Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions

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heterogeneous diffusion problems with low-regularity

solutions

Daniele Antonio Di Pietro, Alexandre Ern

To cite this version:

LOW-REGULARITY SOLUTIONS

DANIELEA.DIPIETRO

1

ANDALEXANDREERN

2

Abstra t. Westudy the onvergen e of theSymmetri Weighted Interior Penaltydis ontinuousGalerkinmethodforheterogeneousdiusionproblems withlow-regularitysolutionsonlybelongingto

W

2,p

with

p ∈ (1, 2]

. In2dwe inferanoptimalalgebrai onvergen erate. In3dwea hievethesameresult for

p >

6

_/

5

,andfor

p ∈ (1,

6

_/

5

]

weprove onvergen ewithoutalgebrai rate.

1. Introdu tion

In this work we analyzethe onvergen e of a dis ontinuous Galerkin (dG) ap-proximationtolow-regularitysolutions ofthemodelproblem

(1)

−∇·(κ∇u) = f

in

Ω,

u = 0

on

∂Ω,

where, for

d ∈ {2, 3}

,

Ω

denotes a bounded onne ted polyhedral domain with boundary

∂Ω

,

f ∈ L

2

_(Ω)

is the for ing term, and

κ ∈ L

∞

_(Ω)

is the diusion oe ient su h that

λ ≤ κ ≤ λ

a.e. in

Ω

for positive real numbers

λ

and

λ

. OwingtotheLaxMilgramLemma, thisproblemiswell-posedintheenergyspa e

V := H

1

0

(Ω)

.

Inpra ti e, thediusion oe ienthas more regularity thanjustbelonging to

L

∞

_(Ω)

. Inwhatfollows,weassumethat thereisa partition

P

Ω

:= {Ω

i

}

1≤i≤N

Ω

of

Ω

su hthat

(i) ea h

Ω

i

,

1 ≤ i ≤ N

Ω

,isanopenpolyhedron;

(ii) therestri tionof

κ

toea h

Ω

i

,

1 ≤ i ≤ N

Ω

,is onstant.

The regularity of the exa t solution for interfa e problems mat hing the above assumptionhasbeenstudiedbyNi aiseandSändig[13 ℄, whereitisproventhat (2) Thereexists

p ∈ (1, 2]

s.t.

u ∈ V

†

:= W

2,p

_(P

Ω

)

, where

W

2,p

_(P

Ω

)

denotes thebroken Sobolev spa e spanned bythose fun tions

v

su hthat

v|

Ω

i

∈ W

2,p

_(Ω

i

)

forall

1 ≤ i ≤ N

Ω

. However,uptodate,the onvergen e analysisofdGmethodsfortheinterfa eproblem(1)hasgenerallyhingedonamore stringentregularityassumptionontheexa tsolution,namely

u ∈ H

3

_/

2

+ǫ

(P

Ω

)

with

ǫ > 0

. Thegoalofthispaper istollthegapbyusingonlytheregularity(2 ). We fullya hievethisgoalin 2d, wherebywe deriveenergynorm errorestimateswith optimalalgebrai onvergen erates. Asimilarresulthasbeenestablishedre ently byWihler andRivière [18℄in the simpler aseof the Lapla eequation in 2d. As

in [18 ℄ our analysis hinges on dis rete stability, strong onsisten y and bounded-ness ofthe dis rete bilinear form, but handling theheterogeneityof the diusion oe ientrequiresspe ial aretoa hieverobustness. Theboundednesspropertyis alsoformulatedin a somewhatdierentway. In3dthesituation ismore deli ate. For

p ∈ (

6

_/

₅

_{, 2]}

we also deriveoptimal algebrai onvergen e rates forthe energy norm error. In this ase, owing to theSobolev embedding, the exa t solution is indeed in

H

1+α

_(P

Ω

)

with

α > 0

. For brevity, wetreatthe2d asewith

p ∈ (1, 2]

and the 3d ase with

p ∈ (

6

_/

₅

_{, 2]}

simultaneously; the analysis readily extends to

p ∈ (

2d

_/

_d+2

_{, 2]}

in any spa edimension. Finally, in 3dwith

p ∈ (1,

6

/

5

]

,wepresent forthesakeof ompletenessa onvergen eproofwithoutalgebrai rates. The anal-ysis, valid in any spa e dimension, follows the ompa tnessargument introdu ed in[5 ℄. Herein,we onsidershape-regularmeshes. Analternativeapproa hbasedon geometri allyrened meshes hasbeeninvestigated, e.g., byWihler, Frauenfelder, andS hwab[17 ℄.

The fo us is here on the Symmetri Weighted Interior Penalty (SWIP) dG method to approximate themodel problem(1) (a ountingfor variationsin sym-metry is straightforward). The SWIP method has been introdu ed in the more general ontext of diusion-adve tion-rea tion problems by Di Pietro, Ern, and Guermond[6 ℄andErn,Stephansen,andZunino[9 ℄. Forthemodelproblem(1 ),the dieren eswithrespe t tothe lassi al Symmetri InteriorPenalty (SIP) method ofArnold[2 ℄layintheuseofdiusion-dependent,weightedaveragetra eoperators and ofa penaltyparameter proportionalto theharmoni average ofthe diusion at interfa es. This allows one to infer energy norm error estimates with multi-pli ative onstant independentof diusion heterogeneity, whi h makesthe SWIP methodparti ularlysuitedtodiusion-adve tionproblemswithsharpinternal lay-ers. The possibility of using non-arithmeti averages in dG methods has been pointed out and used in various ontexts, e.g., by Stenberg [14 ℄ and by Heinri h and o-workers [11 ,10,12 ℄. Theideaof onne tingthea tualvalueoftheweights to the diusion oe ient was originally proposed by Burman and Zunino [4 ℄ in the ontext ofmortaringte hniquesfor asingularlyperturbeddiusion-adve tion equation.

The material is organized as follows. In Ÿ2 we present the dis rete setting. In Ÿ3 we derive algebrai onvergen e rates forexa t solutions in

W

2,p

_(P

Ω

)

with

p ∈ (

2d

_/

_d+2

_{, 2]}

. Finally, the onvergen e for theremaining ases is overed in Ÿ4. Numeri alresultshavealreadybeenpresentedin[8 ,15 ℄forthewell-known2d four- ornerproblem, wherebythe onvergen e ratesderivedhereinhavebeenobserved numeri ally.

2. The dis rete setting

2.1. Meshes and fa es. Let

(T

h

)

h∈H

bea sequen e ofrened simpli ial meshes overing

Ω

exa tly, where

H

denotes a ountable set havingzero as unique a u-mulationpoint. Meshes anpossesshangingnodes. Quiteimportantly,meshesare assumed to be ompatible withthe partition

P

Ω

, that is, su h that for all

h ∈ H

and all

T ∈ T

h

, there exists a unique

Ω

i

of the partition

P

Ω

su h that

T ⊂ Ω

i

. Sin e thediusion oe ientis pie ewise onstant onthe partition

P

Ω

, it is also pie ewise onstantonea h ompatible mesh.

For a mesh element

T ∈ T

h

,

h

T

denotes its diameter and

n

T

itsunit outward normal dened a.e. on

∂T

. The mesh-size is

h := max

T ∈T

T

Figure1. Theset

F

T

fortheelement

T

(shaded) ontainsinthis asethefourmesh fa eswithverti esinboldline

denitionsapplyforevery

h ∈ H

. Foreveryinteger

k ≥ 0

, weintrodu ethespa e P

k

d

(T

h

) :=

v

h

∈ L

2

(Ω) | ∀T ∈ T

h

, v

h

|

T

∈

P

k

d

(T ) ,

whereP

k

d

(T )

inspannedbytherestri tionto

T

ofpolynomialfun tionsin

d

variables of total degree

≤ k

. We say that a ( losed) subset

F

of

Ω

is a mesh fa e if

F

haspositive

(d − 1)

-dimensionalmeasure andifone ofthetwo followingmutually ex lusive onditionsissatised:

(i) Therearedistin tmeshelements

T

1

, T

2

∈ T

h

su hthat

F = ∂T

1

∩∂T

2

;insu h ase,

F

is alledaninterfa eandweset

n

F

:= n

T

1

,theunitnormalve torto

F

pointingfrom

T

1

to

T

2

(theorientationof

n

F

isarbitrarydependingonthe hoi eof

T

1

and

T

2

,butkeptxedin whatfollows);

(ii) Thereis

T ∈ T

h

su hthat

F = ∂T ∩ ∂Ω

;insu h ase,

F

is alled aboundary fa eand weset

n

F

:= n

, theoutwardunitnormalto

∂Ω

.

Interfa es are olle ted in the set

F

i

h

, boundary fa es in

F

b

h

, and mesh fa es in

F

h

:= F

h

i

∪ F

h

b

. Moreover,foreverymeshelement

T ∈ T

h

,theset

F

T

:= {F ∈ F

h

| F ⊂ ∂T }

ontainsthemeshfa es omposingtheboundaryof

T

. Asnonmat hingmeshesare allowed, the ardinal number of

F

T

an be larger than

(d + 1)

; see Figure 1. In what follows,weassumethat

(T

h

)

h∈H

is anadmissiblemesh sequen e,that is,

T

h

isshape-regularintheusualsenseand onta t-regularmeaningthatthereexists

C

independentofthemesh-size

h

su hthat,forall

T ∈ T

h

andall

F ∈ F

T

,

h

T

≤ Ch

F

, thediameterof

F

. Letting

(3)

N

∂

:=

max

h∈H, T ∈T

h

card(F

T

),

onta tregularityimpliesthat

N

∂

isbounded. 2.2. Jumps and weighted averages.

Denition2.1(Jumps). Let

v

beas alar-valuedfun tiondenedon

Ω

andassume that

v

issmoothenoughtoadmitonall

F ∈ F

h

a(possiblytwo-valued)tra e. Then, if

F ∈ F

i

h

with

F = ∂T

1

∩ ∂T

2

,the jump of

v

at

F

isdenedfora.e.

x ∈ F

as

JvK

F

(x) := v|

T

1

(x) − v|

T

2

(x),

whileif

F ∈ F

b

h

with

F = ∂T ∩ ∂Ω

,weset

JvK

F

(x) := v|

T

(x)

.

Denition2.2(Weightedaverages). Let

v

beas alar-valuedfun tiondenedon

Ω

andassumethat

v

issmoothenoughtoadmitonall

F ∈ F

h

a(possiblytwo-valued) tra e. To any interfa e

F ∈ F

i

h

with

F = ∂T

1

∩ ∂T

2

, we assigntwo non-negative realnumbers

ω

T

1

,F

and

ω

T

2

,F

su hthat

Then, theweightedaverage of

v

at

F ∈ F

i

h

isdenedfora.e.

x ∈ F

as

{v}

ω,F

(x) := ω

T

1

,F

v|

T

1

(x) + ω

T

2

,F

v|

T

2

(x).

whileon boundaryfa es

F ∈ F

b

h

with

F = ∂T ∩ ∂Ω

,weset

{v}

ω,F

(x) := v|

T

(x)

. Clearly,theusual(arithmeti )averageatinterfa es orrespondstotheparti ular hoi e

ω

T

1

,F

= ω

T

1

,F

=

1

2

. Hen eforth, we onsider a spe i diusion-dependent hoi efortheweights,namelyforall

F ∈ F

i

h

,

F = ∂T

1

∩ ∂T

2

,

ω

T

1

,F

:=

κ

2

κ

1

+ κ

2

,

ω

T

2

,F

:=

κ

1

κ

1

+ κ

2

,

where

κ

i

= κ|

T

i

,

i ∈ {1, 2}

. Inparti ular, the aseofhomogeneous diusionyields the usual (arithmeti )averages. When

v

is ve tor-valued, theabove average and jump operators a t omponentwise. Whenever no onfusion an arise, both the subs ript

F

andthevariable

x

areomitted.

2.3. Thedis reteproblem. Weaimatapproximatingtheexa tsolution

u

of(1 ) byadGmethod usingthedis retespa e

V

h

:=

P

k

d

(T

h

),

k ≥ 1.

Deneforall

(v

h

, w

h

) ∈ V

h

× V

h

,

a

h

(v

h

, w

h

) :=

Z

Ω

κ∇

h

v

h

·∇

h

w

h

+

X

F ∈F

h

η

γ

κ,F

h

F

Z

F

Jv

h

KJw

h

K

(4)

−

X

F ∈F

h

Z

F

{κ∇

h

v

h

}

ω

·n

F

Jw

h

K −

X

F ∈F

h

Z

F

Jv

h

K{κ∇

h

w

h

}

ω

·n

F

,

where

∇

h

denotes the usual broken gradient operator on

T

h

,

η > 0

is a user-dependentpenaltyparameter(tobe hosenlargeenoughtoensuredis retestability, seeLemma3.4 ),whilethediusion-dependentpenaltyparameter

γ

κ,F

issu hthat forall

F ∈ F

i

h

with

F = ∂T

1

∩ ∂T

2

,

γ

κ,F

:=

2κ

1

κ

2

κ

1

+ κ

2

,

where,as above,

κ

i

= κ|

T

i

,

i ∈ {1, 2}

,whileforall

F ∈ F

b

h

,

F = ∂T ∩ ∂Ω

,

γ

κ,F

:= κ|

T

.

Wenoti ethat theabove hoi e forthepenaltyparameter

γ

κ,F

oninterfa es or-responds to theharmoni mean of the two diusion oe ients oneither side of theinterfa e. Inwhat follows, thetermsin these ond lineof (4 )are respe tively referredtoas onsisten yandsymmetryterms,astheyservetheenfor ementofthe orresponding property at thedis rete level. Thebilinear form

a

h

dened by (4 ) is termed the Symmetri Weighted Interior Penalty (SWIP) bilinear form [6 , 9℄. Whenever

κ

is onstantin

Ω

,theusual (arithmeti )averagesarere overedin the onsisten yandsymmetryterms. Finally,thedis reteproblemis

(5) Find

u

h

∈ V

h

s.t.

a

h

(u

h

, v

h

) =

Z

Ω

2.4. Extension of the dis rete bilinear form. To assert onsisten y for the dis reteproblem (5 ) in theusual strongform, weneedto plug theexa t solution

u

into the rst argumentof the bilinear form

a

h

. This requires in turn to give a meaning to the normalgradient of

u

independently onea h mesh fa e. The fa t that

−∆u = f ∈ L

2

_{(T )}

for all

T ∈ T

h

is insu ient, as it only yields

∇u·n

T

∈

H

−

1

_/

2

(∂T )

. Theregularity(2)isthus ru ial,sin eowingtomesh ompatibility,it impliesforall

v ∈ V

†

,all

T ∈ T

h

,andall

F ∈ F

T

,

(6)

∇v·n

T

∈ L

p

_{(F ).}

Asa result,thedis retebilinearform

a

h

anbeextendedto

V

†h

× V

h

with

V

†h

:= V

†

+ V

h

,

and

V

†

denedby(2 ).

3. Convergen e analysis in2dand in3dfor

p ∈ (

6

_/

₅

_{, 2]}

Inthisse tionweproveoptimal onvergen eratesforthemethod(5 )in2dand in 3d for

p >

6

_/

₅

, that is,

p >

2d

_/

_d+2

. Owing to theSobolevembedding theorem, theregularity(2 )yields

(7)

u ∈ H

1+α

_(P

Ω

)

with

α := 1 + d

 1

2

−

1

p



> 0

.

The error analysis in this se tion pro eeds by establishing onsisten y, dis rete stability, andboundedness fortheSWIP bilinear form

a

h

. The errorismeasured inthefollowingenergynorm: For all

v ∈ V

†h

,

(8)

|||v|||

κ

:=



kκ

1

/

2

∇

h

vk

2

[L

2

_(Ω)]

d

+ |v|

2

_J,κ



1

/

2

,

withjumpseminorm

(9)

|v|

J,κ

:=

X

F ∈F

h

|v|

2

J,κ,F

!

1

/

2

,

|v|

J,κ,F

:=

 γ

κ,F

h

F



1

/

2

kJvKk

L

2

(F )

.

3.1. Te hni al results. This se tion olle ts some useful te hni al results. We re allthefollowinginverseandtra e inequalities(see, e.g.,[3 ,7℄): For all

y

h

∈ V

h

andall

F ∈ F

h

, (10)

ky

h

k

L

q

(F )

≤ C

q

h

(d−1)(

1

q

−

1

2

)

F

ky

h

k

L

2

(F )

,

andthefollowingtra einequality: Forall

y

h

∈ V

h

,all

T ∈ T

h

,andall

F ∈ F

T

,

(11)

h

1

_/

2

F

ky

h

k

L

2

_{(F )}

≤ C

_tr

ky

_h

k

_L

2

_{(T )}

.

The quantity

C

tr

only depends on

d

,

k

, and mesh regularity, while there holds

C

q

≤ max(1, C

∞

)

[16 ℄where

C

∞

onlydependson

d

,

k

,andmeshregularity. For a realnumber

r ∈ (1, +∞)

,weset

β

r

:=

1

2

+ (d − 1)

 1

2

−

1

r



,

andobservethat for

r = 2

,

β

2

=

1

2

. We onsider thefollowingseminorm

Inparti ular,for

r = 2

,

|v|

†,κ,2

=

X

T ∈T

h

X

F ∈F

T

h

F

kκ

1

_/

2

∇v|

T

·n

F

k

2

L

2

_{(F )}

!

1

/

2

.

Themainresultofthisse tionisaboundonthe onsisten yandsymmetryterms in theSWIP bilinearform

a

h

. Inwhat follows,weset

q :=

p

p−1

so that

1

p

+

1

q

= 1

and

q ∈ [2, +∞)

.

Lemma 3.1 (Boundon onsisten yandsymmetryterms). Thereholds: (i) Forall

(v

h

, w) ∈ V

h

× V

†h

,

(12)











X

F ∈F

h

Z

F

{κ∇

h

v

h

}

ω

·n

F

JwK











≤ |v

h

|

†,κ,2

|w|

J,κ

.

(ii) Forall

(v, w

h

) ∈ V

†h

× V

h

, (13)











X

F ∈F

h

Z

F

{κ∇

h

v}

ω

·n

F

Jw

h

K











≤ 2

1

2

−

1

q

C

q

|v|

†,κ,p

|w

h

|

J,κ

.

Proof. (i)Proofof(12 ). Let

(v

h

, w) ∈ V

h

×V

†h

. Forall

F ∈ F

i

h

with

F = ∂T

1

∩∂T

2

, set

ω

i

= ω

T

i

,F

,

κ

i

= κ|

T

i

, and

a

i

= κ

1

_/

2

i

(∇

h

v

h

)|

T

i

·n

F

,

i ∈ {1, 2}

. The Cau hy S hwarzinequalityyields

Z

F

{κ∇

h

v

h

}

ω

·n

F

JwK =

Z

F

(ω

1

κ

1

_/

2

1

a

1

+ ω

2

κ

1

_/

2

2

a

2

)JwK

≤

 1

2

h

F

(ka

1

k

2

L

2

(F )

+ ka

2

k

2

L

2

(F )

)



1

/

2

×



2(ω

2

1

κ

1

+ ω

2

2

κ

2

)h

−1

F

kJwKk

2

L

2

_{(F )}



1

/

2

,

andsin e

2(ω

2

1

κ

1

+ ω

2

2

κ

2

) = γ

κ,F

,itisinferredthat

Z

F

{κ∇

h

v

h

}

ω

·n

F

JwK ≤

 1

2

h

F

(ka

1

k

2

L

2

(F )

+ ka

2

k

2

L

2

(F )

)



1

/

2

|w|

J,κ,F

.

Moreover,forall

F ∈ F

b

h

with

F = ∂T ∩ ∂Ω

and

a = (κ

1

_/

2

∇

h

v

h

)|

T

·n

F

,

Z

F

{κ∇

h

v

h

}

ω

·n

F

JwK ≤ h

1

_/

2

F

kak

L

2

_{(F )}

|w|

_J,κ,F

.

Summingoverthemeshfa es,usingtheCau hyS hwarzinequality,andregrouping thefa e ontributionsofea h meshelementyields(12 ).

(ii) Proof of (13 ). Let

(v, w

h

) ∈ V

†h

× V

h

. For all

F ∈ F

i

h

, letting now

a

i

=

κ

1

/

2

i

(∇

h

v)|

T

i

·n

F

,

i ∈ {1, 2}

,Hölder'sinequalityyields

Z

F

{κ∇

h

v}

ω

·n

F

Jw

h

K ≤

 1

2

h

pβ

p

F

(ka

1

k

p

_L

p

_{(F )}

+ ka

2

k

p

_L

p

_{(F )}

)



1

/

p

× 2

1

/

p



(ω

1

q

κ

q

/

2

1

+ ω

q

2

κ

q

/

2

2

)h

−qβ

p

F

kJw

h

Kk

q

_L

q

_{(F )}



1

/

q

.

Weobservethatsin e

q ≥ 2

,

Moreover,owingtotheinverseinequality(10 )andsin e

1

p

+

1

q

= 1

,

h

−β

p

F

kJw

h

Kk

L

q

_{(F )}

≤ C

_q

h

−β

p

F

h

(d−1)(

1

q

−

1

2

)

F

kJw

h

Kk

L

2

_{(F )}

= C

_q

h

−

1

_/

2

F

kJw

h

Kk

L

2

_{(F )}

.

Hen e,sin e

2

1

p

−

1

2

= 2

1

2

−

1

q

,

Z

F

{κ∇

h

v}

ω

·n

F

Jw

h

K ≤

 1

2

h

pβ

p

F

(ka

1

k

p

L

p

_{(F )}

+ ka

2

k

p

L

p

_{(F )}

)



1

/

p

× 2

1

2

−

1

q

C

q

|w

h

|

J,κ,F

.

Moreover,forall

F ∈ F

b

h

,pro eedingas abovewith

a = (κ

1

_/

2

∇

h

v)|

T

·n

F

yields

Z

F

{κ∇

h

v}

ω

·n

F

Jw

h

K ≤



h

pβ

p

F

kak

p

L

p

_{(F )}



1

/

p

× C

q

|w

h

|

J,κ,F

.

Summingover meshfa es,applyingonelasttimeHölder'sinequality,and regroup-ing the fa e ontributions of ea h mesh element (sin e

1 ≤ 2

1

2

−

1

q

for boundary fa es),weinfer











X

F ∈F

h

Z

F

{κ∇

h

v}

ω

·n

F

Jw

h

K











≤ |v|

†,κ,p

× 2

1

2

−

1

q

C

q

X

F ∈F

h

|w

h

|

q

J,κ,F

!

1

/

q

,

andsin e

q ≥ 2

,weobtain

X

F ∈F

h

|w

h

|

q

J,κ,F

!

1

/

q

≤

X

F ∈F

h

|w

h

|

2

J,κ,F

!

1

/

2

= |w

h

|

J,κ

,

therebyyielding(13 ).



3.2. Consisten y.

Lemma 3.2 (Jumpsofexa tsolution). The exa tsolution

u

issu hthat

JuK = 0

∀F ∈ F

h

,

(14)

Jκ∇uK·n

F

= 0

∀F ∈ F

h

i

.

(15)

Proof. Property (14 ) is lassi al for fun tions in

H

1

0

(Ω)

. To prove (15 ), let

ϕ ∈

C

∞

0

(Ω)

. Sin e

−∇·(κ∇u) = f ∈ L

2

_(Ω)

,

Z

Ω

(−∇·(κ∇u))ϕ =

Z

Ω

κ∇u·∇ϕ.

Furthermore,weobtainusingtheGreentheoremand(6 ), forall

T ∈ T

h

,

Z

T

(−∇·(κ∇u))ϕ =

Z

T

κ∇u·∇ϕ −

Z

∂T

(κ∇u·n

T

)ϕ.

Summing over mesh elementsand a ountingfor the fa t that

ϕ

vanishes on

∂Ω

yields

X

F ∈F

i

h

Z

F

(Jκ∇uK·n

F

)ϕ = 0,

when e the assertion is inferred by hoosing the support of

ϕ

overing a single interfa eandusingadensityargument.



Lemma 3.3 (Consisten y). Forall

w

h

∈ V

h

,

a

h

(u, w

h

) =

Z

Ω

Proof. Plug

u

into therst argument of the bilinear form

a

h

given by(4 ). Inte-grating byparts thersttermyields

(16)

Z

Ω

κ∇u·∇

h

w

h

= −

X

T ∈T

h

Z

T

∇·(κ∇u)w

h

+

X

T ∈T

h

Z

∂T

κ(∇u·n

T

)w

h

.

Rewritingthese ondtermontheright-handsideoftheaboveexpressionasasum overmeshfa esleads to

X

T ∈T

h

Z

∂T

κ(∇u·n

T

)w

h

=

X

F ∈F

i

h

Z

F

J(κ∇u)w

h

K·n

F

+

X

F ∈F

b

h

Z

F

κ(∇u·n)w

h

.

Wenowobservethatforall

F ∈ F

i

h

,

J(κ∇u)w

h

K = {κ∇u}

ω

Jw

h

K + Jκ∇uK{w

h

}

ω

,

where

{w

h

}

ω

:= ω

T

₂

,F

w

h

|

T

₁

+ ω

T

₁

,F

w

h

|

T

₂

. To prove this identity, we set

a

i

=

(κ∇u)|

T

i

,

b

i

= w

h

|

T

i

,

ω

i

= ω

T

i

,F

,

i ∈ {1, 2}

, sothat

J(κ∇u)w

h

K = a

1

b

1

− a

2

b

2

= (ω

1

a

1

+ ω

2

a

2

)(b

1

− b

2

) + (a

1

− a

2

)(ω

2

b

1

+ ω

1

b

2

)

= {κ∇u}

ω

Jw

h

K + Jκ∇uK{w

h

}

ω

,

sin e

ω

1

+ ω

2

= 1

. Asa result,a ountingforboundaryfa es,

X

T ∈T

h

Z

∂T

κ(∇u·n

T

)w

h

=

X

F ∈F

h

Z

F

{κ∇u}

ω

·n

F

Jw

h

K +

X

F ∈F

i

h

Z

F

Jκ∇uK·n

F

{w

h

}

ω

.

Combiningthisexpressionwith(4 )and(16 )yields

a

h

(u, w

h

) = −

X

T ∈T

h

Z

T

∇·(κ∇u)w

h

+

X

F ∈F

h

η

γ

κ,F

h

F

Z

F

JuKJw

h

K

+

X

F ∈F

i

h

Z

F

Jκ∇uK·n

F

{w

h

}

ω

−

X

F ∈F

h

Z

F

JuK{κ∇

h

w

h

}

ω

·n

F

.

Thisyieldstheassertionowingto(14 )(15)andto

−∇·(κ∇u) = f

in

Ω

.



3.3. Stability. Wenowestablishthedis rete oer ivityoftheSWIPbilinearform undertheusualassumptionthat thepenaltyparameter

η

islargeenough. An im-portantpointisthattheminimalthresholdonthepenaltyparameterisindependent ofthediusion oe ient.

Lemma 3.4 (Dis rete oer ivity). Forall

η > C

2

tr

N

∂

,the SWIPbilinearform

a

h

is oer ive on

V

h

with respe ttothe

|||·|||

κ

-norm, i.e.,

∀v

h

∈ V

h

,

a

h

(v

h

, v

h

) ≥ C

sta

|||v

h

|||

2

κ

,

with

C

sta

:= (η − C

2

tr

N

∂

){max(

1

/

2

, η + C

_tr

2

N

∂

)}

−1

. Proof. Let

v

h

∈ V

h

. Werstobservethat

a

h

(v

h

, v

h

) = kκ

1

_/

2

∇

h

v

h

k

2

_[L

2

(Ω)]

d

− 2

X

F ∈F

h

Z

F

{κ∇

h

v

h

}

ω

·n

F

Jv

h

K + η|v

h

|

2

J,κ

,

andboundthese ondtermontheright-handsideusing(12 )to obtain

a

h

(v

h

, v

h

) ≥ kκ

1

_/

2

Owingto thedis retetra e inequality(11 ),wereadilyinfer (17)

|v

h

|

†,κ,2

≤ C

tr

N

1

_/

2

∂

kκ

1

_/

2

∇

h

v

h

k

[L

2

_(Ω)]

d

.

Usingtheinequality

2ab ≤ ǫa

2

_{+ (1/ǫ)b}

2

valid forany

ǫ > 0

yields

a

h

(v

h

, v

h

) ≥ 1 − C

tr

2

N

∂

ǫ
kκ

1

_/

2

∇

h

v

h

k

2

[L

2

(Ω)]

d

+ (η − 1/ǫ) |v

h

|

2

J,κ

.

Itnowsu estotake

ǫ = 2(η + C

2

tr

N

∂

)

−1

toinfertheassertion.



Asastraightforward onsequen eoftheLaxMilgramLemma,Lemma3.4yields thewell-posednessofthedis reteproblem (5 ).

3.4. Boundedness. We onsiderthefollowingadditionalnorm: For all

v ∈ V

†h

,

|||v|||

κ,

†

:= |||v|||

κ

+ |v|

†,κ,p

.

Lemma 3.5 (Boundedness). Thereholds

∀(v, w

h

) ∈ V

†h

× V

h

,

a

h

(v, w

h

) ≤ C

bnd

|||v|||

κ,

†

|||w

h

|||

κ

.

with

C

bnd

= 1 + η + 2

1

2

−

1

q

C

q

+ C

tr

N

1

_/

2

∂

Proof. Let

(v, w

h

) ∈ V

†h

× V

h

and denote by

T

1

, . . . , T

4

the four terms on the right-hand sideof(4 ). UsingtheCau hyS hwarzinequalityyields

|T

1

+ T

2

| ≤ (1 + η)|||v|||

κ

|||w

h

|||

κ

≤ (1 + η)|||v|||

κ,

†

|||w

h

|||

κ

.

Moreover,owingtothebound (13 ),

|T

3

| ≤ 2

1

2

−

1

q

C

q

|v|

†,κ,p

|w

h

|

J,κ

≤ 2

1

2

−

1

q

C

q

|||v|||

κ,

†

|||w

h

|||

κ

.

Finally,usingthebounds (12 )and(17 )leadsto

|T

4

| ≤ |v|

J,κ

|w

h

|

†,κ,2

≤ C

tr

N

1

_/

2

∂

|v|

J,κ

kκ

1

_/

2

∇

h

w

h

k

[L

2

_(Ω)]

d

≤ C

tr

N

1

_/

2

∂

|||v|||

κ

|||w

h

|||

κ

.

Colle tingtheabovebounds yieldstheassertion.



3.5. Convergen e.

Theorem 3.6 (

|||·|||

κ

-normerrorestimate). Assume

η > C

2

tr

N

∂

. There holds (18)

|||u − u

h

|||

κ

≤ C inf

y

h

∈V

h

|||u − y

h

|||

κ,

†

,

with

C = 1 + C

−1

sta

C

bnd

. Moreover, re allingthedenition (7) of

α

,

(19)

|||u − u

h

|||

κ

.

X

T ∈T

h

kκk

p

2

L

∞

_{(T )}

h

pα

T

kuk

p

W

2,p

_{(T )}

!

1

/

p

,

yielding,inparti ular,

|||u − u

h

|||

κ

.

λ

1

_/

2

h

α

_kuk

W

2,p

_(T

h

)

.

Proof. (i)Proofof (18 ). Let

y

h

∈ V

h

. Owing todis retestabilityand onsisten y,

|||u

h

− y

h

|||

κ

≤ C

sta

−1

sup

w

h

∈V

h

\{0}

a

h

(u

h

− y

h

, w

h

)

|||w

h

|||

κ

= C

sta

−1

sup

w

h

∈V

h

\{0}

a

h

(u − y

h

, w

h

)

|||w

h

|||

κ

.

Hen e,owingto boundedness,

|||u

h

− y

h

|||

κ

≤ C

sta

−1

C

bnd

|||u − y

h

|||

κ,

†

.

Estimate (18 )thenresultsfrom thetriangleinequality, thefa t that

|||u − y

h

|||

κ

≤

(ii)Toprove(19 ),weuse(18 )with

y

h

= π

h

u

where

π

h

denotes the

L

2

-orthogonal proje tiononto

V

h

. Forall

T ∈ T

h

,usingtheSobolevembedding

W

1,p

_{(T ) ֒→ L}

2

_{(T )}

sin e

p >

2d

d+2

togetherwith interpolationproperties in

W

2,p

_{(T )}

,it an beshown that

h

−1

T

ku − y

h

k

L

2

(T )

+ k∇

h

(u − y

h

)k

[L

2

_{(T )]}

d

.

h

1+d(

1

2

−

1

p

)

T

kuk

W

2,p

_{(T )}

.

Hen e, sin e

γ

κ,F

≤ min(κ|

T

1

, κ|

T

2

)

for all

F ∈ F

i

h

with

F = ∂T

1

∩ ∂T

2

,

Jy

h

K =

Ju − y

h

K

onall

F ∈ F

h

, and

kvk

L

2

(F )

.

kvk

1

_/

2

L

2

(T )

kvk

1

_/

2

H

1

(T )

forall

T ∈ T

h

,

F ∈ F

T

, and

v ∈ H

1

_{(T )}

,weinfer

|||u − y

h

|||

κ

.

X

T ∈T

h

kκk

L

∞

_{(T )}

h

2+d(1−

2

p

)

T

kuk

2

W

2,p

_{(T )}

!

1

/

2

≤

X

T ∈T

h

kκk

p

2

L

∞

_{(T )}

h

p

_[

1+d(

1

2

−

1

p

)

]

T

kuk

p

W

2,p

_{(T )}

!

1

/

p

,

sin e for non-negative real numbers

(a

T

)

T ∈T

h

,

(

P

T ∈T

h

a

2

T

)

1

_/

2

≤ (

P

T ∈T

h

a

p

T

)

1

_/

_p

. Moreover, sin e

kvk

L

p

(F )

.

kvk

1−

1

p

L

p

_{(T )}

kvk

1

p

W

1,p

_{(T )}

for all

T ∈ T

h

,

F ∈ F

T

, and

v ∈ W

1,p

_{(T )}

,weinfer usinginterpolationpropertiesin

W

2,p

_{(T )}

that

k∇(u − y

h

)|

T

·n

F

k

L

p

_{(F )}

.

h

1−

1

p

T

kuk

W

2,p

_{(T )}

,

when e,using again

(a

T

)

T ∈T

h

,

(

P

T ∈T

h

a

2

T

)

1

_/

2

≤ (

P

T ∈T

h

a

p

T

)

1

_/

_p

,

|u − y

h

|

†,κ,2

.

X

T ∈T

h

kκk

p

2

L

∞

_{(T )}

h

p

_[

1+d(

1

2

−

1

p

)

]

T

kuk

p

W

2,p

_{(T )}

!

1

/

p

.

Theproofis ompletesin e

α = 1 + d(

1

2

−

1

p

)

.



4. Convergen e analysis in3dfor

p ∈ (1,

6

_/

₅

_]

Wetreathere the3d asewith

p ∈ (1,

6

_/

₅

_]

. Inthis ase,theregularity(2 )is in-su ienttoestablish onvergen eratesbypro eedingasinthepreviousse tion. To prove onvergen estillusing admissiblemeshsequen es(andin parti ular, shape-regularmeshes),we onsiderhereadierentanalysiste hnique,inspiredby[5℄and relyingona ompa tnessargument. Inthis ase,aweakerformof onsisten yis in-voked,whi hdoesnotrequiretoextendthedis retebilinearformtothe ontinuous spa e,therebymakingthespa es

V

†

and

V

†h

unne essary.

4.1. Liftinganddis retegradients. Animportantingredientoftheanalysisisa dis retegradientfeaturingsuitable onvergen epropertiesforsequen esofsmooth andofdis retefun tions. Thedis retegradientisdenedintermsoftheweighted liftingoperatorsintrodu ed byDiPietro,Ern, andGuermond[6℄;seealsoAgélas, DiPietro, Eymard, and Masson[1℄. Morepre isely, for anyinteger

l ≥ 0

andall

F ∈ F

h

, wedene the linearoperator

r

l

ω,F

: L

2

(F ) → [

P

l

d

(T

h

)]

d

su h that, forall

Wealsodenethefollowinggloballifting: (21)

R

l

ω,h

(v) :=

X

F ∈F

h

r

l

ω,F

(v).

The

L

2

-normofthegloballifting anbeboundedin termsofthejumpseminorm. Pro eedingasin theproofof(12 )yieldsthat forall

l ≥ 0

andall

v ∈ V

†

,

(22)

kκ

1

_/

2

R

l

ω,h

(JvK)k

[L

2

_(Ω)]

d

≤ C

tr

N

1

_/

2

∂

|v|

J,κ

.

For all

l ≥ 0

andall

v

h

∈ V

h

,wedenethedis retegradient

(23)

G

l

ω,h

(v

h

) := ∇

h

v

h

− R

l

ω,h

(Jv

h

K).

Forfutureuse,wealsointrodu ethefollowingdata-independentnorms,

|||v||| :=



k∇

h

vk

2

[L

2

_(Ω)]

d

+ |v|

2

J



1

/

2

,

|v|

J

:=

X

F ∈F

h

h

−1

_F

kJvKk

2

_L

2

(F )

!

1

/

2

.

For every integer

l ≥ 0

, wealso denote by

π

l

h

the

L

2

-orthogonal proje tiononto P

l

d

(T

h

)

;thesamenotationisusedforthe

L

2

-orthogonal omponentwiseproje tion onto

[

P

l

d

(T

h

)]

d

.

Lemma4.1(Dis reteRelli hKondra hov). Let

(v

h

)

h∈H

beasequen ein

(V

h

)

h∈H

, uniformlyboundedin the

|||·|||

-norm. Then, thereexistsafun tion

v ∈ H

1

0

(Ω)

su h that as

h → 0

,uptoasubsequen e,

v

h

→ v

stronglyin

L

2

_(Ω)

.

Proof. See[5 ,Theorem6.3℄.



Lemma 4.2 (Properties of

G

l

ω,h

). The dis rete gradients

G

l

ω,h

,

l ≥ 0

, enjoy the followingproperties:

(i) For all sequen es

(v

h

)

h∈H

in

(V

h

)

h∈H

uniformly bounded in the

|||·|||

-norm, as

h → 0

,

G

l

ω,h

(v

h

) ⇀ ∇v

weakly in

[L

2

_(Ω)]

d

with

v ∈ H

1

0

(Ω)

provided by Theorem 4.1; (ii) For all

ϕ ∈ C

∞

0

(Ω)

,as

h → 0

,

G

l

ω,h

(π

h

1

ϕ) → ∇ϕ

stronglyin

[L

2

_(Ω)]

d

. Proof. (i)Toprovetheweak onvergen eof

G

l

ω,h

(v

h

)

to

∇v

,let

Φ ∈ [C

∞

0

(Ω)]

d

,set

Φ

h

:= π

h

l

Φ

andobservethat

Z

Ω

G

l

ω,h

(v

h

)·Φ = −

Z

Ω

v

h

∇·Φ +

X

T ∈T

h

Z

∂T

v

h

Φ·n

T

−

X

F ∈F

h

Z

Ω

r

l

ω,F

(Jv

h

K)·Φ

h

= −

Z

Ω

v

h

∇·Φ +

X

F ∈F

h

Z

F

Jv

h

K{Φ − Φ

h

}

ω

·n

F

= T

1

+ T

2

,

wherewehaveusedthedenitionofthe

L

2

-orthogonalproje tiontogetherwith(20 ) and(21 ). As

h → 0

,

T

1

→ −

R

Ω

v∇·Φ

. Forthese ond term,theCau hyS hwarz inequalityyields

T

₂

_{≤ |v}

_h

_|

_J,κ

_×

X

F ∈F

h

h

F

γ

−1

_F

Z

F

|{Φ − Φ

h

}

ω

|

2

!

1

/

2

,

whi h tends to zero owing to the approximation properties of the

L

2

-orthogonal proje tion togetherwith thefa t that

|v

h

|

J,κ

≤ λ

1

_/

2

(ii) Let

ϕ

h

:= π

1

h

ϕ

. Then,

G

l

ω,h

(ϕ

h

) = ∇

h

ϕ

h

− R

ω,h

l

(Jϕ

h

K) = T

1

+ T

2

. Clearly,

T

₁

_{→ ∇ϕ}

stronglyin

[L

2

_(Ω)]

d

as

h → 0

. Moreover,owingto(22 ),itisinferredthat

kR

l

ω,h

(Jϕ

h

K)k

[L

2

_(Ω)]

d

≤ C

_tr

N

1

_/

2

∂

|ϕ

h

|

J,κ

= C

tr

N

1

_/

2

∂

|ϕ

h

− ϕ|

J,κ

,whi htendstozeroas

h → 0

,thereby on ludingtheproof.



4.2. Convergen e. The SWIP bilinear form

a

h

admits the following equivalent formulationon

V

h

× V

h

: For

l ∈ {k − 1, k}

, (24)

a

h

(v

h

, w

h

) =

Z

Ω

κG

l

ω,h

(v

h

)·G

l

ω,h

(w

h

) + j

h

(v

h

, w

h

),

with

j

h

(v

h

, w

h

) := −

R

Ω

κR

l

ω,h

(Jv

h

K)·R

l

ω,h

(Jw

h

K) +

P

F ∈F

h

ηγ

κ,F

h

−1

F

R

F

Jv

h

KJw

h

K

. We annowstateandprovethemain resultofthisse tion.

Theorem 4.3 (Convergen e tominimal regularitysolutions). Let

(u

h

)

h∈H

be the sequen e of approximate solutions generated by solving the dis rete problems (5 ). Then, as

h → 0

, (i)

u

h

→ u

strongly in

L

2

_(Ω)

, (ii)

∇

h

u

h

→ ∇u

strongly in

[L

2

_(Ω)]

d

,(iii)

|u

h

|

J

→ 0

,with

u ∈ V

uniquesolutionto(1 ).

Proof. (i)Aprioriestimate. Were allthedis retePoin aréinequality[5 ,eq. (75)℄, (25)

∀v

h

∈ V

h

,

kv

h

k

L

2

(Ω)

≤ σ

2

|||v

h

|||,

with

σ

2

independent of the mesh-size

h

. Owing to the oer ivity of

a

h

together with(25 ),itisinferredthat

C

sta

λ|||u

h

|||

2

≤ C

sta

|||u

h

|||

2

κ

≤ a(u

h

, u

h

) =

Z

Ω

f u

h

≤ kf k

L

2

(Ω)

ku

h

k

L

2

(Ω)

≤ σ

2

kf k

L

2

(Ω)

|||u

h

|||,

hen e

|||u

h

||| ≤ σ

2

(C

sta

λ)

−1

_{kf k}

L

2

_(Ω)

,thatistosay,thesequen eofdis retesolutions isuniformlyboundedin the

|||·|||

-norm.

(ii) Compa tness. Owing to Theorem 4.1 together with Lemma 4.2i, there exists

u ∈ H

1

0

(Ω)

su h that, as

h → 0

, up to a subsequen e,

u

h

→ u

stronglyin

L

2

_(Ω)

and

G

l

ω,h

(u

h

) ⇀ ∇u

weaklyin

[L

2

_(Ω)]

d

. (iii) Identi ation of the limit. Let

ϕ ∈ C

∞

0

(Ω)

and set

ϕ

h

:= π

1

h

ϕ

. Owing tothe regularityof

ϕ

,itis learthat

|||ϕ − ϕ

h

|||

κ

→ 0

as

h → 0

. Observethat

a

h

(u

h

, ϕ

h

) =

Z

Ω

κG

l

ω,h

(u

h

)·G

l

ω,h

(ϕ

h

) + j

h

(u

h

, ϕ

h

) = T

1

+ T

2

.

As

h → 0

,

T

1

→

R

Ω

κ∇u·∇ϕ

owingtotheweak onvergen eof

G

l

ω,h

(u

h

)

to

∇u

and tothestrong onvergen eof

G

l

ω,h

(ϕ

h

)

to

∇ϕ

provedin Lemma 4.2 . Furthermore, theCau hyS hwarzinequalitytogetherwith(22 )yield

|T

2

| = |j

h

(u

h

, ϕ

h

)| ≤ C

tr

2

N

∂

+ η
|u

h

|

J,κ

|ϕ

h

|

J,κ

≤ C

tr

2

N

∂

+ η
λ

1

_/

2

|u

h

|

J

|ϕ

h

|

J,κ

Sin e

|u

h

|

J

isboundedbypoint(i),andsin e

|ϕ

h

|

J,κ

= |ϕ

h

− ϕ|

J,κ

tendstozeroas

h → 0

,itisinferred that

T

2

→ 0

. Asa result,

(26)

Z

Ω

κ∇u·∇ϕ ← a

h

(u

h

, ϕ

h

) =

Z

Ω

f ϕ

h

→

Z

Ω

f ϕ.

Hen e,bythedensityof

C

∞

0

(Ω)

in

H

1

(iv)Strong onvergen e ofthe gradient andof thejumps. Eqs.(22 )and(24 )yield (27)

∀v

h

∈ V

h

,

a

h

(v

h

, v

h

) ≥ kκ

1

_/

2

G

l

ω,h

(v

h

)k

2

[L

2

_(Ω)]

d

+ η − C

tr

2

N

∂

|v

h

|

2

J,κ

.

Fromtheweak onvergen eof

G

l

ω,h

(u

h

)

to

∇u

, we readily infer theweak onver-gen eof

κ

1

_/

2

G

l

ω,h

(u

h

)

to

κ

1

_/

2

∇u

. Then,owingto(27 )andtoweak onvergen e,

lim inf

h→0

a

h

(u

h

, u

h

) ≥ lim inf

h→0

kκ

1

_/

2

G

l

ω,h

(u

h

)k

2

_[L

2

(Ω)]

d

≥ kκ

1

_/

2

∇uk

2

_[L

2

(Ω)]

d

.

Furthermore,stillowingto(27 ),

lim sup

h→0

kκ

1

/

2

G

l

ω,h

(u

h

)k

2

[L

2

_(Ω)]

d

≤ lim sup

h→0

a

h

(u

h

, u

h

)

= lim sup

h→0

Z

Ω

f u

h

=

Z

Ω

f u = kκ

1

_/

2

∇uk

2

_[L

2

(Ω)]

d

.

This lassi allyprovesthestrong onvergen eof

κ

1

_/

2

G

l

ω,h

(u

h

)

to

κ

1

_/

2

∇u

in

[L

2

_(Ω)]

d

and, hen e, the strong onvergen e of

G

l

ω,h

(u

h

)

to

∇u

in

[L

2

_(Ω)]

d

. Note that

a

h

(u

h

, u

h

) → kκ

1

_/

2

∇uk

2

[L

2

_(Ω)]

d

also. Using(27 )wetheninfer

(η − C

2

tr

N

∂

)|u

h

|

2

J,κ

≤ a

h

(u

h

, u

h

) − kκ

1

_/

2

G

l

ω,h

(u

h

)k

2

[L

2

_(Ω)]

d

,

and, sin e

η > C

2

tr

N

∂

andtheright-hand sidetends tozero,

|u

h

|

J,κ

→ 0

. Toinfer that

|u

h

|

J

→ 0

,simplyobservethat

|u

h

|

J

≤ λ

−

1

_/

2

|u

h

|

J,κ

.



Remark 4.4. When extended to

V

†h

× V

h

, the dis rete bilinear form

a

h

dened by(24 )isnolonger onsistentintheusualniteelementsense;see[5 ,Remark3.3℄. However,(26 )showsthat

a

h

retainsaform ofweakasymptoti onsisten ywhi h su esto infer the onvergen e of themethod when

u

only exhibitsthe minimal regularity.

A knowledgment. TheauthorsaregratefultoS.Ni aise(Universitéde Valen i-ennes)forstimulatingdis ussions. ThisworkwaspartiallysupportedbytheGNR MoMaS(PACEN/CNRS,ANDRA,BRGM,CEA,EdF, IRSN,Fran e).

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2

IFP Energies nouvelles, 1 & 4 avenue de bois Préau, 92852 Rueil-Malmaison Cedex,Fran e

E-mailaddress:dipietrdifpenergiesnouvelles.fr

1

UniversitéParis-Est,CERMICS,E oledesPonts,77455MarnelaValléeCedex 2,Fran e