Cell centered Galerkin methods

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Cell centered Galerkin methods

Daniele Antonio Di Pietro

To cite this version:

Daniele A. Di Pietro

Institut Français du Pétrole, 1 & 4, avenue de Bois Préau, 92852 Rueil

Malmaison, Fran e

Abstra t

Inthisworkweproposeanewapproa htoobtainandanalyzelowestordermethods

for diusive problemsyielding at the sametime onvergen e rates and onvergen e

tominimalregularitysolutions.Theapproa hmergesideasfromFiniteVolumeand

dis ontinuousGalerkin methods ina uniedperspe tive.

Résumé

Dans ette note, on propose une nouvelle appro he pour obtenir et analyser des

méthodes d'ordre bas pour les problèmes diusifs. L'analyse permet d'estimer à

la fois le taux de onvergen e et de prouver la onvergen e vers des solutions à

régularitéminimale.Cetteappro he ombinelesavan éesré entesdanslesméthodes

de Volumes Finisetde Galerkine dis ontinues.

1 Introdu tion

Dans ette note on propose une nouvelle appro he pour obtenir et analyser

desméthodesd'ordrebaspourlesproblèmediusifs.Lesidéessontprésentées

pour le problème modèle

−∇·(ν∇u) = f

in

Ω,

u = 0

on

∂Ω,

(1) où

Ω

denote un ouvert polygonal borné,

f ∈ L

2

_(Ω)

et

ν

est un tenseur uni-formementelliptiqueet onstantpar mor eauxsur une partition

P

Ω

de

Ω

. La méthode est omposée de plusieurs étapes : (i) pour toute fa e du maillage,

on dénit un interpolateur de tra e onsistant pour des fon tions

susam-ment régulières; (ii) à partir des valeurs de tra e, on re onstruit un gradient

par maille qui est aussi onsistant; (iii) le gradient permet dénir un

sous-espa e de l'espa e des fon tions anes par mor eaux, qui est ensuite utilisé

Pour dessolutionssusammentrégulières, laformebilinéairedis rètes'étend

à l'espa e onténant la solution, e qui permet de béné ier des te hniques

d'analyse par estimation d'erreur propres aux méthodes dG. D'autre part,

les te hniques d'analyse par ompa ité typiques des VolumesFinis ( .f.,e.g.,

Eymard, Gallouët et Herbin [5℄) s'appliquent aussi en vertu des résultats

ré- emment obtenus pour les espa es polynomiaux par mor eaux dans [4℄. Par

simpli ité,onutiliseradans laprésentation l'interpolateurbary entrique

pro-posé dans [5℄. Cet interpolateur,simple à mettreen ÷uvre, ne tient toutefois

pas omptedes hétérogénéités du tenseur de diusion. Parmi lesautres hoix

possiblepour mieuxgérerlesopérateurshétérogène,onsignalel'interpolateur

en L introduit par Aavatsmark et al. [1℄ et ré emment analysé dans [2℄. A

titre d'exemple, la méthode sera appliquée aussi à un problème de

diusion-adve tion-réa tion.

2 Dis retization

Letting

a(u, v)

def

=

R

_Ω

∇u·∇v

, the weak formulationof problem (1) reads

Find

u ∈ H

1

0

(Ω)

s.t.

a(u, v) =

Z

Ω

f v

for all

v ∈ H

1

0

(Ω).

(2) Let

H ⊂

R

+

\ {0}

be a ountable set having 0 asan a umulation point and let

{T

h

}

h∈H

dene an admissible polyhedral mesh family in the sense of [4℄ parametrizedbythe meshsize

h

and ompatiblewith

P

Ω

.Let

h ∈ H

begiven. For ea h

T ∈ T

h

, let

x

T

∈ T \ ∂T

be su h that

T

is star-shaped with respe t to

x

T

(the ell enter)and,forallfa es

F ∈ F

h

,set

x

b

F

def

=

R

F

x/|F |

.Thesetof fa es of an element

T ∈ T

h

willbe denoted by

F

T

h

, the set of boundary fa es by

F

b

h

.Forea h internalfa ewe sele tanarbitrarybutxed dire tionforthe normal

n

F

≡ n

T

1

,F

and denoteby

T

1

the element out ofwhi h

n

F

points.The diameters of the generi element

T ∈ T

h

and fa e

F ∈ F

h

are denoted by

h

T

and

h

F

respe tively. Let V

T

h

def

=

R

card(T

h

)

, V

F

h

def

=

R

card(F

h

)

. We all a tra e interpolator

I

a linear bounded operatorfromV

T

h

toX

⊂

V

F

h

su hthat,forallV

T

h

∋ V = {v

T

}

T

∈T

h

, X

∋ I(V) = {I

F

(V)}

F

∈F

h

is su h that

I

F

(V) = 0

if

F ∈ F

b

h

. To ease the presentation, we shall hen eforth refer to the bary entri interpolator of [5℄

dened hereafter. Let, for ea h

F ∈ F

i

h

,

I

F

h

⊂ T

h

be su h that

card(I

h

) = d

anddenoteby

{α

F

T

}

T

∈I

F

h

the familyofrealssu hthat

x

b

F

=

P

T

∈I

F

h

α

F

T

x

T

.The bary entri interpolator is dened as follows:

fullled by the tra e interpolator for any fun tion

w

in a suitablespa e

V

:

∀T ∈ T

h

, ∀F ∈ F

h

T

,

|w(

x

b

F

) − I

F

(Π

V

T

h

(w))| ≤ C

w

h

2

F

,

(4) where

Π

V

Th

: V →

V

T

h

is the operator that maps every element

w ∈ V

onto

Π

V

Th

(w)

def

= {w(x

T

)}

T

∈T

h

. Forthe bary entri interpolator (3)we an take

V

def

= {w ∈ C

₀

2

(Ω) ; ν∇w ∈ H(div; Ω)}.

The global regularity assumptions in

V

are only needed to prove optimal onvergen e ratesandstronglydependonthe hoi eoftheinterpolator.When

ν

isheterogeneous,theL-interpolatorintrodu edin[1℄andanalyzedin[2℄may be preferable, as onlylo alregularityinside ea h

Ω

i

is required. Convergen e tominimalregularity solutionsthatare barely in

H

1

0

(Ω)

is alsoobtained,but noestimate of the onvergen e rate is available inthis ase.

The tra einterpolatoristhe keyingredient to onstru ta onsistent gradient

approximation. Forall

V

∈

V

T

h

, welet ( .f.[5℄)

G

_T

_(V)

def

₌

1

|T |

X

F

∈F

T

h

|F | (I

F

(V) − v

T

) n

T,F

.

For

k ≥ 0

, let

P

k

h

def

= {p

h

∈ L

2

(Ω) ; ∀T ∈ T

h

, p

h|T

∈

P

k

d

(T )}

. Consider the following approximation spa e:

V

h

def

=

n

v

h

∈ P

h

1

; ∃V ∈

V

T

h

_{, ∀T ∈ T}

h

,

v

h

(x

T

) = v

T

and

∇v

h|T

= G

T

(V)

o

.

For every

v

h

∈ V

h

, there holds

v

h|T

(x) = v

T

+ G

T

(V)·(x − x

T

)

for all

T ∈ T

h

and

V

the element of V

T

h

asso iated to

v

h

. The spa e

V

h

is thus a subspa e of

P

1

h

with pie ewise anetest fun tionswhose value at

x

T

ispres ribed and whose gradient depends on the values at neighbouring ell enters, on the

mesh and,possibly, ontheproblemdata viathe interpolator

I

.Wedenethe following norm onthe augmented spa e

V (h)

def

= V + V

h

: Forall

v ∈ V (h)

,

kvk

2

_h

def

=

X

T

∈T

h

k∇vk

2

_[L

2

_{(T )]}

d

+

X

F

∈F

h

1

h

F

kJvKk

2

_L

2

_{(F )}

,

where,for all

ϕ

su h that a(possibly two-valued) tra e isavailableon

F

,

JϕK

def

=







ϕ

|T

1

− ϕ

|T

2

forall

F ∈ F

T

1

h

∩ F

T

2

h

,

ϕ

|T

forall

F ∈ F

T

h

∩ F

h

b

.

The Poin aré inequality proved in [4℄ for the pie ewise broken ane spa e

Lemma 1 (Approximation estimate). For all

w ∈ V

, there exists

C

w

inde-pendent of the meshsize

h

su h that

inf

w

h

∈V

h

kw − w

h

k

h

≤ C

w

h.

(5)

Proof. Use (4)and pro eed asin [5,Theorem 4.2℄ toprove that

X

T

∈T

h

k∇w

h

− ∇wk

2

[L

2

_{(T )]}

d

≤ C

w

h

2

.

Forthe jumpterm,observe that,sin e

w

is ontinuousa ross meshinterfa es anditvanisheson

∂Ω

,

Jw

h

K = Jw

h

− wK

forall

x ∈ F

,

F ∈ F

h

.UsingAgmon's tra e inequality together with Taylorexpansion yieldsthe desiredresult.

ConsiderthefollowingbilinearforminspiredfromtheSIPGmethodofArnold:

Forall

(w, v

h

) ∈ V (h) × V

h

a

h

(w, v

h

)

def

=

Z

Ω

ν∇

h

w·∇

h

v

h

−

X

F∈F

h

Z

F

[{ν∇

h

w}· n

F

Jv

h

K + JwK{ν∇

h

v

h

}· n

F

]

+

X

F

∈F

h

η

h

F

Z

F

n

F

ν· n

F

JwKJv

h

K,

(6)

where

∇

h

denotes the broken gradient,

η > 0

is large enoughto ensure oer- ivity and

{ϕ}

def

=







1

2

(ϕ

|T

1

+ ϕ

|T

2

)

for all

F ∈ F

T

1

h

∩ F

T

2

h

,

ϕ

|T

for all

F ∈ F

T

h

∩ F

h

b

.

The implementation does not require ubatures. Indeed, using linearity, the

termsinsquarebra ketsin(6) anberewrittenas

P

F

∈F

h

|F |{ν∇

h

u

h

}· n

F

Jv

h

(

x

b

F

)K

and

P

F

∈F

h

|F |{ν∇

h

v

h

}· n

F

Ju

h

(

x

b

F

)K

respe tivelyand the penaltyterm an be safelyapproximatedusingthemiddlepointruleby

P

F

∈F

h

η|F |

h

F

Ju

h

(

x

b

F

)KJv

h

(

x

b

F

)K

.

As the dis rete bilinear form

a

h

has been naturally extended to

V (h) × V

h

, the usual dGanalysis te hniques an beapplied.

We onsider the following approximation of (2):

Find

u

h

∈ V

h

s.t.

a

h

(u

h

, v

h

) =

Z

Ω

f v

h

forall

v

h

∈ V

h

.

(7) Problem (7) has only one unknown per ell, as the elimination of fa e

un-knowns has been arried out in the interpolation pro ess. Su h an operation

an be performed in pra ti e when, for a given

V

∈

V

T

h

, the dependen e of

ea h

I

F

(V)

on

V

islimitedtoafew ellvalues(whi h isthe ase, e.g.,forthe bary entri interpolator). On the other hand, when ea h

I

F

(V)

potentially depends on all the values

{v

T

}

T

∈T

h

(as it is the ase for the interpolator ob-tained by expressing fa e unknowns in terms of ell unknowns in the hybrid

Numeri al example for a pure diusion problem on a triangular mesh family (

u

= sin(πx) sin(πy)

,

ν

=

1

d

,

f

= 2π

2

_{sin(πx) sin(πy)}

).

h

ku − u

h

k

L

2

(Ω)

order

ku − u

h

k

h

order 1 /4 2.8891e-02  1.7460e-01  1 /8 6.9933e-03 2.05 8.1249e-02 1.10 1 /16 1.7562e-03 1.99 3.9300e-02 1.05 1 /32 4.4048e-04 2.00 1.9467e-02 1.01 1 /64 1.1098e-04 1.99 9.6534e-03 1.01 1 /128 2.7795e-05 2.00 4.8156e-03 1.00 1 /256 6.9559e-06 2.00 2.4051e-03 1.00

3 Main results and omments

Throughoutthis se tion,

{u

h

}

h∈H

willdenotethe sequen e of solutionsofthe dis reteproblem(7)onthe admissiblemeshfamily

{T

h

}

h∈H

.Pro eedingasin Arnold et al. [3℄, one an prove the following result:

Theorem 2 (Convergen e to regular solutions). Let

u ∈ H

1

0

(Ω)

solve (2) and assume, moreover, that

u ∈ V

. Then, there exists

C

u

independent of the meshsize

h

su h that

ku − u

h

k

h

≤ C

u

h.

Furthermore,the Aubin-Nits he tri kyields

ku − u

h

k

L

2

(Ω)

≤ C

u

h

2

.The above

estimates are numeri ally veried in Table 1. For all

v

h

∈ V

h

and for all

F ∈ F

h

, dene the lifting operator solution of the following problem: Find

r

F

(Jv

h

K) ∈ [P

h

0

]

d

su h that

Z

Ω

r

F

(Jv

h

K)·τ

h

=

Z

F

Jv

h

K{τ

h

}· n

F

∀τ

h

∈ [P

0

h

]

d

_,

and set

R

h

(Jv

h

K)

def

=

P

F

∈F

h

r

F

(Jv

h

K)

. Then, pro eeding asin [4, Theorem 2.2℄ one an prove that, for all sequen es

{v

h

}

h∈H

bounded in the

k·k

h

norm, as

h → 0

, there is

v ∈ H

1

0

(Ω)

su h that

v

h

→ v

in

L

2

_(Ω)

and

∇

h

v

h

− R

h

(Jv

h

K)

weakly onvergesto

∇v

in

[L

2

_(Ω)]

d

.Followingtheguidelinesof[4,Theorem 3.1℄

one an then prove that

Theorem 3 (Convergen e to minimalregularity solutions). Let

u

solve (2). Then,as

h → 0

,(i)

u

h

→ u

in

L

2

_(Ω)

,(ii)

∇

h

u

h

→ ∇u

and

∇

h

u

h

− R

h

(Ju

h

K) → ∇u

in

[L

2

_(Ω)]

d

and (iii)

P

F

∈F

h

h

−1

F

kJu

h

Kk

2

L

2

_{(F )}

→ 0

.

Some omments are of order. The derivation of the method outlined in Ÿ2

remainsvalidtaking inspiration fromany alternativedGmethod forproblem

(1) ( .f. [3℄ for a review). Moreover, the ideas of Ÿ2 virtually extend to any

problem for whi h a dG method has been devised. As an example, for the

diusion-adve tion-rea tion problem

Numeri al example for a diusion-adve tion-rea tion problem on a triangular

mesh family (

k·k

ν,β,h

is the asso iated energy norm,

u

= sin(πx) sin(πy)

,

ν

=

1

d

,

β

= (2π, 0)

,

µ

= 0

,

f

= 2π

2

_{(sin(πx) + cos(πx)) sin(πy)}

).

h

ku − u

h

k

L

2

(Ω)

order

ku − u

h

k

ν,β,h

order 1 /4 3.0457e-02  1.7902e-01  1 /8 7.4629e-03 2.03 8.3564e-02 1.10 1 /16 1.8935e-03 1.98 4.0205e-02 1.06 1 /32 4.7634e-04 1.99 1.9815e-02 1.02 1 /64 1.1978e-04 1.99 9.7981e-03 1.02 1 /128 2.9997e-05 1.99 4.8796e-03 1.00 1 /256 7.4900e-06 2.00 2.4344e-03 1.00 with

β ∈ C

1

_(Ω)

and

µ ≥ 0

su h that

µ +

1

2

∇·β > 0

, the following bilinear formyieldsanoptimally onvergingapproximation( .f.Table2foranumeri al

example):

b

h

(w, v

h

)

def

= a

h

(w, v

h

) −

Z

Ω

wβ·∇

h

v

h

+

X

F

∈F

h

Z

F

w

↑

_Jv

h

K +

Z

Ω

µwv

h

,

where

w

↑ def

_{= (β· n}

F

){w} +

|β· n

₂

F

|

JwK

on every

F ∈ F

i

h

and as

w

↑ def

_{= (β· n)}

⊕

_w

on every

F ∈ F

b

h

. Finally, the ompa tness analysis of [4℄ alsoapply, so that valuable toolsfor the analysis of nonlinear problems are available.

Referen es

[1℄I. Aavatsmark,G.T.Eigestad,B.T.Mallison,andJ.M.Nordbotten, A ompa t

multipointuxapproximationmethodwithimproved robustness,Numer.Methods

PartialDierential Equations 24(2008), no.5,13291360.

[2℄L. Agélas, D. A. Di Pietro, and J. Droniou, The G method for heterogeneous

anisotropi diusion on

general meshes, M2ANMath.Model. Numer.Anal. (2009), To appear.Preprint

availableat http://hal.ar hives-ouvertes.fr/hal-0034273 9/fr.

[3℄D. N. Arnold, F. Brezzi, B. Co kburn, and D. Marini, Unied analysis of

dis ontinuous Galerkin methods for ellipti problems, SIAMJ. Numer.Anal. 39

(2002), no.5,17491779.

[4℄D. A. Di Pietro and A. Ern, Dis rete fun tional analysis tools for

dis ontinuous Galerkin methods with appli ation to the in ompressible

Navier-Stokes equations, Math. Comp. (2009), To appear. Preprint available at

http://hal.ar hives-ouvertes.fr/hal-002 78925 /fr/.

[5℄R. Eymard, T. Gallouët, and R. Herbin, Dis retization of heterogeneous and

anisotropi diusion problems on general non- onforming meshes. SUSHI:

a s heme using stabilization and hybrid interfa es, Preprint available at