Mathematics of Computation

Author(s): Dylan M. Copeland; Jayadeep Gopalakrishnan; Joseph E. Pasciak.
Journal: Math. Comp. 77 (2008), 1941-1965.
MSC (2000): Primary 65F10, 65N30, 78M10, 74G15, 78A30, 35Q60
Posted: March 10, 2008
Retrieve article in: PDF DVI PostScript

Abstract: We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

1.: C. AMROUCHE, C. BERNARDI, M. DAUGE, AND V. GIRAULT, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), pp. 823-864. MR 1626990 (99e:35037)
2.: D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Multigrid in $H({\rm div})$ and $H({\rm curl})$ , Numer. Math., 85 (2000), pp. 197-217. MR 1754719 (2001d:65161)
3.: F. ASSOUS, P. CIARLET, JR., AND S. LABRUNIE, Theoretical tools to solve the axisymmetric Maxwell equations, Math. Methods Appl. Sci., 25 (2002), pp. 49-78. MR 1874449 (2002j:78008)
4.: Z. BELHACHMI, C. BERNARDI, AND S. DEPARIS, Weighted Clement operator and application to the finite element discretization of the axisymmetric Stokes problem, Numer. Math., 105 (2006), pp. 217-247. MR 2262757
5.: C. BERNARDI, M. DAUGE, AND Y. MADAY, Spectral methods for axisymmetric domains, vol. 3 of Series in Applied Mathematics (Paris), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1999. MR 1693480 (2000h:65002)
6.: D. BOFFI, Fortin operator and discrete compactness for edge elements, Numer. Math., 87 (2000), pp. 229-246. MR 1804657 (2001k:65168)
7.: F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, vol. 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
8.: O. CHINELLATO, The Complex-Symmetric Jacobi-Davidson Algorithm and its Application to the Computation of some Resonance Frequencies of Anisotropic Lossy Axisymmetric Cavities, Dissertation ETH No. 16243, Swiss Federal Institute of Technology, Zürich, 2005.
9.: D. M. COPELAND AND J. E. PASCIAK, A least-squares method for axisymmetric div-curl systems, Numer. Linear Algebra Appl., 13 (2006), pp. 733-752. MR 2266104
10.: V. GIRAULT AND P.-A. RAVIART, Finite element approximation of the Navier-Stokes equations, vol. 749 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979. MR 548867 (83b:65122)
11.: J. GOPALAKRISHNAN AND J. E. PASCIAK, The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations, Math. Comp., 75 (2006), pp. 1697-1719. MR 2240631 (2007g:65116)
12.: F. KIKUCHI, Numerical analysis of electrostatic and magnetostatic problems, Sugaku Expositions, 6 (1993), pp. 33-51. MR 1222042
13.: A. KUFNER, Weighted Sobolev spaces, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985.
Translated from the Czech. MR 802206 (86m:46033)
14.: P. LACOSTE AND J. GAY, A new family of finite elements for Maxwell-Fourier's equations, in Mathematical and numerical aspects of wave propagation phenomena (Strasbourg, 1991), SIAM, Philadelphia, PA, 1991, pp. 746-749. MR 1106041 (92c:65135)
15.: B. MERCIER AND G. RAUGEL, Résolution d'un problème aux limites dans un ouvert axisymétrique par éléments finis en , et séries de Fourier en $\theta$ , RAIRO Anal. Numér., 16 (1982), pp. 405-461.
In French. MR 684832 (84g:65154)
16.: P. MONK, Finite element methods for Maxwell's equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447 (2005d:65003)
17.: J.-C. NéDéLEC, Mixed Finite Elements in ${\mathbb{R}}^3$ , Numer. Math., 35 (1980), pp. 315-341. MR 592160 (81k:65125)
18.: C. WEBER, A local compactness theorem for Maxwell's equations, Math. Methods Appl. Sci., 2 (1980), pp. 12-25. MR 561375 (81f:78005)

Dylan M. Copeland
Affiliation: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria
Email: dylan.copeland@ricam.oeaw.ac.at

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611--8105
Email: jayg@math.ufl.edu

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: pasciak@math.tamu.edu

DOI: 10.1090/S0025-5718-08-02102-9
PII: S 0025-5718(08)02102-9
Keywords: Mixed methods, Nedelec elements, axisymmetry, Maxwell's equations, div-curl systems, magnetostatics
Received by editor(s): March 30, 2007
Received by editor(s) in revised form: August 29, 2007
Posted: March 10, 2008
Additional Notes: This work was supported in part by the National Science Foundation through grants DMS-0713833, SCREMS-0619080, DMS-0311902, and DMS-0609544.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS