Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T17:48:02.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Domain Decomposition Algorithms for Time-HarmonicMaxwell Equations with Damping

Published online by Cambridge University Press:  15 April 2002

Ana Alonso Rodriguez  and
Alberto Valli
Ana Alonso Rodriguez
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy. ([email protected])
Alberto Valli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Trento, 38050 Povo (Trento), Italy. ([email protected])
Get access

Abstract

Three non-overlapping domain decomposition methods are proposed for thenumericalapproximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). Foreach method convergence is proved and, for the discrete problem, the rate ofconvergenceof the iterative algorithm is shown to be independent of the number ofdegrees of freedom.

Keywords

Time-harmonic Maxwell equationsdomain decomposition methodsedgefinite elements.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 4 , July 2001 , pp. 825 - 848
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

V.I. Agoshkov and V.I. Lebedev, Poincaré-Steklov operators and the methods of partition of the domain in variational problems, in Vychisl. Protsessy Sist. (Computational processes and systems), G.I. Marchuk, Ed., Nauka, Moscow 2 (1985) 173-227 (in Russian).
A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H(rot;Ω) and the construction of an extension operator. Manuscripta Math. 89 (1996) 159-178.
Alonso, A. and Valli, A., An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68 (1999) 607-631. CrossRef
Alonso, A. and Valli, A., A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg. 143 (1997) 97-112. CrossRef
Alonso, A., Trotta, R.L. and Valli, A., Coercive domain decomposition algorithms for advection-diffusion equations and systems. J. Comput. Appl. Math. 96 (1998) 51-76. CrossRef
L.C. Berselli, Some topics in fluid mechanics. Ph.D. thesis, Dipartimento di Matematica, Università di Pisa, Italy (1999).
Berselli, L.C. and Saleri, F., New substructuring domain decomposition methods for advection-diffusion equations. J. Comput. Appl. Math. 116 (2000) 201-220. CrossRef
Bjørstad, P.E. and Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23 (1986) 1097-1120. CrossRef
A. Bossavit, Électromagnétisme, en vue de la modélisation. Springer-Verlag, Paris (1993).
J.-F. Bourgat, R. Glowinski, P. Le Tallec and M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain Decomposition Methods, T.F. Chan et al., Eds., SIAM, Philadelphia (1989) 3-16.
Bramble, J.H., Pasciak, J.E. and Schatz, A.H., An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp. 46 (1986) 361-369. CrossRef
Buffa, A. and Ciarlet, P., On, Jr. traces for functional spaces related to Maxwell's equations Part I: An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 9-30. 3.0.CO;2-2>CrossRef
Buffa, A. and Ciarlet, P., On, Jr. traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Meth. Appl. Sci. 24 (2001) 31-48. 3.0.CO;2-X>CrossRef
M. Cessenat, Mathematical methods in electromagnetism: Linear theory and applications. World Scientific Pub. Co., Singapore (1996).
Collino, P., Delbue, G., Joly, P. and Piacentini, A., A new interface condition in the non-overlapping domain decomposition method for the Maxwell equation. Comput. Methods Appl. Mech. Engrg. 148 (1997) 195-207. CrossRef
B. Després, P. Joly and J.E. Roberts, A domain decomposition method for the harmonic Maxwell equation, in Iterative Methods in Linear Algebra, R. Beaurvens and P. de Groen, Eds., North Holland, Amsterdam (1992) 475-484.
Kim, S., Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numer. Math. 79 (1998) 231-259. CrossRef
R. Leis, Exterior boundary-value problems in mathematical physics, in Trends in Applications of Pure Mathematics to Mechanics 11 , H. Zorski, Ed., Pitman, London (1979) 187-203.
Marini, L.D. and Quarteroni, A., A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. CrossRef
Monk, P., A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. CrossRef
Nédélec, J.C., Mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 35 (1980) 315-341. CrossRef
Nédélec, J.C., A new family of mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 50 (1986) 57-81. CrossRef
A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Oxford University Press, Oxford (1999).
Santos, J.E., Global and domain-decomposed mixed methods for the solution of Maxwell's equations with application to magnetotellurics. Numer. Methods. Partial Differ. Equations 14 (1998) 407-437. 3.0.CO;2-O>CrossRef
A. Toselli, Domain decomposition methods for vector field problems. Ph.D. thesis, Courant Institute, New York University, New York (1999).