|
An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations
Author(s):
Ana
Alonso;
Alberto
Valli.
Journal:
Math. Comp.
68
(1999),
607-631.
MSC (1991):
Primary 65N55, 65N30;
Secondary 35Q60
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
The time-harmonic Maxwell equations are considered in the low-frequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.
References:
- 1.
- R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
- 2.
- A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H(rot;
 ) and the construction of an extension operator, Manuscr. Math. 89 (1996), 159-178. MR 96k:46057 - 3.
- A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations, Comput. Meth. Appl. Mech. Engrg. 143 (1997), 97-112. MR 98b:78020
- 4.
- A. Alonso and A. Valli, Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via the Fredholm alternative theory, Math. Meth. Appl. Sci., to appear.
- 5.
- C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains, preprint R 96001, Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Paris, 1996.
- 6.
- M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Meth. Appl. Sci. 12 (1990), 365-368. MR 91c:35028
- 7.
- M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Springer-Verlag, Berlin, 1988. MR 91a:35078
- 8.
- M. K\v{r}í\v{z}ek and P. Neittaanmäki, On time-harmonic Maxwell equations with nonhomogeneous conductivities: solvability and FE-approximation, Apl. Mat. 34 (1989), 480-499. MR 90m:35177
- 9.
- R. Leis, Exterior boundary-value problems in mathematical physics, in Trends in Applications of Pure Mathematics to Mechanics, Vol. 11, H. Zorski ed., Pitman, London, 1979, pp. 187-203. MR 81d:78016
- 10.
- P. Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), 243-261. MR 94b:65134
- 11.
- P. Monk, Analysis of a finite element method for Maxwell's equations, SIAM J. Numer. Anal. 29 (1992), 714-729. MR 93k:65096
- 12.
- J.C. Nédélec, Mixed finite elements in
 , Numer. Math. 35 (1980), 315-341. MR 81k:65125 - 13.
- J.C. Nédélec, A new family of mixed finite elements in
 , Numer. Math. 50 (1986), 57-81. MR 88e:65145 - 14.
- A. Quarteroni, G. Sacchi Landriani and A. Valli, Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements, Numer. Math. 59 (1991), 831-859. MR 93b:65190
- 15.
- J. Saranen, On generalized harmonic fields in domains with anisotropic nonhomogeneous media, J. Math. Anal. Appl. 88 (1982), 104-115; Erratum: J. Math. Anal. Appl. 91 (1983), 300. MR 84d:78004a, MR 84d:78004b
- 16.
- J. Saranen, On electric and magnetic problems for vector fields in anisotropic nonhomogeneous media, J. Math. Anal. Appl. 91 (1983), 254-275. MR 85i:78004
- 17.
- A. Valli, Orthogonal decompositions of
 , preprint UTM 493, Dipartimento di Matematica, Università di Trento, 1996.
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(1991):
65N55, 65N30,
35Q60
Retrieve articles in all Journals with MSC
(1991):
65N55, 65N30,
35Q60
Additional Information:
Ana
Alonso
Affiliation:
Dipartimento di Matematica, Università di Trento, 38050 Povo (Trento), Italy
Email:
alonso@science.unitn.it
Alberto
Valli
Affiliation:
Dipartimento di Matematica, Università di Trento, 38050 Povo (Trento), Italy
Email:
valli@science.unitn.it
DOI:
10.1090/S0025-5718-99-01013-3
PII:
S 0025-5718(99)01013-3
Keywords:
Domain decomposition methods,
Maxwell equations
Received by editor(s):
December 2, 1996
Received by editor(s) in revised form:
July 30, 1997
Additional Notes:
Partially supported by H.C.M. contract CHRX 0930407
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org