Goal oriented, anisotropic, a posteriori error estimates for the Laplace ...

4 downloads 0 Views 89KB Size Report
pour la mécanique des fluides. application la prédiction haute-fidélité du bang sonique. PhD. Thesis, Paris VI (2008). 26. Micheletti, S., Perotto, S.: Reliability ...
Goal oriented, anisotropic, a posteriori error estimates for the Laplace equation Frederic Alauzet, Wissam Hassan and Marco Picasso

Abstract A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ratio. Error estimates are presented in the natural H 1 seminorm or in the framework of goal oriented error control. The proposed estimator relies on anisotropic interpolation estimates derived by Formaggia and Perotto [19, 20] and on Zienckiewicz-Zhu [37, 36] post-processing techniques, thus avoiding the use of numerical techniques in order to compute approximations of the Hessian of the solution. All the constant involved in the error estimates are independent of the mesh size and aspect ratio, which enables the use of anisotropic, adaptive finite elements.

1 Introduction A posteriori error estimates aim to link the error between the true solution u and the finite element approximation uh with a computable quantity - the so-called error estimator η . Then, the error estimator can be used as a refining - or coarsening criteria in adaptive finite element algorithms. The subject was initiated by Babuska and Rheinboldt [4] and mesh adaptation is nowadays a classical feature in finite element software, see for instance [32, 16, 5]. Interpolation estimates are usually needed in order to derive a posteriori error estimates. In the simplest case, namely continuous, piecewise linear finite elements on triangles, Lagrange interpolation [13] is needed to prove a priori error estimates whereas Cl´ement interpolation [14] is needed to derive a posteriori error estimates. Frederic Alauzet INRIA Rocquencourt Wissam Hassan IACS EPFL Marco Picasso IACS EPFL and INRIA Rocquencourt

1

2

Frederic Alauzet, Wissam Hassan and Marco Picasso

In the classical setting of isotropic meshes, both interpolation estimates hold under the so-called regularity assumption which requires that ∃C > 0

∀h > 0

∀K ∈ Th

hK ≤ C. ρK

Hereabove, Th denotes a mesh of the calculation domain Ω into triangles K with diameter hK less than h and ρK is the largest circle contained in K. However, in practice, anisotropic finite elements are used with success in order to solve complex problems such as fluid flow around bodies, see for instance [22, 9, 21, 2]. Recently, the theory of finite elements was updated in order to comply with the use of anisotropic finite elements. Hereafter, we will consider the contributions of Formaggia and Perotto [19, 20], however similar results have been obtained for Lagrange [7, 22, 12] and Cl´ement [23] interpolation. In the classical setting of isotropic meshes satisfying the regularity assumption, the interpolation estimate for the Lagrange interpolation operator rh with polynomial degree one write [13]: ∃C > 0 Z

K

∀h > 0

|∇(v − rh v)|2 ≤ C

h4K ρK2

Z

K

∀K ∈ Th ∀v ∈ H 2 (K)  2 2  2 2  2 2 ! ∂ v ∂ v ∂ v , + + 2 ∂ x1 ∂ x2 ∂ x1 ∂ x22

where C does not depend on the mesh aspect ratio. If, for instance, v depends only on x2 and if the mesh is refined only in the x2 direction, then the right hand side of the above estimate blows up since the mesh aspect ratio increases. On the other side, anisotropic interpolation estimates [19, 20] are as follows: Z

K

|∇(v − rh v)|2 ≤ C

4 Z λ1,K

(rT1,K H(v)r1,K )2

2 + 2λ1,K

Z

2 λ2,K

K

K

2 (rT1,K H(v)r2,K )2 + λ2,K

Z

K

(rT2,K H(v)r2,K )2

!

.

Hereabove H(v) is the Hessian matrix  ∂ 2v ∂ 2v  ∂ x2 ∂ x1 ∂ x2  , 1 H(v) =   ∂ 2v ∂ 2v  ∂ x1 ∂ x2 ∂ x22 

r1,K and r2,K denote orthogonal unit vectors in the direction of maximum and minimum streching, respectively, while λ1,K and λ2,K denote the stretching amplitudes in the direction of maximum and minimum streching, respectively. The precise definition of these quantities is proposed in the next section.

Goal oriented, anisotropic, a posteriori error estimates for the Laplace equation

3

If, for instance, v depends only on x2 and if the mesh is refined only in the x2 direction, then r1,K = (1 0)T and r2,K = (0 1)T and the above anisotropic interpolation estimate reduces to Z Z  2 2 ∂ v 2 |∇(v − rh v)|2 ≤ Cλ2,K 2 K K ∂ x2 so that convergence is achieved as soon as λ2,K goes to zero, no matter what the aspect ratio λ1,K /λ2,K is. The same argument holds when the isovalues of v and the mesh are rotated with any angle. Based on anisotropic interpolation estimates, a priori and a posteriori error estimates have been revisited and adaptive algorithms having meshes with large aspect ratio have been used with success in CFD, see for instance [22, 9, 21, 2, 15, 25]. Most of the paper involving anisotropic adaptive meshes deal with an estimate of the Hessian matrix. In [28] an anisotropic error estimator involving the gradient matrix was proposed for elliptic and parabolic problems in the energy norm. A lower bound was proved in [29, 26] for the Laplace problem. Goal oriented, anisotropic a posteriori error estimates involving gradients were proposed in [15] for advectiondiffusion-reaction, but only an upper bound was proved. In this paper we prove a lower bound for goal oriented, anisotropic a posteriori error estimates in the frame of the Laplace equation.

2 The Laplace equation with anisotropic finite elements Given a polygonal domain Ω ⊂ R2 , given f ∈ L2 (Ω ), we are searching for u : Ω → R such that −∆ u = f

in Ω , on ∂ Ω .

u=0

(1)

For any 0 < h < 1, let Th be a conforming triangulation of Ω into triangles K with diameter hK less than h. Let Vh be the usual finite element space of continuous, piecewise linear functions on the triangles of Th , zero valued on ∂ Ω . The simplest finite element approximation of (1) therefore consists in seeking uh ∈ Vh such that Z



∇uh · ∇vh =

Z



f vh

∀vh ∈ Vh .

(2)

We now describe the mesh anisotropy using the framework of Formaggia and Perotto [19, 20]. Again, alternative descriptions are available [7, 22, 12, 23]. For any triangle K of the mesh, let TK : Kˆ → K be the affine transformation which maps the reference triangle Kˆ into K. Let MK be the Jacobian of TK that is x = TK (ˆx) = MK xˆ + tK .

4

Frederic Alauzet, Wissam Hassan and Marco Picasso

Since MK is invertible, it admits a singular value decomposition MK = RTK ΛK PK , where RK and PK are orthogonal and where ΛK is diagonal with positive entries. In the following we set   T   r λ 0 and RK = 1,K , (3) ΛK = 1,K 0 λ2,K rT2,K with the choice λ1,K ≥ λ2,K . A simple example of such a transformation is x1 = H xˆ1 , x2 = hxˆ2 , with H ≥ h, thus       H0 1 0 MK = , λ1,K = H, λ2,K = h, r1,K = , r2,K = . 0 h 0 1 A geometrical interpretation of the decomposition MK = RTK ΛK PK is the following. Consider the case when Kˆ is the unit equilateral reference triangle and consider the set of points lying on the unit circle, that is the points xˆ satisfying xˆ T xˆ = 1. Since xˆ = MK−1 (x − tK ), we have 1 = (x − tK )T MK−T MK−1 (x − tK ) = (x − tK )T RTK ΛK−2 RK (x − tK ), thus the unit circle is mapped into an ellipse with directions r1,K and r2,K , the amplitude of stretching being λ1,K and λ2,K . In the frame of anisotropic meshes, the classical minimum angle condition is not required. However, for each vertex, the number of neighbouring vertices should be bounded from above, uniformely with respect to the mesh size h. Also, for each triangle K of the mesh, there is a restriction related to the patch ∆K , the set of triangles having a vertex common with K. More precisely, the diameter of the reference patch TK−1 (∆K ) must be uniformly bounded independently of the mesh geometry. This assumption excludes some too distorted reference patches. It should be noted that the restriction about the diameter reference patch was not present in the two original papers of Formaggia and Perotto [19, 20]. A similar condition can also be found in the papers of Kunert [23]. This restriction has been added in the paper [27] and is needed in order to prove Cl´ement interpolation estimates, since quantities are involved in the reference patch. This assumptions implies that the local geometric quantities λi,K , ri,K , i = 1, 2, vary smoothly on neighbouring triangles. In practice, no restrictions have been added in order to satisfy this condition and the anisotropic mesh generators that have been used [8, 17, 18] seem to satisfy the uniform boundedness of the reference patch.

3 A posteriori error estimates in the energy norm Let us introduce the anisotropic error estimator proposed in [28] and corresponding to the error in the energy norm k∇(u − uh )kL2 (Ω ) . For all K ∈ Th , let

Goal oriented, anisotropic, a posteriori error estimates for the Laplace equation

ΠK f =

1 |K|

Z

5

f

K

be the L2 (K) projection of f onto the constants. Let ℓi , i = 1, 2, 3 be the triangle three edges, let [·] denote the jump of the bracketed quantity across ℓi , with the convention [·] = 0 for an edge ℓi on the boundary ∂ Ω . Then, the anisotropic error estimator corresponding to the energy norm on triangle K is defined by ! 1/2 3  1 |ℓ | i k [∇uh · n] kL2 (ℓi ) ωK (u − uh ). (ηKEN )2 = kΠK f kL2 (K) + ∑ 2 i=1 λ1,K λ2,K (4) Here n is the edge unit normal (in arbitrary direction), and ωK (v) is defined for all v ∈ H 1 (Ω ) by     2 2 (5) rT1,K GK (v)r1,K + λ2,K rT2,K GK (v)r2,K , ωK2 (v) = λ1,K where GK (v) denotes the 2 × 2 matrix defined by

  Z ∂v 2 ∂v ∂e dx dx   ∆ ∂ x1 ∆K ∂ x1 ∂ x2 K .   GK (v) =  Z Z 2   ∂v ∂e ∂v dx dx ∆K ∂ x1 ∂ x2 ∆K ∂ x2 Z



(6)

The following upper and lower bounds in the energy norm have been proved in [29]. Similar results can be found in [26]. Theorem 1. There exists a constant C1 independent of the mesh size and aspect ratio such that Z   2 (7) k f − ΠK f k2L2 (K) . |∇(u − uh )|2 ≤ C1 ∑ (ηKEN )2 + ∑ λ1,K Ω

K∈Th

K∈Th

Moreover, if the mesh is such that there exists a constant C2 independent of the mesh size and aspect ratio such that

ωK2 (u − uh ) ≤ C2 2 λ2,K K∈Th



Z



|∇(u − uh )|2 .

(8)

then, there exists a constant C3 independent of the mesh size and aspect ratio such that Z  2 (9) k f − ΠK f k2L2 (K) . ∑ (ηKEN )2 ≤ C3 |∇(u − uh )|2 + ∑ λ1,K K∈Th



K∈Th

Remark 1. In the isotropic setting, λ1,K ≃ λ2,K ≃ hK , assuming that f is smooth enough, estimates (7) and (9) yield

6

Frederic Alauzet, Wissam Hassan and Marco Picasso

Z



|∇(u−uh )|2 ≤ C1



ηK2 +h.o.t.

and

K∈Th



ηK2 ≤ C3

K∈Th

Z



|∇(u−uh )|2 +h.o.t.,

where h.o.t. denotes a high order term that behaves as O(h4 ) and where

ηK2 = h2K kΠK f k2L2 (K) +

1 3 ∑ |ℓi |k [∇uh · n] k2L2 (ℓi ) 2 i=1

is the classical explicit, residual based error estimator studied for instance in [3, 34]. In the isotropic setting, assumption (8) is not necessary but, in turn, the constants C1 and C3 hereabove depend on the mesh aspect ratio. Remark 2. The estimator (4) is not a usual error estimator since u is still involved. However, if we can guess u − uh , then (4) can be used to derive a computable quantity. This idea has been used in [28, 29] and an efficient anisotropic error indicator has also been obtained replacing the derivatives

∂ (u − uh ) ∂ uh ∂ uh in (6) by − Πh , i=1,2, ∂ xi ∂ xi ∂ xi

(10)

where Πh is an approximate L2 (Ω ) projection onto Vh . More precisely, from constant values of ∂ uh /∂ xi on triangles, we build values at vertices P using the formula     ∂ uh ∂ uh 1 |K| Πh (P) = i = 1, 2. ∑ ∂ xi ∂ xi |K ∑ |K| K∈T K∈Th

P∈K

h

P∈K

Approximating ∂ (u − uh )/∂ xi by (I − Πh )∂ uh /∂ xi is at the base of the celebrated Zienkiewicz-Zhu error estimator [37, 36] and can be justified theoretically whenever superconvergence occurs, that is when ∇u− Πh ∇uh is better than O(h). For instance, it is proved in [31, 1] that the Zienkiewicz-Zhu error estimator is asymptotically exact on parallel meshes, see also [10] for 3D results. Superconvergence has also be obtained for 2D midly structured meshes in [35] but excludes for instance the chevron pattern, for which the Zienkiewicz-Zhu is not aymptotically exact, see [31]. On general unstructured meshes, the Zienkiewicz-Zhu error estimator is only proved to be equivalent to the true error, see for instance [31, 11] for isotropic meshes and [24] for anisotropic meshes. Numerical results show that the good properties of the Zienkiewicz-Zhu error estimator are underestimated by theoretical results. Remark 3. Assumption (8) is true provided there exists a constant C independent of the mesh aspect ratio such that, for all K ∈ Th ,     2 2 (11) rT1,K GK (u − uh )r1,K ≤ Cλ2,K rT2,K GK (u − uh )r2,K , λ1,K

in other words, when the error gradient in the direction of maximum stretching is less than the error gradient in the direction of minimum stretching. This is for

Goal oriented, anisotropic, a posteriori error estimates for the Laplace equation

instance the case when     2 2 rT1,K GK (u − uh )r1,K = λ2,K rT2,K GK (u − uh )r2,K λ1,K

∀K ∈ Th ,

7

(12)

that is to say when the error in both the directions of minimum and maximum stretching is equidistributed. This is precisely the goal of the adaptive algorithm described in [28, 29]. Numerical results reported in [28, 29] have shown that the effectivity index is aspect ratio independent for adapted meshes.

4 Goal oriented a posteriori error estimates We now present an error estimator for goal oriented a posteriori error estimates based on first order derivatives rather than second order derivatives. This error estimator has already been introduced in [15] for advection-diffusion problems but only an upper bound was proved. Hereafter, we propose a lower bound proceeding as in [29]. We refer [30, 6] for goal oriented, isotropic a posteriori error estimates. In order to simplify the presentation we consider the linear functional J defined for all v ∈ L1 (Ω ) by Z J(v) =

Ω0

v,

where Ω0 ⊂ Ω . Our goal is now to control J(u − uh ) and we introduce the dual problem: find z ∈ H01 (Ω ) such that Z



∇z · ∇v = J(v)

∀v ∈ H01 (Ω ).

(13)

We also need the corresponding finite element approximation of z namely zh ∈ Vh such that Z ∇zh · ∇vh = J(vh ) ∀v ∈ Vh . (14) Ω

The error estimator corresponding to the goal oriented error J(u − uh ) on triangle K is now defined by ! 1/2  1 3 |ℓi | GO 2 (ηK ) = kΠK f kL2 (K) + ∑ k [∇uh · n] kL2 (ℓi ) ωK (z − zh ), 2 i=1 λ1,K λ2,K (15) and proceeding as in [29], we can prove the following. Proposition 1. There is a constant C independent of the mesh size and aspect ratio such that ! J(u − uh ) ≤ C



K∈Th

(ηKGO )2 +



K∈Th

k f − ΠK f kL2 (K) ωK (z − zh ) .

(16)

8

Frederic Alauzet, Wissam Hassan and Marco Picasso

Proof. Let Ih be Cl´ement’s interpolant [14]. From Proposition 3.2 in [19], there exists a constant C depending only on the reference element Kˆ such that, for all v ∈ H 1 (Ω ), for all K ∈ Th kv − Ih vkL2 (K) ≤ CωK (v).

(17)

Moreover, proceeding as in the proof of Proposition 2 in [20], there exists a constant C depending only on the reference element Kˆ such that, for all v ∈ H 1 (Ω ), for all K ∈ Th , for i = 1, 2, 3 kv − Ih vkL2 (ℓi ) ≤ C



|ℓi | λ1,K λ2,K

1/2

ωK (v).

Using (1) (2) (13) (14) we have Z

Z

J(u − uh ) = f (z − zh − vh ) − ∇uh · ∇(z − zh − vh ) Ω ZΩ  Z 1 = ∑ (ΠK f + ∆ uh )(z − zh − vh ) + [∇uh · n] (z − zh − vh ) , 2 ∂K K K∈T h

for all vh ∈ Vh . We then choose vh = Ih (z − zh ), use Cauchy-Schwarz inequality and the above anisotropic interpolation estimates to obtain the result. The proof of the following result is as in [29]. Proposition 2. There exists a function ϕ ∈ H01 (Ω ) and a constant C independent of the mesh size and aspect ratio such that, for all K ∈ Th we have Z

ℓi

[∇uh · n] ϕ =



|ℓi | λ1,K λ2,K

1/2 Z

ℓi

2

[∇uh · n]

1/2

ωK (z − zh ),

i = 1, 2, 3, (18)

Z

(ΠK f )ϕ =

Z

|∇ϕ |2 ≤ C

K

K

Z

K

(ΠK f )2

ωK2 (z − zh ) . 2 λ2,K

1/2

ωK (z − zh ),

(19) (20)

We then prove what follows. Proposition 3. There exists a constant C independent of the mesh size and aspect ratio such that   ω (z − z ) K h GO 2 k∇(u−u )k + f k k f − ) ≤ C η λ Π . (21) ( 2 2 K 1,K h L (K) ∑ ∑ K L (K) λ 2,K K∈T K∈T h

h

Proof. Using the definition of the error estimator, conditions (18) (19) and the definition of ΠK , we have

Goal oriented, anisotropic, a posteriori error estimates for the Laplace equation



(ηKGO )2 =





Z

( f ϕ − ∇uh · ∇ϕ ) −



Z

∇(u − uh ) · ∇ϕ −

K∈Th

K∈Th

=

K

K∈Th K

=



Z

K∈Th K

fϕ +

1 2

Z

∂K

[∇uh · n] ϕ − Z



K∈Th K



Z

K∈Th K



Z

K∈Th K

9

( f − ΠK f )ϕ

( f − ΠK f )(ϕ − ΠK ϕ )

( f − ΠK f )(ϕ − ΠK ϕ ).

Using Cauchy-Schwarz estimate we therefore obtain



(ηKGO )2 ≤

K∈Th



K∈Th



k∇(u − uh )kL2 (K) k∇ϕ kL2 (K)  + k f − ΠK f kL2 (K) kϕ − ΠK ϕ kL2 (K) . (22)

Using the properties of ΠK and again Lemma 2.2 of [19], we have Z

K

(ϕ − ΠK ϕ )2 = λ1,K λ2,K

Z



(ϕˆ − Πˆ K ϕˆ )2 ≤ Cˆ λ1,K λ2,K

Z



ˆ ϕˆ |2 ≤ Cˆ λ 2 |∇ 1,K

Z

K

|∇ϕ |2 ,

where Cˆ depends only on the reference element, so that (22) yields   GO 2 ˆ ( k∇(u − u )k + C f k k f − ) ≤ η λ Π 2 2 K 1,K h L (K) ∑ K ∑ L (K) k∇ϕ kL2 (K) . K∈Th

K∈Th

It suffices to use (20) to obtain (21). From the three above Propositions, we can now state the main result of the paper, which can be compared to Theorem 1. Theorem 2. There exists a constant C1 independent of the mesh size and aspect ratio such that   2 k f − ΠK f k2L2 (K) J(u − uh ) ≤ C1 ∑ (ηKGO )2 + ∑ λ1,K K∈Th

K∈Th

+



K∈Th

 2 kz − zh k2L2 (K) . (23) λ1,K

Moreover, if the mesh is such that there exists a constant C2 independent of the mesh size and aspect ratio such that

ω 2 (z − zh ) ≤ C2 ∑ Kλ2 2,K K∈Th

Z



|∇(z − zh )|2 .

(24)

then, there exists a constant C3 independent of the mesh size and aspect ratio such that

10

Frederic Alauzet, Wissam Hassan and Marco Picasso



(ηKGO )2 ≤ C3

K∈Th

Z



|∇(u − uh )|2 +

Z



|∇(z − zh )|2 +



K∈Th

 2 k f − ΠK f k2L2 (K) . (25) λ1,K

The three following remarks are similar to Remarks 1, 2 and 3. Remark 4. In the isotropic setting, λ1,K ≃ λ2,K ≃ hK , assuming that f is smooth enough, estimates (23) and (25) write J(u − uh ) ≤ C1



ηK2 + h.o.t.

K∈Th

and



K∈Th

ηK2 ≤ C3

Z



|∇(u − uh )|2 +

Z



 |∇(z − zh )|2 + h.o.t.,

(26)

where h.o.t. denotes a high order term that behaves as O(h4 ) and where

ηK2 =

1 3 h2K kΠK f k2L2 (K) + ∑ |ℓi |k [∇uh · n] k2L2 (ℓi ) 2 i=1

!1/2

k∇(z − zh )kL2 (K) .

In general, the last term in the above definition is estimated using interpolation results [6] or post-processing [33]. In the isotropic setting, assumption (24) is not necessary but, in turn, the constants C1 and C3 hereabove depend on the mesh aspect ratio. Moreover, whenever u and z are smooth enough, then (26) writes



ηK2 ≤ C4 h2 + h.o.t.,

K∈Th

where C4 is independent of h, thus the error estimator is of optimal O(h2 ) order. Remark 5. The estimator (4) is not a usual error estimator since z is still involved. However, if we can guess z − zh , then (4) can be used to derive a computable quantity. Following [28, 29], we propose to replace the derivatives

∂ (z − zh ) ∂ zh ∂ zh in (6) by − Πh , i=1,2, ∂ xi ∂ xi ∂ xi

(27)

where Πh is an approximate L2 (Ω ) projection onto Vh . Remark 6. Assumption (24) is true provided there exists a constant C independent of the mesh aspect ratio such that, for all K ∈ Th ,     2 2 λ1,K rT1,K GK (z − zh )r1,K ≤ Cλ2,K rT2,K GK (z − zh )r2,K ,

in other words, when the dual error gradient in the direction of maximum stretching is less than the dual error gradient in the direction of minimum stretching. This is

Goal oriented, anisotropic, a posteriori error estimates for the Laplace equation

for instance the case when     2 2 rT1,K GK (z − zh )r1,K = λ2,K rT2,K GK (z − zh )r2,K λ1,K

11

∀K ∈ Th ,

that is to say when the dual error in both the directions of minimum and maximum stretching is equidistributed. This condition should be enforced in the adaptive algorithm for goal oriented anisotropic meshes.

References 1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142(1-2), 1–88 (1997) 2. Alauzet, F.: High-order methods and mesh adaptation for Euler equations. Internat. J. Numer. Methods Fluids 56(8), 1069–1076 (2008) 3. Babuˇska, I., Dur´an, R., Rodr´ıguez, R.: Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal. 29(4), 947–964 (1992) 4. Babuˇska, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978) 5. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general-purpose object-oriented finite element library. ACM Trans. Math. Software 33(4), Art. 24, 27 (2007) 6. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001) 7. Borouchaki, H., George, P.L., Hecht, F., Laug, P., Saltel, E.: Delaunay mesh generation governed by metric specifications. I. Algorithms. Finite Elem. Anal. Des. 25(1-2), 61–83 (1997). Adaptive meshing, Part 1 8. Borouchaki, H., Laug, G.: The BL2D Mesh Generator : Beginner’s Guide, User’s and Programmer’s Manual. Technical report RT-0194, Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt, 78153 Le Chesnay, France (1996) 9. Bourgault, Y., Picasso, M., Alauzet, F., Loseille, A.: On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows. Internat. J. Numer. Methods Fluids 59(1), 47–74 (2009) 10. Brandts, J., Kˇr´ızˇ ek, M.: Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23(3), 489–505 (2003) 11. Carstensen, C.: All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable. Math. Comp. 73(247), 1153–1165 (electronic) (2004) 12. Chen, L., Sun, P., Xu, J.: Optimal anisotropic meshes for minimizing interpolation errors in L p -norm. Math. Comp. 76(257), 179–204 (electronic) (2007) 13. Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pp. 17–351. North-Holland, Amsterdam (1991) 14. Cl´ement, P.: Approximation by finite element functions using local regularization. RAIRO Analyse Num´erique 9(R-2), 77–84 (1975) 15. Ded`e, L., Micheletti, S., Perotto, S.: Anisotropic error control for environmental applications. Appl. Numer. Math. 58(9), 1320–1339 (2008) 16. Demkowicz, L.: Computing with hp-adaptive finite elements. Vol. 1. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL (2007). One and two dimensional elliptic and Maxwell problems, With 1 CD-ROM (UNIX) 17. Distene S.A.S., Pˆole Teratec - BARD-1, Domaine du Grand Ru´e, 91680 Bruy`eres-le-Chatel, France: MeshAdapt : A mesh adaptation tool, User’s manual Version 3.0 (2003) 18. Dobrzynski, C., Frey, P.J., Mohammadi, B., Pironneau, O.: Fast and accurate simulations of air-cooled structures. Comput. Methods Appl. Mech. Engrg. 195(23-24), 3168–3180 (2006)

12

Frederic Alauzet, Wissam Hassan and Marco Picasso

19. Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001) 20. Formaggia, L., Perotto, S.: Anisotropic error estimates for elliptic problems. Numer. Math. 94(1), 67–92 (2003) 21. Frey, P.J., Alauzet, F.: Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Engrg. 194(48-49), 5068–5082 (2005) 22. Habashi, W.G., Dompierre, J., Bourgault, Y., Ait-Ali-Yahia, D., Fortin, M., Vallet, M.G.: Anisotropic mesh adaptation: towards user-independent, mesh-independent and solverindependent CFD. I. General principles. Internat. J. Numer. Methods Fluids 32(6), 725–744 (2000) 23. Kunert, G.: An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86(3), 471–490 (2000) 24. Kunert, G., Nicaise, S.: Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes. M2AN Math. Model. Numer. Anal. 37(6), 1013–1043 (2003) 25. Loseille, A.: Adaptation de maillage anisotrope 3d multi-´echelles et cibl´ee a` une fonctionnelle pour la m´ecanique des fluides. application la pr´ediction haute-fid´elit´e du bang sonique. PhD Thesis, Paris VI (2008) 26. Micheletti, S., Perotto, S.: Reliability and efficiency of an anisotropic Zienkiewicz-Zhu error estimator. Comput. Methods Appl. Mech. Engrg. 195(9-12), 799–835 (2006) 27. Micheletti, S., Perotto, S.: Space-time adaption for advection-diffusion-reaction problems on anisotropic meshes. MOX Report 25/2007 (2007) 28. Picasso, M.: An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: application to elliptic and parabolic problems. SIAM J. Sci. Comput. 24(4), 1328–1355 (electronic) (2003) 29. Picasso, M.: Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives. Comput. Methods Appl. Mech. Engrg. 196(1-3), 14–23 (2006) 30. Rannacher, R.: Adaptive Galerkin finite element methods for partial differential equations. J. Comput. Appl. Math. 128(1-2), 205–233 (2001). Numerical analysis 2000, Vol. VII, Partial differential equations 31. Rodr´ıguez, R.: Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Equations 10(5), 625–635 (1994) 32. Schmidt, A., Siebert, K.G.: Design of adaptive finite element software, Lecture Notes in Computational Science and Engineering, vol. 42. Springer-Verlag, Berlin (2005). The finite element toolbox ALBERTA, With 1 CD-ROM (Unix/Linux) 33. Suttmeier, F.T.: Reliable, goal-oriented postprocessing for FE-discretizations. Numer. Methods Partial Differential Equations 21(2), 387–396 (2005) 34. Verf¨urth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner (1996) 35. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp. 73(247), 1139–1152 (electronic) (2004) 36. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24(2), 337–357 (1987) 37. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique. Internat. J. Numer. Methods Engrg. 33(7), 1331–1364 (1992)