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Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Elements nis mixtes de Raviart Thomas
de type Petrov-Galerkin
Francois Dubois
Isabelle Gre et Charles Pierre (UPPA, Pau)
Conference in honor of Abderrahmane Bendali
Universite de Pau et des Pays de l'Adour
mardi 12 decembre 2017CNAM Paris et Universite Paris-Sud, Orsay.
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Freres et surs en mathematiques...
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus These d'Etat d'Abderrahmane a l'X, 10 janvier 1984
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Je me souviens...
P.-A. Raviart (le jour-m^eme) :
\tu n'as m^eme pas mis de cravatte !"
J.-C. Nedelec (en prive, plus tard) :
\dans les articles, il n'y a pas l'information... tout est dans la these de Bendali !"
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Une reference indispensable pour les champs de vecteurs... JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 107, 537-560 (1985)
A Variational Approach for the
Vector Potential Formulation of the Stokes
and Navier-Stokes Problems in
Three Dimensional Domains
A. BENDALI
,bole Polyrechnique. Cenlre de MarhPmaiiques Appliqukes, 91128 Palaiseau. France
J. M. DOMINGUEZ
Lahoratoire d'Anal.vse Nurrkrique, ChicersilP Pierre & Marie Curie.
4, place Jussieu. 7.5230 Paris Cedes OC? France
AND
S. GALLIC
C.E. Limeil, C.E.A., Set-rice MA., E.M.. B.P. 27.
94190 Villeneuce. Si. Georges, France
A few well-posed variational problems are constructed, whose solutions are vec- tor potentials of the velocity Field of the three-dimensional Stokes problem. A choice of the adequate boundary conditions to be imposed is performed in a systematic manner even in the cast of not simply connected domains. The test functions do not have to be divergence free in the formulations given here. It is then seen that a similar approach is well suited to the treatment of the non-linear
Navier-Stokes problem.
( 1985 Academic Press. Inc
I. INTRODUCTION
When solving boundary-value problems (arising from hydrodynamics or electromagnetism, for instance), one usually has to work with divergence- free functions. Unfortunately a divergence-free condition is delicate to implement numerically. When the problem is two dimensional, it is possible to overcome this difficulty through the introduction of a stream function cp satisfying: V x cp = u, where u is the divergence-free field. Then after imposing a suitable boundary condition on q, the problem reduces to 537
0022.247)(/85 $3.00
109 10: 1.12
Copyright 0 1985 by Academic Press, Inc.
All rights of reproduction in any form reserved.
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Mixed nite elements for the Poisson equation14Continuous problem:u2L2( ),p2H(div) p=ru,divp+f= 0Z pqdx+Z udivqdx= 0 for each vector eldq2H(div)Z (divp+f)vdx= 0 for any scalar eldv2L2(
Discrete problem
u T2L2
TP0,pT2HT(div)RT; Raviart-Thomas (1977)p
T=X a2T1p a'a Z b a(x)nbd =ab;a2 T1;b2 T1 a(x) =K+Kx;x2K2 T2 u
T(x) =uK;x2K2 T2
N e+Nascalar unknowns
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Mixed nite elements for the Poisson equation(ii)p
T=X b2T1p b'b,uT2P0, Z p
T'adx+Z
u
Tdiv'adx= 0,8a2 T1X
b2T1 Z a'bdx p b+uKuL= 0 ca= (K;L);a2 T1 Z K divpTdx+Z K fdx= 0,8K2 T2 X a2T1\@Kp ad +Z K fdx= 0,K2 T2A nite volume method ?
No! The mass-matrix induces a
nonlo calgradient op erator!
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Mass lumping of Baranger, Maitre and Oudin (1996)
p T=X b2T1p b'b,uT2P0, X b2T1 Z a'bdx p b+uKuL= 0,a2 T1
Replace the mass matrix
R 'a'bdxby a correct approximationthenpa=uLuK Ka+La withKa=12 cotanKa ca= (K;L);a2 T1 X a2T1\@Kp ad +Z K fdx= 0,K2 T2VF4 schemeof Herbin (1995)
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Petrov-Galerkin nite volumes17Reformulate the mixed nite element method to enforce the explicitation of the gradient on an edge in terms of the values in the triangles.
Continuous problem:u2L2(
),p2H(div), (p;q) + (u;divq) = 0,8q2H(div) (divp;v) =(f;v),8v2L2(
Discrete problem:uT2L2
TP0,pT2HT(div)RT
test functionsv2L2
T,q2H?
T(div)
Discrete functional space for
test functions H?
T(div)
generated by vector elds'?a:H?
T(div) =span('?a;a2 T1)
conforming in the spaceH(div) :'?a2H(div) the familyf'?a;a2 T1grepresent exactly the algeb raicdual basis of the Ravia rt-Thomasfamily for theL2scalar product:( 'a; '?b) = 0;8a6=b2 T1.
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Petrov-Galerkin nite volumes(ii)8
:u
T2L2T(
);pT2HT(div) (pT;q) + (uT;divq) = 0;8q2H?
T(div)
(divpT;v) + (f;v) = 0;8v2L2T(
A nite volume scheme!p
T=X b2T1p b'b,uT2P0,@ca= (K;L);a2 T1 ('a; '?a)pa=(uT;div'?a), a2 T1X a2T1\@Kp ad + (f;1)K= 0,K2 T2
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Butter
y Petrov-Galerkin nite volume scheme SupportV(SN) of the dual Raviart-Thomas basis function'?SN then ('a; '?a)pa=(uT;div'?a) = linear combination ofuK;uL;uM;uP;uQ;uR,a2 T1
Presented at the
3th conference \Finite Volumes for Complex Applications" (Porquerolles, 2002)
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Butter
y Petrov-Galerkin nite volume scheme(ii)Numerical tests with
Sophie Borel
(DEA Orsa y,2002)
Christophe Le Potier
(CEA Sacla y)
Mahdi Tekitek
(DEA Orsa y,2003).
Various schemes for boundary conditions
Recovering exact low degree polynomial solutions
Experimental convergence obtained for two-dimensional test cases No complete mathematical understanding of the convergence
Presented by Mahdi Tekitek at the
4th conference \Finite Volumes for Complex Applications" (Marrakech, 2005)
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Working with Isabelle and Charles (Pau, June 2014)21
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
A new variant of the same framework
R2, bounded, convex,@
polyhedral
Right hand sidef2L2(
Dirichlet problemu=fin
,u= 0on@ Mixed variational formulation of the continuous problem: u2L2( ),p2H(div) (p;q) + (u;divq) = 0,8q2H(div) (divp;v) =(f;v),8v2L2(
Discrete problem:uT2L2
TP0,pT2HT(div)RT
test functionsv2L2
T,q2H?
T(div)
H
T(div) =< '?a;a2 T1>
?a2H(div) ('a; '?b) = 0;8a6=b2 T1. with
Isab elleGre
and
Cha rlesPierre
(P au,2013-2017).
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus A new variant of the same framework(ii)Raviart-Thomas basis function'afor the edgea= (S;N) a(x) =8 :12jKj(xW) =14jKjrjxWj2;x2K
12jLj(xE) =14jLjrjxEj2;x2L
0elsewhere:
Dual Raviart-Thomas
basis function '?ahypothesis:supp('?a)supp('a) thenp=X a2T1p a'aandpa=uLuK('?a; 'a) ?a2H(div) then'?an= 0 on the four edges NWSEN.
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus A new variant of the same framework(iii)0 = ('?a; 'NW) =14jKjZ K '?arjxSj2dx 0 = Z @K'?anjxSj2d Z K div'?ajxSj2dx Z a'?anjxSj2d Z K div'?ajxSj2dx
Impose that
these t wointegral a reb othequa lto zero
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus A new variant of the same framework(iv)Boundary term : Z a'?anjxSj2d = 0
Flux of'?aon the edgea:'?ana1jajg(s)
with a universal functiongdened on [0;1] such that g(s) =g(1s);8s2[0;1] Z 1 0 g(s)ds= 1, scalingZ a'?and = 1 Z 1 0 s2g(s)ds= 0.
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus A new variant of the same framework(v)Two-dimensional term : Z K div'?ajxSj2dx= 0
We impose that the previous relation is true
for the three vertices of the triangle K
IntroduceKdiv'?ain the triangleK
We searchKsuch thatZ
K
KjxAj2dx= 0
for each vertex of the triangleK
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus
Recovering the VF4 scheme
('?a; 'a) =14jKjZ K '?arjxWj2dx14jLjZ L '?arjxEj2dx
14jKjZ
@K'?anjxWj2d
14jLjZ
@L'?anjxEj2d
14jKjZ
a'?anjxWj2d +14jLjZ a'?anjxEj2d 12 cotanKa+cotanLaafter some elementary geometry be careful !
0 < <2;82 T1
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Construction a possible dual basis vector eld28Dual basis function'?
K;j(x) =1det(dFK;a)dFK;abrK,
Ksolution of a Neumann problem in the reference elementbK
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Construction a possible dual basis vector eld(ii)Giveng: [0;1]!Rsuch that g(s) =g(1s),R1
0g(s)ds= 1,R1
0s2g(s)ds= 0.
We impose moreoverg(0) =g(1) = 0
exampleg(s) = 30s(s1)(421s(1s))
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Construction a possible dual basis vector eld(iii)Neumann problem in bK=f(bx;by);bx0;by0;bx+by1g Datum eg2H1=2(@bK) on@bK: e g=gonba= [0;1] f0g,eg= 0 elsewhere
Ane functionFK;asuch that
bK3bx7!x=FK;a(bx)2Kis one to one
Right hand side
fK(bx) = 2jKjK(x).
SolutionKof theNeumann p roblem
K=fKinbK,@K@n=egon@bK
sinceZ b
KfKdx=Z
@Kegd = 1.
SolutionK2H2(bK)
andkKk2;^KCbK kfKk0;bK+kegk1=2;@bK
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Satisfy the sucient stability conditions31Discrete stability Hypothesis: the interpolation operator and the familyUof meshes H
T(div)3qX
a2T1q a'a7!qX a2T1q a'?a2H?
T(div)
satisfy the conditions
Akqk20(q;q);8q2HT(div);8T 2 U
kqk0Bkqk0;8q2HT(div);8T 2 U (divq;divq)Ckdivqk20;8q2HT(div);8T 2 U kdivqk0Dkdivqk0;8q2HT(div);8T 2 U.
Then we have the following
unifo rmdisc reteinf-sup stabilit y condition for the Petrov Galerkin mixed formulation : inf
2VT;kkV=1sup
2V?
T;kkV1
Error estimate
: there exists a const antC>0 such that kuuTk0+kppTkdivC hTkfk1:
Finite volumesRavia rt-ThomasButter
y new va riantconstruction stabilit yC D A B conclusion b onus Satisfy the sucient stability conditions (coecient C
Divergence estimate
normalization conditionZquotesdbs_dbs12.pdfusesText_18
3 Definite, indefinite, singular, plural