On considère la discrétisation des équations de Maxwell, telle qu'elle a été proposée dans [3,2,1]. Les approximations numériques du champ électromagnétique et du multiplicateur de Lagrange associé à la divergence du champ sont réalisées à l'aide de l'élément fini de Taylor–Hood P2-iso-P1, et complétées de fonctions-test singulières, lorsque le domaine de calcul est non convexe, à bord non régulier. Le but de la Note est de prouver l'existence d'une condition inf-sup discrète. On peut également appliquer ce résultat à la discrétisation du système de Stokes en vitesse-pression [7].
We consider the discretization of Maxwell equations, proposed in [3,2,1]. The electromagnetic field and the Lagrange multiplier related to its divergence are approximated numerically by the P2-iso-P1 Taylor–Hood Finite Element. Singular test-functions are added when the domain is non-convex, with a non-smooth boundary. The aim of this Note is to establish a discrete inf-sup condition. The result can be applied to the discretization of the velocity-pressure Stokes system [7].
Accepté le :
Publié le :
Patrick Ciarlet 1 ; Vivette Girault 2
@article{CRMATH_2002__335_10_827_0,
author = {Patrick Ciarlet and Vivette Girault},
title = {Condition \protect\emph{inf-sup} pour l'\'el\'ement fini de {Taylor{\textendash}Hood} {<strong>\protect\emph{P}\protect\textsubscript{2}</strong>-iso-<strong>\protect\emph{P}\protect\textsubscript{1}</strong>,} {3-D} ; application aux \'equations de {Maxwell}},
journal = {Comptes Rendus. Math\'ematique},
pages = {827--832},
publisher = {Elsevier},
volume = {335},
number = {10},
year = {2002},
doi = {10.1016/S1631-073X(02)02564-5},
language = {fr},
}
TY - JOUR AU - Patrick Ciarlet AU - Vivette Girault TI - Condition inf-sup pour l'élément fini de Taylor–Hood P2-iso-P1, 3-D ; application aux équations de Maxwell JO - Comptes Rendus. Mathématique PY - 2002 SP - 827 EP - 832 VL - 335 IS - 10 PB - Elsevier DO - 10.1016/S1631-073X(02)02564-5 LA - fr ID - CRMATH_2002__335_10_827_0 ER -
%0 Journal Article %A Patrick Ciarlet %A Vivette Girault %T Condition inf-sup pour l'élément fini de Taylor–Hood P2-iso-P1, 3-D ; application aux équations de Maxwell %J Comptes Rendus. Mathématique %D 2002 %P 827-832 %V 335 %N 10 %I Elsevier %R 10.1016/S1631-073X(02)02564-5 %G fr %F CRMATH_2002__335_10_827_0
Patrick Ciarlet; Vivette Girault. Condition inf-sup pour l'élément fini de Taylor–Hood P2-iso-P1, 3-D ; application aux équations de Maxwell. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 827-832. doi : 10.1016/S1631-073X(02)02564-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02564-5/
[1] Résolution des équations de Maxwell instationnaires avec charge dans un domaine singulier bidimensionnel, C. R. Acad. Sci. Paris, Série I, Volume 330 (2000), pp. 391-396
[2] Resolution of the Maxwell equations in a domain with reentrant corners, Math. Mod. Num. Anal, Volume 32 (1998), pp. 359-389
[3] On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys, Volume 109 (1993), pp. 222-237
[4] Mixed and Hybrid Finite Element Methods, Springer, New York, 1991
[5] Basic error estimates for elliptic problems (P.G. Ciarlet; J.-L. Lions, eds.), Handbook of Numerical Analysis, II, North-Holland, 1991, pp. 17-351
[6] A coercive bilinear form for Maxwell's equations, J. Math. Anal. Appl, Volume 157 (1991), pp. 527-541
[7] Finite Element Methods for Navier–Stokes Equations, Springer Series in Computational Mathematics, Springer, Berlin, 1986
[8] V. Girault, L.R. Scott, A quasi-local interpolation operator preserving the discrete divergence, soumis à Calcolo
[9] Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comput, Volume 54 (1990), pp. 483-493
[10] Error estimate for a mixed finite element approximation of the Stokes problem, RAIRO Anal. Numér, Volume 18 (1984), pp. 175-182
[11] A local compactness theorem for Maxwell's equations, Math. Meth. Appl. Sci, Volume 2 (1980), pp. 12-25
- A mixed
- Max-norm estimates for Stokes and Navier-Stokes approximations in convex polyhedra, Numerische Mathematik, Volume 131 (2015) no. 4, pp. 771-822 | DOI:10.1007/s00211-015-0707-8 | Zbl:1401.76087
- Incompressibility in the multimodel Arlequin framework, International Journal for Numerical Methods in Engineering, Volume 94 (2013) no. 4, pp. 374-399 | DOI:10.1002/nme.4454 | Zbl:1352.74367
- Multigrid discretization and iterative algorithm for mixed variational formulation of the eigenvalue problem of electric field, Abstract and Applied Analysis, Volume 2012 (2012), p. 25 (Id/No 190768) | DOI:10.1155/2012/190768 | Zbl:1256.78001
- A two-scale discretization scheme for mixed variational formulation of eigenvalue problems, Abstract and Applied Analysis, Volume 2012 (2012), p. 29 (Id/No 812914) | DOI:10.1155/2012/812914 | Zbl:1246.65220
- A divergence-free finite element method for a type of 3D Maxwell equations, Applied Numerical Mathematics, Volume 62 (2012) no. 6, pp. 802-813 | DOI:10.1016/j.apnum.2011.06.009 | Zbl:1241.78043
- Solving Maxwell's equations in singular domains with a Nitsche type method, Journal of Computational Physics, Volume 230 (2011) no. 12, pp. 4922-4939 | DOI:10.1016/j.jcp.2011.03.013 | Zbl:1220.78009
- A domain decomposition method for the parallelization of a three-dimensional Maxwell solver based on a constrained formulation, Mathematics and Computers in Simulation, Volume 81 (2011) no. 11, pp. 2371-2388 | DOI:10.1016/j.matcom.2011.03.001 | Zbl:1230.78035
- NUMERICAL ANALYSIS OF THE GENERALIZED MAXWELL EQUATIONS (WITH AN ELLIPTIC CORRECTION) FOR CHARGED PARTICLE SIMULATIONS, Mathematical Models and Methods in Applied Sciences, Volume 19 (2009) no. 11, p. 1959 | DOI:10.1142/s0218202509004017
- Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements, Numerische Mathematik, Volume 113 (2009) no. 4, pp. 497-518 | DOI:10.1007/s00211-009-0246-2 | Zbl:1180.78048
- Computing electromagnetic eigenmodes with continuous Galerkin approximations, Computer Methods in Applied Mechanics and Engineering, Volume 198 (2008) no. 2, pp. 358-365 | DOI:10.1016/j.cma.2008.08.005 | Zbl:1194.78053
- Numerical Solution to Maxwell’s Equations in Singular Waveguides, Computational Science – ICCS 2007, Volume 4490 (2007), p. 235 | DOI:10.1007/978-3-540-72590-9_34
- Efficient Numerical Solution of Cahn–Hilliard–Navier–Stokes Fluids in 2D, SIAM Journal on Scientific Computing, Volume 29 (2007) no. 6, p. 2241 | DOI:10.1137/050648110
- Time-dependent Maxwell's equations with charges in singular geometries, Computer Methods in Applied Mechanics and Engineering, Volume 196 (2006) no. 1-3, pp. 665-681 | DOI:10.1016/j.cma.2006.07.007 | Zbl:1121.78305
- Augmented formulations for solving Maxwell equations, Computer Methods in Applied Mechanics and Engineering, Volume 194 (2005) no. 2-5, pp. 559-586 | DOI:10.1016/j.cma.2004.05.021 | Zbl:1063.78018
- Maximum-norm stability of the finite element Stokes projection, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 84 (2005) no. 3, pp. 279-330 | DOI:10.1016/j.matpur.2004.09.017 | Zbl:1210.76051
- Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: The singular complement method., Journal of Computational Physics, Volume 191 (2003) no. 1, pp. 147-176 | DOI:10.1016/s0021-9991(03)00309-7 | Zbl:1033.65086
Cité par 17 documents. Sources : Crossref, zbMATH
Commentaires - Politique