An adaptive finite element method for the wave equation based on anisotropic a posteriori error estimates in the L2(H1) norm

(1)

HAL Id: inria-00435786

https://hal.inria.fr/inria-00435786v2

Submitted on 6 Jan 2010

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

equation based on anisotropic a posteriori error

estimates in the L2(H1) norm

Marco Picasso

To cite this version:

Marco Picasso. An adaptive finite element method for the wave equation based on anisotropic a

posteriori error estimates in the L2(H1) norm. [Research Report] RR-7115, INRIA. 2009, pp.22.

�inria-00435786v2�

(2)

a p p o r t

d e r e c h e r c h e

IS

S

N

0

2

4

9 -6

3

9

9 IS

R

N

IN

R

IA

/R

R

--7

1

5 --F

R

+

E

N

G

Domaine 1

An adaptive finite element method for the wave

equation based on anisotropic a posteriori error

estimates in the L

2 (H

1 ) norm.

Marco Picasso

N° 7115

(3)

(4)

Centre de recherche INRIA Paris – Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

estimates in the

L

2 _(H

1 ₎

norm.

Mar o Pi asso

Domaine: Mathématiquesappliquées, al ul etsimulation Équipe-ProjetGamma

and

Institut d'AnalyseetCal ulS ientique,Station 8,E olePolyte hnique FédéraledeLausanne,1015Lausanne,Switzerland

Rapport dere her he n°7115Novembre200922pages

Abstra t: Anadaptiveniteelementalgorithmispresentedforthewave equa-tionintwospa edimensions. Thegoaloftheadaptivealgorithmisto ontrolthe errorin thesamenormasforparaboli problems, namelythe

L

2 _{(0, T ; H}

1 _(Ω))

norm, where

T

denotes the nal time and

Ω

the omputationaldomain. The mesh aspe tratio anbelargewheneverneeded,thusallowingagivenlevelof a ura ytoberea hedwithfewerverti esthanwith lassi alisotropi meshes. The renement and oarsening riteria are based on anisotropi , a posteriori errorestimatesandonanellipti re onstru tion.

Anumeri alstudyof theee tivityindexonnon-adaptedmeshes onrms the sharpness of the error estimator. Numeri al results on adapted meshes indi ate that the error indi ator slightly underestimates the true error. We onje ture thatthe missing information orresponds tothe interpolationerror between su essive meshes. It is observed that the error indi ator be omes sharpagainwhen onsideringthedampedwaveequationwithalargedamping oe ient,thuswhentheparaboli hara terofthePDEbe omespredominant.

Key-words: Waveequation,Adaptiveniteelements,Anisotropi aposteriori errorestimates

(5)

d'erreur a posteriori anisotropes dans la norme

L

2 (H

1 )

.

Résumé: Unalgorithmed'élémentsnisadaptatifsestprésentépourl'équation desondesàdeuxdimensionsd'espa e. Le but del'algorithmeest de ontrler l'erreurdans lanormenaturelle desproblèmesparaboliques,àsavoirlanorme

L

2 _{(0, T ; H}

1 _(Ω))

. Le rapport d'aspe t des triangles peut être grand si né es-saire, e qui permet d'obtenir unniveaude pré isiondonné ave relativement peudesommets. Les ritèresderanementet déranementsontbaséssurun estimateurd'erreuraposteriorianisotrope.

Lesrésultatsnumériquessurdesmaillagesnon-adaptésmontrentquel'estimateur d'erreur est pré is. Les résultats numériques sur des maillages adaptés mon-trent que l'estimateur d'erreur sous-estime légèrement l'erreur. Nous pensons que ette dis répan e provient de l'erreur d'interpolation entre maillages qui n'est pas ontenuedans l'estimateur. Les résultats montrent que ette erreur d'interpolation entre maillages dimunue lorsqu'un terme d'amortissement est ajouté dans l'équation des ondes, 'est-à-dire lorsque le ara tère parabolique del'équationaugmente.

Mots- lés : Equation des ondes, Eléments nis adaptatifs, Estimations d'erreuraposteriorianisotropes

(6)

1 Introdu tion

A posteriori error estimates and adaptive nite elements have already been widely onsideredforsolvingellipti ,paraboli andrstorderhyperboli prob-lems. However,hyperboli problemsofse ond orderhavebeenmu hless stud-ied,see forinstan e [15,5,6,1,2,8℄.

Re ently, adaptivenite elementswith large aspe t ratio havebeen intro-du ed in order to redu e the number of verti es required for a given level of a ura y[3,7℄. Ellipti problemswere onsideredin [21,22℄,with goalto on-trol theerrorinthenatural

H

1 _(Ω)

norm,while the

L

2 _{(0, T ; H}

1 _(Ω))

normwas usedforparaboli problemsin [20,10,17℄.

When onsidering hyperboli problems of se ond order, the natural norm is the

C

0 _{([0, T ]; H}

1

0 (Ω)) ∩ C

1 ([0, T ]; L

2 (Ω))

norm. However,asexplainedin [8℄, the

C

0 _{([0, T ]; L}

2 _{(Ω)) ∩ C}

1 _{([0, T ]; H}

−1

_(Ω))

normseemstobemoreappropriatein orderto deriveaposteriorierrorestimates.

Inthispaper,aposteriorierrorestimatesarederivedinthe

L

2 _{(0, T ; H}

1 _(Ω))

normforthewaveequationin twospa edimensions,whi hallowstheadaptive algorithmsdevelopedforparaboli problemsin[20,10,17℄tobeusedwithvery few modi ations. For this purpose, the ellipti re onstru tion te hnique [18℄ introdu edforparaboli problemswillbeused.

Theoutlineisthefollowing. Thewaveequation andthe ontinuous, pie e-wise linear nite element dis retization in spa e is presented in the next se -tion. An aposterioriupperbound isproposed in se tion3fortheerrorin the

L

2 _{(0, T ; H}

1 _(Ω))

norm,usingtheellipti re onstru tionte hniqueintrodu edfor paraboli problemsin[18℄. Anordertwotimedis retizationissele tedand nu-meri alresultsonnon-adaptedmeshesarereportedinse tion4. Itisobserved that the error indi atoris sharp provided the errordue to timedis retization be omes negligible, whi h is the ase when the time step

τ

is of the order of the spa e step

h

. An adaptivealgorithm is presented in se tion 5in order to ontrol the errorin the

L

2 _{(0, T ; H}

1 _(Ω))

norm. Numeri al results show a dis- repan ybetweenthe errorindi atorand thetrueerror. Wesuspe t that this dis repan y orresponds to the interpolation error betweensu essive meshes whi h is not a ounted in the present paper - and di ult to ompute sin e non- ompatibleanisotropi meshesare involved. Inorder tovalidate this on-je ture,theadaptivealgorithmis usedto solvethedampedwaveequation. It is observedthat thelargerthedamping term,the better theee tivityindex. This observation is onsistentwith thefa t that wehaveobservedin previous papers[20,10, 17℄ that theinterpolationerror betweensu essivemesheswas notrelevantforparaboli problems. Wetherefore on ludethatthisisnottrue forthewaveequation.

2 The wave equation and its nite element

dis- retization

Let

T

bethenal time, let

Ω

beapolygon. Given

f : Ω × (0, T ) → R

, given initial onditions

u

0 , v

0 : Ω → R

,weareinterestedinnding

u : Ω × (0, T ) → R

(7)

su hthat

∂

2 _u

∂t

2 − ∆u = f

in

Ω × (0, T ),

(1)

u = 0

on

∂Ω × (0, T ),

(2)

u(·, 0) = u

0 ,

∂u

∂t

(·, 0) = v

0

in

Ω.

(3) Following for instan e Chap. 3, Se t. 8 of [16℄, the weak formulation of the aboveproblem onsists, given

f ∈ L

2 _{(0, T ; L}

2 _(Ω))

,

u

0 ∈ H

1

0 (Ω)

,

v

0 ∈ L

2 _(Ω)

, in nding

u ∈ L

∞

(0, T ; H

0

1 (Ω)),

∂u

∂t

∈ L

∞

_{(0, T ; L}

2 _(Ω)),

∂

2 u

∂t

2 ∈ L

2 _{(0, T ; H}

−1

_(Ω)),

su hthat

u(·, 0) = u

0

in

H

1

0 (Ω)

,

∂u

∂t

(·, 0) = v

0

in

L

2 _(Ω)

and

∂

2 _u

∂t

2 , v

+

Z

Ω

∇u · ∇v =

Z

Ω

f v,

(4) for all

v ∈ H

1

0 (Ω)

,

a.e. t ∈ (0, T )

, where

< ·, · >

denotes the duality pairing between

H

−1

_(Ω)

and

H

1

0 (Ω)

. FromChap. 3,Se t. 8,Theorem8.1andRemark 8.2 of [16℄, su h asolution exists and is unique. Moreover,using a paraboli regularizationte hnique,itisprovedin Theorem8.2of[16℄that

u ∈ C

0 ([0, T ]; H

0

1 (Ω)),

∂u

∂t

∈ C

0 _{([0, T ]; L}

2 _(Ω)).

We now onsider a nite element dis retization in spa e. We will derive an aposteriorierror bound in the asewhen the same mesh is used between initialandnaltime. Therefore,theinterpolationerrorduetotheuseofseveral mesheswillnotbe onsideredfromthetheoreti alpointofview. However,the adaptive algorithm presented in se tion 5 will obviously make use of several mesheswheneverneeded. Moreover,numeri alexperimentsseemto showthat this interpolationerror should not be negle tedin the framework of the wave equation,eventhoughpreviousstudieshaveshown thattheinterpolation error isnotrelevantforparaboli problems,providedthenumberofremeshingsdoes notdependonthemesh sizeandtime step,see[20,10, 17℄. Theestimationof thisinterpolationerrorisnotanobvioustasksin enon- onforminganisotropi meshesareinvolvedandisthereforebeyondthes opeofthepresentpaper.

For any

h > 0

, let

T

h

be a onformal nite element triangulation with triangles

K

havingdiameter

h

K

≤ h

. Wearelookingforwardtousingtriangles with large aspe t ratio, thus the usual minimum angle ondition will not be satised. Inthe sequelweadopt thenotationsof[11, 12℄ but thoseof[13,14℄ ould be used aswell. Let

T

K

: ˆ

K → K

be the mapping from the referen e triangle

K

ˆ

tothe urrentelement

K

andlet

∆

K

betheunionoftheneighbouring triangles sharing a vertex with

K

. The following two assumptions must be satised: i)thenumberoftrianglesbelongingto

∆

K

mustbebounded above, uniformelywithrespe tto

h

andii)thediameterof

T

−1

K

(∆

K

)

shouldbebounded above,uniformelywithrespe tto

h

,whi hex ludesmesheswithlarge urvature, seeforinstan e[20,19℄forexamples.

(8)

Let

V

h

be the usual subspa eof

H

1

0 (Ω)

orresponding to ontinuous fun -tions, pie ewise linear and vanishing on the boundary. We denote by

r

h

:

C

0 _{( ¯}

_{Ω) → V}

h

the usual Lagrange interpolant. Then, assuming that the ini-tialdata

u

0

and

v

0

are

C

0 _{( ¯}

_Ω)

,thenite elementdis retization of (4)isto nd

u

h

∈ H

2 (0, T ; V

h

)

su hthat

u

h

(0) = r

h

u

0

,

∂u

h

(0)/∂t = r

h

v

0

and

Z

Ω

∂

2 _u

h

∂t

2 v

h

+

Z

Ω

∇u

h

· ∇v

h

=

Z

Ω

f v

h

,

(5)

forall

v

h

∈ V

h

,

a.e. t ∈ (0, T )

.

Inthesequelwederiveanisotropi aposteriorierrorestimatesin thesame normasforparaboli problems,namelythe

L

2 _{(0, T ; H}

1 _(Ω))

norm,whi hallows thesameadaptivealgorithmasin [20℄tobeused.

3 Anisotropi a posteriori error estimates

In order to ontrol theerror in the

L

2 _{(0, T ; H}

1 _(Ω))

norm, we introdu e asin [18℄theellipti re onstru tion

U ∈ L

2 _{(0, T ; H}

1

0 (Ω))

dened by

Z

Ω

∂

2 _u

h

∂t

2 v +

Z

Ω

∇U · ∇v =

Z

Ω

f v,

(6)

Ourgoalis to ontrol

u − u

h

= u − U + U − u

h

.

Werstre allsomeinterpolationestimatesdueto[11,12℄. Foranytriangle

K

of the mesh, let

T

K

: ˆ

K → K

be theane transformationwhi hmaps the referen etriangle

K

ˆ

into

K

. Let

M

K

betheJa obianof

T

K

thatis

x

_{= T}

_K

_(ˆ

x

_{) = M}

_K

x

_ˆ

_{+ t}

_K

_.

Sin e

M

K

isinvertible,itadmitsasingularvaluede omposition

M

K

= R

T

K

Λ

K

P

K

, where

R

K

and

P

K

are orthogonaland where

Λ

K

isdiagonal withpositive en-tries. Inthefollowingweset

Λ

K

=

λ

1,K

₀

_λ

0 2,K

and

R

K

=

r

T

1,K

r

T

2,K

,

(7)

withthe hoi e

λ

1,K

≥ λ

2,K

. Finally,foranytriangle

K

ofthemesh,thethree edgeswillbedenotedby

ℓ

i,K

,

i = 1, 2, 3

.

ThefollowinginterpolationestimatesfortheClémentinterpolant

R

h

anbe foundin [11,12,22℄.

Lemma1 Thereisa onstant

C

independentofthemeshsizeandaspe tratio su hthat,for all

v ∈ H

1 _(Ω)

,for all

K ∈ T

h

,for all

i = 1, 2, 3

:

kv − R

h

vk

2 L

2 (K)

+

λ

1,K

λ

2,K

|ℓ

i,K

|

kv − R

h

vk

2 L

2 (ℓ

i,K

)

≤ Cω

2 K

(v),

(8) where

ω

K

(v)

isdenedby

ω

2 K

(v) = λ

2 1,K

r

T

_1,K

_G

_K

_(v)r

_1,K

+ λ

2 2,K

r

T

_2,K

_G

_K

_(v)r

_2,K

,

(9)

and

G

K

(v)

denotes the

2 × 2

matrixdenedby

G

K

(v) =







Z

∆

K

∂v

∂x

1

2 dx

Z

∆

K

∂v

∂x

1 ∂v

∂x

2 dx

Z

∆

K

∂v

∂x

1 ∂v

∂x

2 dx

Z

∆

K

∂v

∂x

2

2 dx







.

(10)

We are now in position to ontrol the error

u − u

h

= u − U + U − u

h

. Con erning

U − u

h

,wehavethefollowingresult.

Lemma2 Let

u

h

,

U

bedenedby (5) , (6) , respe tively. Then, thereexists

C

independent ofthe meshsizeandaspe tratiosu hthat

Z

T

0 Z

Ω

|∇(U − u

h

)|

2 ≤ C

Z

T

0 X

K∈T

h

η

K,1

2 ,

(11) where

η

2 K,1

isdenedby

η

K,1

2 =

f −

∂

2 _u

h

∂t

2 L

2 (K)

+

1

2

3 X

i=1

_|ℓ

i,K

|

λ

1,K

λ

2,K

1/2

k [∇u

h

· n] k

L

2 (ℓ

i,K

)

!

× ω

K

(U − u

h

).

(12) Here

[·]

denotes the jump of the bra keted quantity a ross edge

ℓ

i,K

, with the onvention

[·] = 0

for anedge

ℓ

i,K

onthe boundary

∂Ω

.

Proof. Using(6)and(5) ,wehave

Z

Ω

|∇(U − u

h

)|

2 =

Z

Ω

f −

∂

2 _u

h

∂t

2 (U −u

h

−v

h

)−

Z

Ω

∇u

h

·∇(U −u

h

−v

h

)

∀v

h

∈ V

h

.

Integrating by parts the diusion term over ea h triangle

K

, hoosing

v

h

=

R

h

(U − u

h

)

, the Clément interpolant of

U − u

h

and using the interpolation estimatesofLemma 1yieldstheresult.

Remark1 The estimator (12 )is not ausualerror estimator sin e

U − u

h

is stillinvolved. However,if we anguess

U − u

h

,then(11 ) an beusedtoderive a omputable quantity. This idea has been used in [20 , 22 ℄ and an e ient anisotropi error indi ator hasalso been obtainedrepla ing thederivatives

∂(U − u

h

)

∂x

i

in (10 )by

Π

h

∂u

h

∂x

i

−

∂u

h

∂x

i

,

i=1,2, (13)

where

Π

h

is the approximate

L

2 _(Ω)

proje tion onto

V

h

dened for ea h vertex

P

of

T

h

by







Π

h

∂u

h

∂x

1 (P )

Π

h

∂u

h

∂x

2 (P )







=

X

1 K∈T

h

P ∈K

|K|







X

K∈T

h

P ∈K

|K|

∂u

h

∂x

1 |K

X

K∈T

h

P ∈K

|K|

∂u

h

∂x

2 |K







.

(10)

Wenow ontrol

u − U

withrespe tto

U − u

h

.

Lemma3 Let

u

,

u

h

,

U

be denedby (4), (5) , (6), respe tively. Assume that

f ∈ H

2 _{(0, T ; L}

2 _(Ω))

so that

u

h

∈ H

4 _{(0, T ; V}

h

)

and

U ∈ H

2 _{(0, T ; H}

1

0 (Ω))

. We have, for

0 ≤ t ≤ T

:

Z

Ω

∂

∂t

(u − U )(t)

2 +

Z

Ω

|∇(u − U )(t)|

2 ≤ 2

Z

Ω

∂

∂t

(u − U )(0)

2 + 2

Z

Ω

|∇(u − U )(0)|

2 + 2t

Z

t

0 Z

Ω

∂

2 ∂t

2 (U − u

h

)

2 .

(14)

Proof. Using(4)and(6) wehave

Z

Ω

∂

2 ∂t

2 (u − U )v +

Z

Ω

∇(u − U ) · ∇v =

Z

Ω

∂

2 ∂t

2 (u

h

− U )v,

forall

v ∈ H

1

0 (Ω)

,

a.e. t ∈ (0, T )

. Wethen hoose

v = ∂/∂t(u − U )

to obtain

1

2 d

dt

Z

Ω

∂

∂t

(u − U )

2 + |∇(u − U )|

2 !

=

Z

Ω

∂

2 ∂t

2 (u

h

− U )

∂

∂t

(u − U ).

Weset

y(t) =

Z

Ω

∂

∂t

(u − U )

2 + |∇(u − U )|

2 !

,

anduseCau hy-S hwarzinequalitytoobtain

1

2 d

dt

y(t) ≤

∂

2 ∂t

2 (u

h

− U )

_L

₂

_(Ω)

y

1/2

(t).

(15) Sin e

d

dt

y

1/2

₌

1

2 y

−1/2

dy

dt

,

wethereforeobtain,integrating(15)betweentime

0

and

t

:

y

1/2

(t) ≤ y

1/2

(0) +

Z

t

0 ∂

2 ∂t

2 (u

h

− U )

_L

₂

_(Ω)

.

TakingthesquareandusingYoung'sinequalityyieldstheresult.

Itnowremainstoestimate

∂/∂t(u − U )(0)

,

∇(u − U )(0)

and

∂

2 _/∂t

2 _{(U − u}

h

)

.

Lemma4 Assumethatthe polygon

Ω

is onvexandthat

f ∈ H

2 _{(0, T ; L}

2 _(Ω))

. Let

u

,

u

h

,

U

be dened by (4), (5) , (6), respe tively. Then, there exists

C

1

independent of themesh size(butdependingon the aspe t ratio)su hthat

Z

T

0 Z

Ω

∂

2 ∂t

2 (U − u

h

)

2 ≤ C

1 Z

T

0 X

K∈T

h

η

K,2

2 ,

(16)

(11)

where

η

2 K,2

isdenedby

η

2 K,2

= h

4 K

∂

2 _f

∂t

2 −

∂

4 _u

h

∂t

4

2 L

2 (K)

+ h

3 K

∇

∂

2 _u

h

∂t

2 .n

2 L

2 (∂K)

.

(17) Moreover, if

u

0

,

v

0 ∈ H

2 _(Ω)

,then thereexists

C

2

independentof themesh size andaspe tratio,

C

3

and

C

4

independentofthe meshsize(butdependingonthe aspe t ratio)su hthat

Z

Ω

∂

∂t

(u − U )(0)

2 +

Z

Ω

|∇(u − U )(0)|

2 ≤ C

2 X

K∈T

h

η

K,3

2 + C

3 X

K∈T

h

η

K,4

2 + C

4 h

2 Z

Ω

|D

2 u

0 |

2 + |D

2 v

0 |

2 .

(18) where

η

2 K,3

and

η

2 K,4

are denedby

η

K,3

2 =

f (0) −

∂

2 _u

h

∂t

2 (0)

L

2 (K)

+

1

2

3 X

i=1

_|ℓ

i,K

|

λ

1,K

λ

2,K

1/2

k [∇u

h

(0) · n] k

L

2 (ℓ

i,K

)

!

× ω

K

((u − U )(0)),

(19)

η

K,4

2 = h

4 K

∂f

∂t

(0) −

∂

3 _u

h

∂t

3 (0)

2 L

2 (K)

+ h

3 K

∇

∂u

h

∂t

(0) · n

2 L

2 (∂K)

.

(20)

Proof. Wemimi k[18℄inorderto obtain(17). From(5)and(6)wehave

Z

Ω

∇(U − u

h

) · ∇v

h

= 0

∀v

h

∈ V

h

.

Sin e

f ∈ H

2 _{(0, T ; L}

2 _(Ω))

,

u

h

∈ H

4 _{(0, T ; V}

h

)

and

U ∈ H

2 _{(0, T ; H}

1

0 (Ω))

, thus, dierentiatingtheaboveequationtwi e withrespe tto

t

,weobtain

Z

Ω

∇

∂

2 ∂t

2 (U − u

h

) · ∇v

h

= 0

∀v

h

∈ V

h

.

(21) Let

ϕ ∈ L

2 _{(0, T ; H}

1

0 (Ω))

bedenedby

Z

Ω

∇ϕ · ∇v =

Z

Ω

∂

2 ∂t

2 (U − u

h

)v

∀v ∈ H

1

0 (Ω).

Wethushave,using(21),

Z

Ω

∂

2 ∂t

2 (U − u

h

)

2 =

Z

Ω

∇ϕ · ∇

∂

2 ∂t

2 (U − u

h

)

=

Z

Ω

∇(ϕ − v

h

) · ∇

∂

2 ∂t

2 (U − u

h

),

forall

v

h

∈ V

h

. Sin e

Ω

isa onvexpolygon,

U ∈ H

2 _{(0, T ; H}

2 _(Ω))

,thuswe an integratebyparts thediusion termoverea htriangle

K

to obtain

Z

Ω

∂

2 ∂t

2 (U − u

h

)

2 =

X

K∈T

h

Z

K

∆

∂

2 ∂t

2 (U −u

h

)(ϕ−v

h

)+

1

2 Z

∂K

∇

∂

2 ∂t

2 (U − u

h

) · n

(ϕ−v

h

)

!

.

(12)

Sin e

Ω

isa onvexpolygon,

ϕ ∈ L

2 _{(0, T ; H}

2 _(Ω))

,wearethusallowedto hoose

v

h

= r

h

ϕ

, the Lagrangeinterpolantof

ϕ

. Moreover,there exists

C

depending onlyonthesize of

Ω

su hthat

kϕk

H

2 _(Ω)

≤ C

∂

2 ∂t

2 (U − u

h

)

L

2 (Ω)

.

Therefore, using standart interpolation results, there exists

C

independent of themeshsize (butdependingontheaspe tratio)su hthat

Z

Ω

∂

2 ∂t

2 (U − u

h

)

2 ≤ C

X

K∈T

h

4 K

∆

∂

2 ∂t

2 (U − u

h

)

2 L

2 (K)

+

1

2 h

3 K

∇

∂

2 ∂t

2 (U − u

h

) · n

2 L

2 (∂K)

!

.

Finally,sin e

−∆

∂

2 _U

∂t

2 =

∂

2 _f

∂t

2 −

∂

4 _u

h

∂t

4

and

∇

∂

2 ∂t

2 U · n

= 0,

then(16)isproved.

Wenow derive anupperbound for

(u − U )(0)

. From (6) wehave, for all

v ∈ H

1

0 (Ω)

,

Z

Ω

∇(u − U ) · ∇v =

Z

Ω

∂

2 ∂t

2 (u

h

− u)v

=

Z

Ω

∂

2 _u

h

∂t

2 v − f v − ∇u

h

· ∇v

+

Z

Ω

∇(u − u

h

) · ∇v,

thususing(5) weobtain:

Z

Ω

∇(u − U ) · ∇v

=

Z

Ω

∂

2 _u

h

∂t

2 (v − v

h

) − f (v − v

h

) − ∇u

h

· ∇(v − v

h

)

+

Z

Ω

∇(u − u

h

) · ∇v.

(22)

Atinitialtime,theaboveequationwrites

Z

Ω

∇(u−U )(0)·∇v =

Z

Ω

∂

2 _u

h

∂t

2 (0)(v − v

h

) − f (0)(v − v

h

) − ∇u

h

(0) · ∇(v − v

h

)

+

Z

Ω

∇(u

0 − r

h

u

0 ) · ∇v.

(23)

We then hoose

v = (u − U )(0)

,

v

h

= R

h

(u − U )(0)

, where

R

h

is Clément's interpolant,usetheinterpolationestimatesofLemma 1toobtain

Z

Ω

|∇(u − U )(0)|

2 ≤ C

X

K∈T

h

η

K,3

2 +

Z

Ω

|∇(u

0 − r

h

u

0 )|

2 ,

(13)

Dierentiating (22)withrespe tto

t

yields,at initialtime

Z

Ω

∇

∂

∂t

(u−U )(0)·∇v =

Z

Ω

∂

3 _u

h

∂t

3 (0)(v − v

h

) −

∂f

∂t

(0)(v − v

h

) − ∇

∂u

h

∂t

(0) · ∇(v − v

h

)

+

Z

Ω

∇(v

0 − r

h

v

0 ) · ∇v,

(24) forall

v ∈ H

1

0 (Ω)

and

v

h

∈ V

h

. Let

ϕ ∈ H

1

0 (Ω)

bedenedby

Z

Ω

∇ϕ · ∇v =

Z

Ω

∂

∂t

(U − u)(0)v

∀v ∈ H

1

0 (Ω).

Wethen hose

v = ∂/∂t(u − U )(0)

anduse(24)toobtain

Z

Ω

∂

∂t

(u − U )(0)

2 =

Z

Ω

∂

3 _u

h

∂t

3 (0)(ϕ − v

h

) −

∂f

∂t

(0)(ϕ − v

h

) − ∇

∂u

h

∂t

(0) · ∇(ϕ − v

h

)

+

Z

Ω

∇(v

0 − r

h

v

0 ) · ∇ϕ.

We then hoose

v

h

= r

h

ϕ

, the Lagrange interpolant of

ϕ

, and use standart interpolationresultstoprove(18).

Wenowstatethemain resultofthepaper.

Theorem1 Assume that the polygon

Ω

is onvex, that

u

0 , v

0 ∈ H

2 _(Ω)

and

f ∈ H

2 _{(0, T ; L}

2 _(Ω))

. Let

u

,

u

h

,

U

bedenedby (4),(5), (6),respe tively. Let the errorestimators

η

K,1

,

η

K,2

,

η

K,3

,

η

K,4

bedened by (12), (17), (19) , (20), respe tively. Then,thereexists

C

1

independentofthemeshsizeandaspe tratio,

C

2

independentof the meshsize(butdepending onthe aspe t ratio)su h that

Z

T

0 Z

Ω

|∇(u − u

h

)|

2 ≤ C

1 Z

T

0 X

K∈T

h

η

2 K,1

+

X

K∈T

h

η

2 K,3

!

+ C

2 Z

T

0 X

K∈T

h

η

2 K,2

+

X

K∈T

h

η

2 K,4

+ h

2 Z

Ω

|D

2 _u

0 |

2 + |D

2 v

0 |

2 !

.

(25) Proof. Wehave

Z

T

0 Z

Ω

|∇(u − u

h

)|

2 ≤ 2

Z

T

0 Z

Ω

|∇(u − U )|

2 + 2

Z

T

0 Z

Ω

|∇(U − u

h

)|

2 .

FromLemma3wehave

Z

T

0 Z

Ω

|∇(u−U )|

2 ≤ 2T

Z

Ω

∂

∂t

(u − U )(0)

2 +2T

Z

Ω

|∇(u − U )(0)|

2 +T

2 Z

T

0 Z

Ω

∂

2 ∂t

2 (U − u

h

)

2 .

(14)

4 Time dis retization and numeri al results with

non adapted meshes

The time dis retization onsidered here orresponds to the impli itorder two Newmark s heme. Given an integer

N

, we set

τ = T /N

thetime step,

t

n

₌

nτ

and introdu e

u

n

h

an approximation of

u

h

(t

n

₎

,

n = 0, ..., N

. The initial onditionsare set to

u

0 h

= r

h

u

0

,

v

0 h

= r

h

v

0

. Therst approximation

u

1 h

∈ V

h

satises

Z

Ω

u

1 h

− u

0 h

− τ v

0 h

τ

2 v

h

+

Z

Ω

1

4 ∇u

1 h

+

1

4 ∇u

0 h

· ∇v

h

=

Z

Ω

f (t

0 _)v

h

,

forall

v

h

∈ V

h

. Then,forea h

n = 1..., N

,we ompute

u

n+1

h

∈ V

h

su hthat

Z

Ω

u

n+1

_h

− 2u

n

h

+ u

n−1

h

τ

2 v

h

+

Z

Ω

1

4 ∇u

n+1

h

+

1

2 ∇u

n

h

+

1

4 ∇u

n−1

h

·∇v

h

=

Z

Ω

f (t

n

)v

h

,

for all

v

h

∈ V

h

. This s heme is

O(h + τ

2 ₎

onvergentin the

L

2 _{(0, T ; H}

1 _(Ω))

norm. Therefore, setting

τ = O(h)

, the error due to time dis retization be- omes asymptoti ally negligible so that the error due to spa e dis retization onlyshould bere overed. Theinterestedreadershould notethat the Stormer-Numerovs heme[23℄ ouldbeusedto obtaina

O(h + τ

4 ₎

onvergents heme. Wenow report numeri alresults using uniform meshes and onstant time stepswhenusing

Z

T

0 X

K∈T

h

η

K,1

2

(26)

aserrorindi ator. Goingba ktotheupperbound(25),it anbenoti edthat theterms

Z

T

0 X

K∈T

h

η

2 K,2

+

X

K∈T

h

η

2 K,4

are of higher order and anthus be disregarded. This is not the ase of the terms

X

K∈T

h

η

2 K,3

and

h

2 Z

Ω

|D

2 _u

0 |

2 + |D

2 v

0 |

2 .

Sin e atrapezoidalquadrature formulais usedto approximate (26), The

η

K,3

term yields a ontribution whi h is very similar to that of (26). Numeri al resultsreportedhereafterindi atethatusing(26)isindeedsu ienttorepresent orre tlytheerror.

(15)

Inorderto studythequalityofourerrorindi ator,thefollowingtwo ee -tivityindi esareintrodu ed

ei

ZZ

=







Z

T

0 X

K∈T

h

(I − Π

h

)

∂u

h

∂x

1

2 +

(I − Π

h

)

∂u

h

∂x

2

2 !

Z

T

0 Z

Ω

|∇(u − u

hτ

)|

2 





1/2

,

ei

A

=







Z

T

0 X

K∈T

h

η

2 K,1

Z

T

0 Z

Ω

|∇(u − u

hτ

)|

2 





1/2

,

where

Π

h

is dened in Remark1,

η

K,1

isdened in (12)and

u

hτ

denotesthe ontinuous,pie ewiselinearapproximationintimedened by

u

hτ

(x, t) =

t − t

n−1

τ

u

n

h

(x) +

t

n

_{− t}

τ

u

n−1

h

(x)

t

n−1

≤ t ≤ t

n

,

x ∈ Ω.

(27) Therstnumeri alexample orrespondstothe asewhen

Ω =]0, 1[

2 ,

T = 0.4,

f (x

1 , x

2 , t) = 0,

u

0 (x

1 , x

2 ) = exp

−1000∗(x

1 −0.5)

2 ,

v

0 (x

1 , x

2 ) = 0.

(28)

Homogeneousboundary onditionsapplyontheverti alsidesof

Ω

whereas Neu-manhomogeneousboundary onditionsapplyonthehorizontalsides. Therefore thesolutionisone-dimensionalandisgivenbyd'Alembertformula

u(x

1 , x

2 , t) =

1

2 u

0 (x

1 − t, x

2 ) + u

0 (x

1 + t, x

2 )

.

The omputed solutionat time

0

and

0.3

is reported in Fig. 1 when using a uniform mesh with mesh

h

x

= 0.005

,

h

y

= 0.05

and atime step

τ = 0.0005

. Numeri alexperimentswithseveraluniformmeshesand onstanttimestepsare reportedinTable1.

In rows 1-6, it is observed that when setting

h = O(τ

2 ₎

then the error is dividedbyfourea htimethetimestepisdividedbytwothus

Z

T

0 Z

Ω

|∇(u − u

hτ

)|

2 !

1/2

= O(h + τ

2 ).

Itshould benotedthat

ei

ZZ

, theee tivity indexofZZ,isnot losetoone in rows1-3duetothefa tthattheerrorduetospa edis retizationisofthesame orderthantheerrordueto timedis retization.

Inordertoinsurethattheerrorduetotimedis retizationbe omesnegligible withrespe ttotheerrorduetospa edis retizationwe hoose

τ = O(h)

inrows 7-9. Weobservethattheerrorisstilloforderoneandthat

ei

ZZ

now onverges toonewhen

h

onvergestozero,asforellipti problems.

(16)

h

x

h

y

τ

e

ei

ZZ

ei

A

N

vert

0.01 0.1 0.01 1.70 0.23 0.60 1292 0.0025 0.025 0.005 0.46 0.19 0.52 20001 0.000625 0.00625 0.0025 0.11 0.19 0.51 325302 0.01 0.1 0.001 0.83 0.51 1.34 1292 0.0025 0.025 0.0005 0.10 0.86 2.41 20001 0.000625 0.00625 0.00025 0.022 0.98 2.67 325302 0.01 0.1 0.001 0.83 0.51 1.34 1292 0.005 0.05 0.0005 0.27 0.71 2.00 5070 0.0025 0.025 0.00025 0.10 0.87 2.44 20001 0.00125 0.0125 0.000125 0.046 0.96 2.63 80391

Table1: Example28. Nonadaptedmeshes.

τ

2 _{= O(h)}

and

τ = O(h)

. Notation :

e =

_R

_T

0 R

Ω

|∇(u − u

hτ

)|

2 1/2

,

N

vert

is the number of verti es of the nal mesh. -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

t = 0

t = 0.3

Figure1: Example28. Nonadaptedmeshes.

h

x

= 0.005

,

h

y

= 0.05

,

τ = 0.0005

. Computedsolution

u

h

(x

1 , 1, t)

withrespe tto

x

1

attime

t = 0

and

t = 0.3

.

Inthese ondnumeri alexamplewe hangethefollowingparametersin(28) :

T = 1.4,

u

0 (x

1 , x

2 ) = exp

−1000∗

(x

₁

−0.5)

2 _+(x

2 −0.5)

2 ,

(29) andhomogeneousboundary onditionsapplyonthewholeboundaryof

Ω

. For this test ase, theexa tsolution isnot known and multiple ree tions anbe observedwhentheinitialwavehitstheboundary,seeFig. 2.

(17)

Figure2: Example29. Nonadaptedmeshes. Numeri alsolutionona

100 × 100

uniformmeshattime

0.3

,

0.6

,

0.9

,

1.2

.

5 An anisotropi , adaptive nite element

algo-rithm

Wenowusetheadaptivealgorithmdes ribedin[20℄using

Z

T

0 X

K∈T

h

η

2 K,1

aserrorindi ator. Thegoalistondanisotropi triangulations

T

n

h

,

n = 1, ..., N

su hthattherelativeestimatederroris losetoapresettoleran e

T OL

,namely :

(1 − α)T OL ≤

Z

T

0 X

K∈T

h

η

2 K,1

!

1/2

Z

T

0 Z

Ω

|∇u

hτ

|

2 !

1/2

≤ (1 + α)T OL.

(30)

Hereabove,

0 < α < 1

is a parameter ae ting the numberof remeshings, in generalwe hoose

α = 0.25

. Asu ient onditiontosatisfy(30)istobuild,for ea h

n = 1, ..., N

,ananisotropi triangulation

T

n

h

su hthat

(1 − α)

2 T OL

2 Z

t

n

t

n−1

Z

Ω

|∇u

hτ

|

2 ≤

Z

t

n

t

n−1

X

K∈T

h

η

K,1

2 ≤ (1 + α)

2 T OL

2 Z

t

n

t

n−1

Z

Ω

|∇u

hτ

|

2 .

(18)

We thenpro eedasin [20℄ to build su h an anisotropi mesh, using theBL2D mesh generator[9℄.

Example(28)is onsidered. Unless otherwisespe ied,the initial triangu-lation is a uniform

100 × 10

mesh. Results are reported when using several valuesof

T OL

with

τ = O(T OL

−1

₎

sothattheerrorduetotimedis retization be omes asymptoti ally negligible. The results of Table 2 orrespond to the hoi e

α = 0.25

in (30)whi hwastheparameterusedin[20℄. Asexpe ted,the error is divided bytwo ea h time

T OL

is, the number of verti es of thenal mesh is multipliedby four andthe numberofremeshings isroughly onstant. However, unlikethe resultsobtainedwith non adapted meshes,the ee tivity index of ZZ does notgo to one. The resultswith

T OL = 0.125

arereported in Fig. 3to 5. As seenin Fig. 5, thedis repan y betweenthe trueand esti-matederrorin reaseswith respe tto time. Wesuspe t thatthemissing error is due to interpolation betweenmeshes and thereforeweperform experiments with

α = 0.5

in (30) , or even by imposing the times at whi h remeshing is performed. The resultsare reported in Table3. Clearly, the ee tivityindex of ZZ at nal time depends on the number of remeshings. Therefore, unlike whathasbeenobservedforparaboli problems[20,17℄,we onje turethatthe interpolationerrorbetweenmeshes annotbenegle tedforthewaveequation. Estimatingthiserrorisnotaneasytaskfornon ompatibleanisotropi meshes andisbeyondthes opeofthepresentpaper. However,thisinterpolationerror should be onsideredin aforth oming ontribution.

T OL

τ

e

ei

ZZ

_ei

A

_N

vert

N

mesh

max ratio

av ratio

0.125 0.001 0.27 0.45 1.23 2382 28 145 27

0.0625 0.0005 0.10 0.64 1.73 9781 27 245 29 0.03125 0.00025 0.051 0.64 1.77 36865 39 226 30 0.015625 0.000125 0.023 0.71 1.96 182914 45 285 25

Table2: Example28. Adaptivealgorithm. Resultswithrespe tto

T OL

when

α = 0.25

. Caption:

N

mesh

isthenumberofgeneratedmeshes,

max ratio

(resp.

av ratio

)is the maximum (resp. average) aspe t ratio

λ

1,K

/λ

2 K

of thenal mesh.

(19)

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1

t = 0

t = 0.3

Figure 3: Example 28. Adaptive algorithm with

T OL = 0.125

,

α = 0.25

. Computedsolution

u

h

(x

1 , 1, t)

withrespe tto

x

1

attime

t = 0

and

t = 0.3

.

Figure 4: Example 28. Adaptive algorithm with

T OL = 0.125

,

α = 0.25

. Adaptedmesh attime

t = 0

(1305verti es)and

t = 0.3

(2164verti es).

(20)

0.01 0.1 1 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 trueerror errorestimator

ei

ZZ

remeshingtime

♦

♦♦

♦ ♦

♦

♦ ♦ ♦

♦ ♦ ♦ ♦

♦ ♦ ♦

♦ ♦

♦

Figure 5: Example28. Adaptivealgorithmwith

T OL = 0.125

,

α = 0.25

. True andestimatederrorwithrespe ttotime.

e

ei

ZZ

_ei

A

_N

vert

N

mesh

max ratio

av ratio

0.049 0.77 2.06 60179 23 217 26 0.051 0.64 1.77 36865 39 226 30 0.071 0.76 2.03 33281 20 204 34 0.064 0.68 1.85 15817 40 355 56

Table3: Example28. Adaptedmeshes. Inuen e ofthenumberofremeshings with

T OL = 0.03125

and

τ = 0.00025

;row1: adaptivealgorithmwith

α = 0.5

; row 2: adaptivealgorithm with

α = 0.25

; row3: 20 imposed remeshingsat time

0.02

,

0.04

,...

0.38

; row 4 : 40 imposed remeshings at time

0.01

,

0.02

,...

0.39

.

Inorderto he kthe onje turethattheinterpolationerrorbetweenmeshes annot be negle ted, onservativeinterpolation [4℄ rather than linear interpo-lation is onsideredto interpolatethe omputed solutionafter remeshing. The resultsarereportedinTable4and learlyshowthattheee tivityindexofZZ is losertoonewith onservativeinterpolationratherthanlinearinterpolation.

T OL

τ

e

ei

ZZ

_ei

A

_N

vert

N

mesh

max ratio

av ratio

0.125 0.001 0.20 0.66 1.77 3797 18 111 24

0.0625 0.0005 0.089 0.74 2.00 9260 25 245 32 0.03125 0.00025 0.045 0.73 2.00 27711 37 339 43 0.015625 0.000125 0.021 0.78 2.15 86381 46 799 52

T OL

when

α = 0.25

whenusingthe onservativeinterpolationmethodof[4℄tointerpolate the omputedsolutionafter remeshing.

(21)

Still in order to he kthe onje ture that the interpolation error between meshes annotbenegle ted,thedampedwaveequationis onsidered

∂

2 _u

∂t

2 − ∆u + β

∂u

∂t

= f.

(31) Clearly, when the damping oe ient

β ≥ 0

in reases, then the behaviourof theequation ressembles that of a paraboli problem, thus we onje ture that the interpolation errorbetweenmeshes should de rease. Still onsidering the example(28) of the previousse tion, the exa tsolution an be omputed by meansof Fourierseries. The adaptivealgorithm is run with several valuesof

β

and the results are reported in Table 5. Clearly, when

β = 100

, then the ee tivity index of ZZ is lose to one and the ee tivity index of our error estimatoris lose to 2.7 whi h orresponds to the value already observed for variousellipti andparaboli problems,whi hshowsthattheinterpolationerror betweenmeshesbe omesunimportant.

β

e

ei

ZZ

_ei

A

_N

vert

N

mesh

max ratio

av ratio

0. 0.27 0.45 1.23 2382 28 145 27

10. 0.098 0.68 1.86 2264 30 141 27

100. 0.060 0.99 2.73 233 8 175 44

Table 5: Example 28. Adaptive algorithm. Resultswhen solvingthedamped waveequation(31)withseveralvaluesof

β

with

T OL = 0.125

and

τ = 0.001

.

Example (29) of the previous se tion is now onsidered. Eventhough the exa tsolutionisnotknown, onvergen eoftheadaptivealgorithmwithrespe t to

T OL

is observed. The adapted meshes at time

0.3

,

0.6

,

0.9

and

1.2

are reportedinFig. 6. PlotsofthesolutionalongthediagonalarereportedinFig. 7and8forseveralvaluesof

T OL

. TheresultsaresummarizedinTable6.

(22)

Figure 6: Example 29. Adaptivealgorithmwith

T OL = 0.125

. Adaptedmesh attime

0.3

(23632verti es),

0.6

(37132verti es),

0.9

(38187verti es),

1.2

(52132 verti es). -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

100 × 100

uniform

T OL = 0.25

T OL = 0.125

T OL = 0.0625

Figure 7: Example 29. Adaptive algorithm. Computed solution

u

h

(x

1 , x

2 , t)

alongthediagonal

x

1 = x

2

attime

0.3

. Convergen ewithrespe tto

T OL

when

(23)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

100 × 100

uniform

T OL = 0.25

T OL = 0.125

T OL = 0.0625

Figure 8: Example 29. Adaptive algorithm. Computed solution

u

h

(x

1 , x

2 , t)

alongthediagonal

x

1 = x

2

attime

1.2

. Convergen ewithrespe tto

T OL

when

τ = 0.001

.

T OL

τ

N

vert

N

mesh

max ratio

av ratio

0.25 0.001 7172 104 12 2

0.125 0.001 50264 146 38 3

0.0625 0.001 176427 160 58 4

T OL

.

6 Con lusions

Anaposteriorierrorestimatein the

L

2 _{(0, T ; H}

1 _(Ω))

normis proposedforthe waveequation. Numeri alresultson non-adaptedmeshesshowthat the error indi ator

η

K,1

dened by (12) is sharp even with meshes having large aspe t ratio. An adaptivealgorithm alreadypresented forparaboli problemsis on-sidered. Anumeri alstudyoftheee tivityindex onadapted meshesshowsa dis repan ybetweenthetrueerrorandtheerrorindi ator. Wesuspe tthatthis error orrespondstotheinterpolationerrorintrodu edwhenremeshingo urs. Experiments of the damped waveequation indeed show that the dis repan y betweenerrorandestimatorde reaseswhen thedamping oe ientin reases. Thisisina ordan ewiththefa tthattheinterpolationerrorduetoremeshing doesnotneed tobe onsideredforparaboli problemsprovided thenumberof remeshingsdoesnotdependon

h

(or

T OL

)and

τ

,see[20,10,17℄.

An estimation of the interpolation error due to remeshing should be the subje tofafuturework.Thisisadi ulttasksin enon- ompatibleanisotropi meshesareinvolved.

(24)

A knowledgements

GeorgiosAkrivisisa knowledgedfor onstru tiveremarks. Frédéri Alauzetis a knowledgedforprovidingtheprogram orrespondingto onservative interpo-lation [4℄.

Referen es

[1℄ S.Adjerid. Aposterioriniteelementerrorestimationforse ond-order hy-perboli problems.Comput.MethodsAppl.Me h.Engrg.,191(41-42):4699 4719,2002.

[2℄ S.Adjerid. Aposteriorierrorestimationforthemethodoflumpedmasses applied to se ond-order hyperboli problems. Comput. Methods Appl. Me h. Engrg.,195(33-36):42034219,2006.

[3℄ F. Alauzet, P. J. Frey, P. L. George, and B. Mohammadi. 3D transient xed pointmesh adaptationfortime-dependentproblems: appli ationto CFDsimulations. J. Comput.Phys.,222(2):592623,2007.

[4℄ F. Alauzet and M. Mehrenberger. P1- onservativesolution interpolation onunstru turedtriangularmeshes. RapportdeRe her heINRIANo8604, 2009.

[5℄ W.BangerthandR.Ranna her.Finiteelementapproximationofthe a ous-ti waveequation: error ontrolandmeshadaptation.East-WestJ.Numer. Math.,7(4):263282,1999.

[6℄ W.BangerthandR.Ranna her.Adaptiveniteelementte hniquesforthe a ousti waveequation. J.Comput.A oust.,9(2):575591,2001.

[7℄ Y. Belhamadia, A. Fortin, and E. Chamberland. Three-dimensional anisotropi meshadaptationforphase hangeproblems. J.Comput.Phys., 201(2):753770,2004.

[8℄ C.Bernardi and E. Süli. Time and spa eadaptivity forthese ond-order waveequation. Math. Models Methods Appl.S i.,15(2):199225,2005.

[9℄ H.Borou hakiandP.Laug. TheBL2Dmeshgenerator: Beginner'sguide, user'sandprogrammer'smanual. Te hni alReport RT-0194,Institut Na-tional de Re her he en Informatique et Automatique (INRIA), Ro quen- ourt,78153LeChesnay,Fran e,1996.

[10℄ E. Burman and M.Pi asso. Anisotropi ,adaptive nite elementsfor the omputation of a solutal dendrite. Interfa es Free Bound., 5(2):103127, 2003.

[11℄ L. Formaggia and S. Perotto. New anisotropi a priori error estimates. Numer.Math., 89(4):641667,2001.

[12℄ L.FormaggiaandS.Perotto. Anisotropi errorestimatesforellipti prob-lems. Numer.Math., 94:6792,2003.

(25)

[13℄ G. Kunert. An aposterioriresidual errorestimatorforthenite element method onanisotropi tetrahedralmeshes. Numer. Math., 86(3):471490, 2000.

[14℄ G. Kunert and R. Verfürth. Edge residuals dominate a posteriori error estimates for linear nite elementmethods on anisotropi triangular and tetrahedralmeshes. Numer. Math.,86(2):283303,2000.

[15℄ X.D.LiandN.-E.Wiberg.Implementationandadaptivityofaspa e-time nite element method for stru tural dynami s. Comput. Methods Appl. Me h.Engrg., 156(1-4):211229,1998.

[16℄ J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and appli ations. Vol. I. Springer-Verlag, New York, 1972. Translated fromtheFren hbyP.Kenneth,DieGrundlehrendermathematis hen Wis-sens haften,Band181.

[17℄ A. Lozinski, V. Pra hittham, and M. Pi asso. An anisotropi error esti-mator forthe rank-ni olsonmethod: appli ation to aparaboli problem. SIAMJ. S i. Comp.,31(4):27572783,2009.

[18℄ C.MakridakisandR.H.No hetto. Ellipti re onstru tionandaposteriori errorestimatesforparaboli problems.SIAMJ.Numer.Anal.,41(4):1585 1594(ele troni ),2003.

[19℄ S. Mi heletti, S. Perotto, and M. Pi asso. Stabilized nite elements on anisotropi meshes: apriorierrorestimatesfortheadve tion-diusionand theStokesproblems. SIAMJ.Numer. Anal.,41(3):11311162,2003.

[20℄ M.Pi asso. An anisotropi errorindi atorbasedonZienkiewi z-Zhuerror estimator : appli ation to ellipti and paraboli problems. SIAM J. S i. Comp.,24:13281355,2003.

[21℄ M. Pi asso. Numeri al study of the ee tivity index for an anisotropi errorindi atorbasedonZienkiewi z-Zhuerrorestimator. Comm. Numer. MethodsEngrg., 19(1):1323,2003.

[22℄ M. Pi asso. Adaptivenite elementswith largeaspe t ratio basedonan anisotropi errorestimatorinvolvingrstorderderivatives.Comput. Meth-odsAppl.Me h.Engrg., 196(1-3):1423,2006.

[23℄ J.StoerandR. Bulirs h. Introdu tiontonumeri al analysis,volume12of Texts in Applied Mathemati s. Springer-Verlag,New York,third edition, 2002. Translated from the German by R. Bartels, W. Gauts hi and C. Witzgall.

(26)

Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence Cedex

Centre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier

Centre de recherche INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, avenue Halley - 59650 Villeneuve d’Ascq

Centre de recherche INRIA Nancy – Grand Est : LORIA, Technopôle de Nancy-Brabois - Campus scientifique

615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex

Centre de recherche INRIA Rennes – Bretagne Atlantique : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex

Centre de recherche INRIA Saclay – Île-de-France : Parc Orsay Université - ZAC des Vignes : 4, rue Jacques Monod - 91893 Orsay Cedex

Centre de recherche INRIA Sophia Antipolis – Méditerranée : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

http://www.inria.fr