

osc²_h≤ρ₁osc²_H+ρ₂a_h,γ(u_h−u_H, u_h−u_H) ∀γ≥γ₀. (2.11)

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We further assume that the error betweenuandu_H in the norm a_h,γ or in the norma_H,γ are comparable, namely we assume that

∃C₄≥0 : a_h,γ(u−u_H, u−u_H)≤

1 +C₄ γ

a_H,γ(u−u_H, u−u_H) ∀γ≥γ₀. (2.12) Finally we suppose the quasi-orthogonality relation: there existh₀>0 and Λ_h>0 such that

Λ_h→1 ash→0, and

a_h,γ(u−u_h, u−u_h)≤Λ_ha_h,γ(u−u_H, u−u_H)−a_h,γ(u_h−u_H, u_h−u_H) ∀h < h₀, γ≥γ₀. (2.13) Note that this last condition is satisﬁed if we have a Galerkin orthogonality relation and a symmetric forma_h,γ as the next lemma shows:

Lemma 2.2. Assume that the Galerkin orthogonality relation

a_h,γ(u−u_h, v_h) = 0 ∀v_h∈V_h, (2.14) holds and that a_h,γ is symmetric

a_h,γ(u, v) =a_h,γ(v, u), ∀u, v∈V +V_h. Then (2.13)holds withΛ_h= 1.

Proof. By the symmetry property ofa_h,γ we see that

a_h,γ(u−u_H, u−u_H)−a_h,γ(u−u_h, u−u_h)−a_h,γ(u_h−u_H, u_h−u_H) = 2a_h,γ(u−u_h, u_h)−2a_h,γ(u−u_h, u_H).

The conclusion follows from (2.14) becauseu_H∈V_H⊂V_h.

Remark 2.3. For a method that does not use a parameterγ (like conforming methods for instance, see the end of this section or [9]) we can formally takeγ= +∞. In that case the results stated below remain valid and all terms of the form ^C_γ with some positive constantCcan be simply replaced by 0.

All these data allow to prove the convergence of the algorithm:

Theorem 2.4. There exist a constant γ₁≥γ₀>0suﬃciently large, a mesh size h₀ suﬃciently small and two constants κ >0 and0< μ <1such that for all h≤h₀ andγ≥γ₁, one has

a_h,γ(u−u_h, u−u_h) +κosc²_h≤μ(a_H,γ(u−u_H, u−u_H) +κosc²_H), (2.15) whereh < H are two consecutive mesh parameters.

Proof. Using the error reduction estimate (2.10) and the coerciveness properties (2.3) and (2.5), we obtain

T∈TˆH

η_T(u_H)²≤C₂(a_h,γ(u_h−u_H, u_h−u_H) + osc²_H) +C₃

γ a_h,γ(u−u_h, u−u_h), (2.16) withC₃ =C₃/α₀.

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Applying the upper bound (2.6) to the rough mesh parameterH, and secondly using the marking proce- dures (2.7) and (2.8), we have

θ²a_H,γ(u−u_H, u−u_H)≤θ²

C₁

T∈TH

η_T(u_H)²+c₁osc²_H

≤C₁

T∈TˆH

η_T(u_H)²+c₁

T∈TˆH

osc_H(T)².

Using now the estimate (2.16), we arrive at

θ²a_H,γ(u−u_H, u−u_H)≤C₁C₂a_h,γ(u_h−u_H, u_h−u_H) +C₅osc²_H+C₁C₃

γ a_h,γ(u−u_h, u−u_h), or equivalently

a_h,γ(u_h−u_H, u_h−u_H)≥ θ²

C₁C₂a_H,γ(u−u_H, u−u_H)− C₃

γC₂a_h,γ(u−u_h, u−u_h)−C₆osc²_H. (2.17) Now using the quasi-orthogonality relation (2.13) and introducing a parameter β ∈ (0,1] ﬁxed suﬃciently small later on, we obtain

a_h,γ(u−u_h, u−u_h)≤Λ_ha_h,γ(u−u_H, u−u_H) + (β−1)a_h,γ(u_h−u_H, u_h−u_H)−βa_h,γ(u_h−u_H, u_h−u_H).

The last term of this right-hand side is estimated by invoking (2.17), and therefore a_h,γ(u−u_h, u−u_h)≤Λ_ha_h,γ(u−u_H, u−u_H) + (β−1)a_h,γ(u_h−u_H, u_h−u_H)

− θ²β

C₁C₂a_H,γ(u−u_H, u−u_H) +C₃β

γC₂a_h,γ(u−u_h, u−u_h) +βC₆osc²_H. Using the estimate (2.12), we arrive at

1−C₃β γC₂

a_h,γ(u−u_h, u−u_h)≤

Λ_h

1 + C₄ γ

− θ²β C₁C₂

a_H,γ(u−u_H, u−u_H) + (β−1)a_h,γ(u_h−u_H, u_h−u_H) +βC₆osc²_H. Choosingγ₁ large enough so that 1−^C_γC³^β

2 >0 forγ≥γ₁,i.e., γ₁≥1 +C₃β

C₂ , (2.18)

the last estimate is equivalent to

a_h,γ(u−u_h, u−u_h)≤ Λ_h

1 +^C_γ⁴

−_C^θ₁²_C^β 2

1−^C_γC³^β 2

a_H,γ(u−u_H, u−u_H) (2.19)

+ β−1 1−^C_γC³^β

2

a_h,γ(u_h−u_H, u_h−u_H) + βC₆ 1−^C_γC³^β

2

osc²_H.

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To take into account the oscillating terms, we multiply (2.11) byκ:= ^1−β

ρ2

1−^C_γC³^β

2

and ﬁnd

κosc²_h≤κρ₁osc²_H+ 1−β 1−^C_γC³^β

2

a_h,γ(u_h−u_H, u_h−u_H). (2.20)

The sum of the estimates (2.19) and (2.20) yields

a_h,γ(u−u_h, u−u_h) +κosc²_h≤Λ_h

1 +^C_γ⁴

−_C^θ₁²_C^β 2

1−^C_γC³^β 2

a_H,γ(u−u_H, u−u_H) +

⎛

⎝κρ₁+ βC₆ 1−^C_γC³^β

2

⎞

⎠osc²_H.

This estimate leads to the conclusion if we can choseγ₁ large enough as well ash₀ andβ small enough so that there exists 0< μ <1 such that

Λ_h

1 +^C_γ⁴

−_C^θ₁²_C^β 2

1−^C_γC³^β 2

≤μ,

κρ₁+ βC₆ 1−^C_γC³^β

2

≤μκ.

These two estimates are equivalent to (using the deﬁnition ofκ) Λ_h

1 +C₄

γ

− θ²β C₁C₂ ≤μ

1−C₃β γC₂

, (2.21)

ρ₁+C₆βρ₂

1−β ≤μ. (2.22)

To guarantee the estimate (2.22), we simply choseβ small enough such that ρ₁+C₆βρ₂

1−β <1, which is equivalent to

β

1−β < 1−ρ₁ C₆ρ₂ ,

which is always possible since the left-hand side of this estimate tends to zero asβ goes to zero.

Hence with such a choice ofβ, the estimate (2.22) holds with 1> μ≥ρ₁+^C_1−β⁶^βρ². Now we go on with the estimate (2.21), which is equivalent to

Λ_h

1 +C₄ γ

+C₃βμ

γC₂ ≤μ+ θ²β

C₁C₂· (2.23)

Asμ <1 this estimate holds if

Λ_h+1 γ

Λ_hC₄+C₃β C₂

< μ+ θ²β

C₁C₂ (2.24)

is valid. As Λ_htends to 1 ashgoes to 0, we ﬁxh₀small enough such that for allh≤h₀we have Λ_h≤1 + θ²β

3C₁C₂· (2.25)

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Similarly we ﬁxγ₁large enough in order to guarantee that 1

γ

≤ θ²β

3C₁C₂ ∀γ≥γ₁. (2.26)

Indeed this estimate is equivalent to

Λ_hC₄+^C_C³^β

θ²β 2 3C₁C₂ ≤γ ∀γ≥γ₁,

and by (2.25) it holds if

1 + _3C^θ²₁^β_C 2

C₄+^C_C³^β 2

θ²β 3C₁C₂ ≤γ₁. (2.27)

This shows that (2.26) holds withγ₁satisfying this estimate (2.27).

The estimates (2.25) and (2.26) lead to Λ_h+1

γ

≤1 + 2θ²β 3C₁C₂, which shows (2.24) if μ >1−_3C^θ²₁^β_C

2.

The proof is complete.

Remark 2.5. In general, there is no reason that the minimal stability parameterγ₀ satisﬁes (2.18) and (2.27), hence the convergence is only guaranteed for a larger threshold parameter γ₁. Nevertheless numerical tests below reveal that the reduction factor of the error (approximated value of μ) does not vary signiﬁcantly with respect to γ. Note further that from the above proof we can see that if the constants C₃ in (2.10) and C₄ in (2.12) are equal to zero then we can choseγ₁=γ₀.

The remainder of this paper is to prove the convergence of some adaptive methods applied to some approximated schemes of some boundary value problems; the convergence being obtained by checking the assumptions (2.3), (2.5), (2.6), (2.10), (2.11), (2.12) and (2.13).

For instance in the framework of the paper [23], reaction-convection-diﬀusion problems are approximated by subspaces V_h ⊂ V and with a_h,γ =a. Hence the coerciveness assumptions (2.3), (2.5) follows directly from the coerciveness of a, (2.6) is standard (see [3,28]), (2.10) is proved in Lemma 3.1 of [23], (2.11) is proved in Lemma 3.2 of [23], (2.12) is immediate and ﬁnally Lemma 2.1 of [23] is devoted to the proof of (2.13).

Remark 2.6. As said before the two main applications of our abstract framework concern DG methods where γ is the usual jump penalty parameter. We do not try to apply our framework to other methods where some penalization parameters are used (like SUPG methods for instance).

3. A

^POSTERIORI

error estimators for a discontinuous Galerkin method for diffusion problems

Here we revisit the results from [10] and show the convergence of an adequate adaptive algorithm at least in dimension 2 by using some recent results from [19].

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More precisely we consider the two-dimensional diﬀusion equation in a bounded domain Ω of

R

² ^{with a}

polygonal boundary Γ and homogeneous mixed boundary conditions:

−div (a∇u) =f in Ω, u= 0 on Γ_D,

a∇u·n= 0 on Γ_N, (3.1)

where Γ = ¯Γ_D∪Γ¯_N and Γ_D∩Γ_N =∅.

We suppose thatais piecewise constant, namely we assume that there exists a partitionP of Ω into a ﬁnite set of Lipschitz polygonal domains Ω₁, . . . ,Ω_J such that, on each Ω_j, a=a_j, where a_j is a positive constant.

For simplicity, we assume that Γ_Dhas a non-vanishing measure. We further assume that Ω is simply connected and that Γ is connected.

The variational formulation of (3.1) involves the bilinear form a(u, v) =

Ω

a∇u· ∇v.

Given f ∈L²(Ω), the weak formulation consists in ﬁndingu∈H_D¹(Ω) :={u∈H¹(Ω) :u= 0 on Γ_D} such that

a(u, v) = (f, v) =

Ω

fv, ∀v∈H_D¹(Ω). (3.2)

Remark 3.1. In [10] we imposed non-homogeneous boundary conditions, for the sake of simplicity we have restrict ourselves to homogeneous boundary conditions, nevertheless all the results stated below holds for non- homogeneous boundary conditions

Here to approximate problem (3.1) (or more precisely its variational formulation (3.2)), we use a discontinuous Galerkin scheme. Following [4,18,19], we consider the following discontinuous Galerkin approximation of our continuous problem: we consider a triangulation Th made of triangles T whose edges are denoted by e and assume that this triangulation is shape-regular, i.e., for any element T, the ratio h_T/ρ_T is bounded by a constantσ >0 independent ofT∈ T_h and of mesh-sizeh= max_T∈T_h h_T, whereh_T is the diameter ofT andρ_T the diameter of its largest inscribed ball. We further assume thatT_h is conforming with the partitionP of Ω, i.e., anyT ∈ T_h is included in one and only one Ω_i. With each edgeeof the triangulation, we associate a ﬁxed unit normal vectorn_e, andn_T stands for the outer unit normal vector ofT. For boundary edgese⊂∂Ω∩∂T, we setn_e =n_T. E_h represents the set of edges of the triangulation, and we assume that the Dirichlet part of the boundary Γ_D can be written as union of edges. We also need to distinguish between edges included into Ω, Γ_Dor Γ_N, in other words, we set

E_h,int ={e∈ E_h:e⊂Ω}, Eh,D ={e∈ Eh:e⊂Γ_D}, E_h,N ={e∈ E_h:e⊂Γ_N}.

For shortness, we also writeEh,ID =Eh,int∪ Eh,D. In the sequel,a_T denotes the value of the piecewise constant coeﬃcientarestricted to the elementT. Finally forT ∈ T_h,ω_T denotes the patch consisting of all the triangles of T_h having a nonempty intersection with T. Similarly for an edge e, ω_e denotes the patch consisting of all the triangles ofT_h havingeas edge.

In the following, theL²-norm on a domainDwill be denoted by · _D; the index will be dropped ifD= Ω.

We use · _s,D and | · |_s,D to denote the standard norm and semi-norm onH^s(D) (s ≥0), respectively. The energy norm is deﬁned byv²=a(v, v), for any v∈H¹(Ω).

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Problem (3.2) is approximated in the (discontinuous) ﬁnite element space:

V_h=

v_h∈L²(Ω)|v_h|T ∈P_l(T), T ∈ T_h

, (3.3)

wherel is a ﬁxed positive integer. The spaceV_his equipped with the norm

q_h,γ:=

⎛

⎝a^1/2∇_hq²_Ω+γ

e∈Eh,ID

h⁻¹_e q

²_e

⎞

⎠

1/2

,

whereγ is a positive parameter ﬁxed below and for anyq∈V_h, we deﬁne its broken gradient∇_hqin Ω by (∇_hq)_|T =∇q_|T, ∀T ∈ T_h.

As usual we need to deﬁne some jumps and means through anye∈ E_h of the triangulation. Fore∈ E_hsuch that e⊂Ω, we denote byT⁺ and T⁻ the two elements ofT_h containinge. Letq∈V_h, we denote byq^±, the traces ofq taken fromT^±, respectively. Then we deﬁne the mean ofqoneby

q

=q⁺+q⁻

2 ·

Forv∈[V_h]², we denote similarly

v

= v⁺+v⁻

2 ·

The jump of qoneis now deﬁned as follows:

q

=q⁺n_T++q⁻n_T−. Remark that the jump

q

of qis vector-valued.

For a boundary edgee,i.e.,e⊂∂Ω, there exists a unique element T⁺∈ Th such thate⊂∂T⁺. Therefore the mean and jump ofq are deﬁned by

q

=q⁺ and q

=q⁺n_T+. With these notations, we deﬁne the bilinear forma_h,γ(., .) as follows:

a_h,γ(u_h, v_h) :=

T∈Th

Ta∇u_h· ∇v_h−

e∈Eh,ID

e(

a∇hv_h

· u_h

+

a∇hu_h

· v_h

)

+γ

e∈Eh,ID

h⁻¹_e

e

u_h

· v_h

, ∀u_h, v_h∈V_h,

where the positive parameter γ is chosen large enough to ensure coerciveness of the bilinear form a_h,γ on V_h (see,e.g., Lem. 2.1 of [18]).

The discontinuous Galerkin approximation of problem (3.2) reads now: ﬁndu_h∈V_h, such that a_h,γ(u_h, v_h) =

Ω

fv_h, ∀v_h∈V_h. (3.4)

3.1. The a posteriori error analysis based on Raviart-Thomas finite elements

Error estimators can be constructed in many different ways as, for example, using residual type error estimators which measure locally the jump of the discrete flux [18,19]. Here, introducing the flux j = a∇u as auxiliary variable, we locally define an error estimator based on a H(div)-conforming approximation of this variable. Hence the discrete flux approximationj_h will be searched in a H(div)-conforming spaceRT_h based

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on the Raviart-Thomas finite elements. This means that our error estimate of the conforming part of the error is based on the error betweena∇hu_h and an approximating fluxj_h ofj that we search in the Raviart-Thomas finite element space

RT_h=

v_h∈H(div,Ω)|v_h|T ∈RT_l−1(T), T ∈ Th .

For a similar approach where the ﬂuxes are computed using local Neumann problems, see for instance [6,22].

On a triangleT, an elementpofRT_l−1(T) is characterized by the degrees of freedom given by

•

e

p·n q, ∀q∈Pl−1(e), ∀e⊂∂T,

•

T

p·q, ∀q∈[Pl−2(T)]². Therefore we ﬁx the discrete ﬂuxj_h by setting

e

j_h·n_Tq=

e

g_T,e q, ∀q∈P_l−1(e), ∀e⊂∂T, (3.5)

T

j_h·q=

T

a∇u_h·q−a_Tl_∂T(q), ∀q∈[P_l−2(T)]², (3.6) where for alle⊂∂T,g_T,e is deﬁned by

g_T,e=

a∇hu_h

−γh⁻¹_e u_h

·n_T ife∈ Eh,int, g_T,e=a∇_hu_h·n_T −γh⁻¹_e u_h ife∈ E_h,D,

g_T,e= 0 ife∈ Eh,N, and the linear forml_∂T is given by

l_∂T(q) = 1 2

e⊂∂T\Γ

e

u_h

·q+

e⊂∂T∩ΓD

e

u_hq·n_T.

Denote by Π_l−1 the L²-projection on W_h =

w_h∈L²(Ω)|w_h|T ∈P_l−1(T), T ∈ T_h

. Then, we have the following main property (see Lem. 3.1 of [10])

divj_h=−Π_l−1f. (3.7)

We now recall the estimator introduced in [10]: it consists in three parts: a conforming part that only involves the diﬀerence between the discrete ﬂux approximationj_h anda∇u_h:

η_CF,T(u_h) =a^−1/2(a∇u_h−j_h)T. (3.8) The nonconforming part is built by using the Oswald interpolation operator ofu_h, namely the unique element w_h ∈ V_h∩H_D¹(Ω) deﬁned in the following natural way (see Thm. 2.2 of [18]): to each node n of the mesh in Ω∪Γ_N, corresponding to Lagrangian-type degree of freedom ofV_h∩H_D¹(Ω), the value of w_h is the average of the values ofu_h at this noden,i.e.,w_h(n) = ^n∈T^|T^|u^h|T⁽ⁿ⁾

n∈T|T| . Then the non conforming estimator is simply η_NC,T(u_h) =a^1/2∇(w_h−u_h)T. (3.9) Finally we introduce the estimator corresponding to jumps ofu_h:

η_J,T(u_h)²=1 2

e∈Eh,int∩T

η_J,e(u_h)²+

e∈Eh,D∩T

η_J,e(u_h)², η_J,e(u_h)²= γ h_e

u_h ²_e.

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The estimator onT is then deﬁned by

η_T(u_h)²=η_CF,T(u_h)²+η_NC,T(u_h)²+η²_J,T(u_h)². The oscillating terms depending on the datumf is deﬁned as usual by

osc²_h=

T∈Th

h²_Ta⁻¹_T f−Π_l−1f²_T.

Now using the results from Section2, we describe a convergent algorithm for the above estimators.

3.2. Checking the assumptions from Section 2

SinceV_h⊂V =H_D¹(Ω) anda_h,γ=a, the coerciveness estimates (2.3) and (2.5) are not direct but they are respectively proved in Lemma 3.1 and Proposition 4.2 of [19]. The estimate (2.12) is proved as follows: ﬁrst Proposition 4.1 of [19] shows that there existsC >0 independent of handγ such that

a_h,γ(u−u_H, u−u_H)≤a_H,γ(u−u_H, u−u_H) +Cγ

e∈Eh,ID

h⁻¹_e u_H

²_e.

But Theorem 3.2 of [19] guarantees that

γ

e∈Eh,ID

h⁻¹_e u_H

²_e≤C

γ∇_H(u−u_H)². As Proposition 4.2 of [19] implies that

∇_H(u−u_H)²≤2a_H,γ(u−u_H, u−u_H),

for γ≥γ₁ and γ₁ large enough (depending on the order l and on the shape regularity constant σ), the three above estimates lead to (2.12).

Sincea_h,γis symmetric and (2.14) holds (see the identity (3.2) of [19]), the quasi-orthogonality estimate (2.13) holds with Λ_h= 1 due to Lemma2.2.

The estimate (2.6) was proved in Theorem 3.4 of [10] since (2.5) holds.

The oscillation reduction estimate (2.11) follows from Lemma 3.2 of [23] if the marking strategy from Deﬁ- nition 2.1is used and if the successive meshes are constructedvia the procedure REFINE of Morin, Nochetto and Siebert [23–25].

It remains the error reduction estimate (2.10): ﬁrst in the proof of Theorem 3.6 of [10] we see that η_CF,T(u_H)²≤C(a)

e⊂T

h_TJ_e,n(u_H)²_e+1

γη_J,T² (u_H)

,

whereC(a) is a positive constant depending onaand the jump terms are the usual ones deﬁned by J_e,n(u_H) =

⎧⎨

⎩

a∇u_H·n_e

for interior edges ofTH,

0 for Dirichlet boundary edges ofTH,

∇u_H·n_e for Neumann boundary edges ofTH.

On the other hand using Theorem 2.2 of [18], there existsC >0 independent ofγ andH such that η_NC,T(u_H)²≤C

γη_J,T² (u_H).

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These two estimates imply that

T∈TˆH

η_T(u_H)²≤C(a)

⎛

⎝

e∈EˆH

h_TJ_e,n(u_H)²_e+

e∈EH,ID

h⁻¹_e u_H

²_e

⎞

⎠,

for someC(a)>0 depending only onaand shape regularity constantσ. The ﬁrst term of this right hand side is a part of the estimator from [19] and by the estimate (4.31) of [19] we have

e∈EˆH

h_TJ_e,n(u_H)²_e≤C

⎛

⎝∇h(u_h−u_H)²+

e∈Eh,ID

h⁻¹_e u_h

²_e

⎞

⎠,

and consequently

T∈TˆH

η_T(u_H)² ≤C(a)

⎛

⎝∇h(u_h−u_H)²+

e∈EH,ID

h⁻¹_e u_H

²_e+

e∈Eh,ID

h⁻¹_e u_h

²_e

⎞

⎠

≤2C(a)

⎛

⎝∇h(u_h−u_H)²+

e∈EH,ID

h⁻¹_e

u_H−u_h

²_e+

e∈Eh,ID

h⁻¹_e u_h

²_e

⎞

⎠.

For the two ﬁrst terms of this right-hand side using the deﬁnition of the norm · _h,γ, we have

∇h(u_h−u_H)²+

e∈EH,ID

h⁻¹_e

u_H−u_h

²_e≤Cu_h−u_H²_h,γ.

For the third term we notice that

e∈Eh,ID

h⁻¹_e u_h

²_e=

e∈Eh,ID

h⁻¹_e

u−u_h ²_e≤ 1

γu−u_h²_h,γ.

The three above estimates lead to the estimate (2.10).

3.3. Some numerical tests

In order to illustrate the performance of our estimator η_h and the convergence of the adaptive algorithm, for two benchmark examples we show the meshes obtained after some iterations, as well as the experimental convergence orders of the error

EOC_e= 2lg^u−u_u−u^H^H,γ

hh,γ

lg_DOF^DOF^h

H

, the eﬀectivity indices

Eﬀ =η_h/u−u_h_h,γ

and the reduction factors of the error (approximated value of the constant√μappearing in Thm.2.4) RFE =u−u_h_h,γ/u−u_H_H,γ,

calculated during the different iterations. We use the iterative algorithm of bulk type described in Definition2.1 with θ₁=θ₂=θ= 0.75, 0.8 or 0.9 and refine the triangles of ˆT_H by a standard refinement procedure with a limitation on the minimal angle.