Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron

Annalisa Buffa; Martin Costabel; Monique Dauge

doi:10.1016/S1631-073X(03)00138-9

Partial Differential Equations

Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron
[Régularité anisotrope pour le Laplacien et l'opérateur de Maxwell dans un polyèdre]

Présenté par : Philippe G. Ciarlet

Annalisa Buffa ¹ ; Martin Costabel ² ; Monique Dauge ²

¹ IMATI-CNR, Pavia, Italy
² Irmar, Université de Rennes 1, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 7, pp. 565-570.

Résumé
Abstract

Nous choisissons d'étudier le problème de Dirichlet pour le Laplacien et le problème de Maxwell électrique, comme représentants de classes plus larges de problèmes intéressant la modélisation de phénomènes physiques stationnaires. Nous énonçons des résultats de régularité dans deux familles d'espaces de Sobolev à poids : l'une, classique, isotrope, et l'autre, nouvelle, anisotrope, où l'on tient compte de l'hypoellipticité le long des arêtes d'un domaine polyédral.

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account.

Reçu le : 2003-02-18
Accepté le : 2003-02-22
Publié le : 2003-04-01

DOI : 10.1016/S1631-073X(03)00138-9

Affiliations des auteurs :

Annalisa Buffa ¹ ; Martin Costabel ² ; Monique Dauge ²

¹ IMATI-CNR, Pavia, Italy
² Irmar, Université de Rennes 1, France

@article{CRMATH_2003__336_7_565_0,
     author = {Annalisa Buffa and Martin Costabel and Monique Dauge},
     title = {Anisotropic regularity results for {Laplace} and {Maxwell} operators in a polyhedron},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {565--570},
     publisher = {Elsevier},
     volume = {336},
     number = {7},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00138-9},
     language = {en},
}

TY  - JOUR
AU  - Annalisa Buffa
AU  - Martin Costabel
AU  - Monique Dauge
TI  - Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 565
EP  - 570
VL  - 336
IS  - 7
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00138-9
LA  - en
ID  - CRMATH_2003__336_7_565_0
ER  -

%0 Journal Article
%A Annalisa Buffa
%A Martin Costabel
%A Monique Dauge
%T Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron
%J Comptes Rendus. Mathématique
%D 2003
%P 565-570
%V 336
%N 7
%I Elsevier
%R 10.1016/S1631-073X(03)00138-9
%G en
%F CRMATH_2003__336_7_565_0

Annalisa Buffa; Martin Costabel; Monique Dauge. Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron. Comptes Rendus. Mathématique, Volume 336 (2003) no. 7, pp. 565-570. doi : 10.1016/S1631-073X(03)00138-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00138-9/

Bibliographie
Cité par

[1] T. Apel; S. Nicaise The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci., Volume 21 (1998), pp. 519-549

[2] F. Assous; P. Ciarlet; P.-A. Raviart; E. Sonnendrücker Characterization of the singular part of the solution of Maxwell's equations in a polyhedral domain, Math. Methods Appl. Sci., Volume 22 (1999), pp. 485-499

[3] M. Birman; M. Solomyak L²-theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys, Volume 42 (1987) no. 6, pp. 75-96

[4] A.-S. Bonnet-Ben Dhia; C. Hazard; S. Lohrengel A singular field method for the solution of Maxwell's equations in polyhedral domains, SIAM J. Appl. Math., Volume 59 (2000), pp. 2028-2044

[5] A. Buffa, M. Costabel, and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains, Technical report, IMATI-CNR, 2003, in preparation

[6] M. Costabel; M. Dauge General edge asymptotics of solutions of second order elliptic boundary value problems I & II, Proc. Roy. Soc. Edinburgh Sect. A, Volume 123 (1993), pp. 109-184

[7] M. Costabel; M. Dauge Singularities of electromagnetic fields in polyhedral domains, Arch. Rational Mech. Anal., Volume 151 (2000), pp. 221-276

[8] M. Costabel; M. Dauge Weighted regularization of Maxwell equations in polyhedral domains, Numer. Math., Volume 93 (2002), pp. 239-277

[9] M. Dauge Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math., Springer-Verlag, Berlin, 1988

[10] B.Q. Guo The h-p version of the finite element method for solving boundary value problems in polyhedral domains (M. Costabel; M. Dauge; S. Nicaise, eds.), Boundary Value Problems and Integral Equations in Nonsmooth Domains, Luminy, 1993, Dekker, New York, 1995, pp. 101-120

[11] V.A. Kondrat'ev Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., Volume 16 (1967), pp. 227-313

[12] V.G. Maz'ya; J. Rossmann Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten, Math. Nachr., Volume 138 (1988), pp. 27-53

[13] S.A. Nazarov; B.A. Plamenevsky Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter Exp. Math., 13, de Gruyter, Berlin, 1994

Cité par Sources :

Commentaires - Politique