An iterative substructuring method for Maxwell's equations in two dimensions
Authors:
Andrea Toselli, Olof B. Widlund and Barbara I. Wohlmuth
Journal:
Math. Comp. 70 (2001), 935949
MSC (2000):
Primary 65N30, 65N55, 65F10, 78M10
Published electronically:
March 1, 2000
MathSciNet review:
1710632
Fulltext PDF Free Access
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Abstract: Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner. For conforming finite element approximations of , it is known that the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures, and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure. In this paper, the same result is established for similar iterative methods for loworder Nédélec finite elements, which approximate in two dimensions. Results of numerical experiments are also provided.
 1.
Robert
A. Adams, Sobolev spaces, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. Pure and
Applied Mathematics, Vol. 65. MR 0450957
(56 #9247)
 2.
Ana
Alonso and Alberto
Valli, An optimal domain decomposition
preconditioner for lowfrequency timeharmonic Maxwell equations,
Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607
(99i:78002), http://dx.doi.org/10.1090/S0025571899010133
 3.
Douglas
N. Arnold, Richard
S. Falk, and R.
Winther, Preconditioning in
𝐻(𝑑𝑖𝑣) and applications, Math. Comp. 66 (1997), no. 219, 957–984. MR 1401938
(97i:65177), http://dx.doi.org/10.1090/S0025571897008260
 4.
, Multigrid in H(div) and H(curl), Numer. Math. to appear.
 5.
, Multigrid preconditioning in on nonconvex polygons, Comput. Appl. Math. 17 (1998), 303315. CMP 99:12
 6.
J.
H. Bramble, J.
E. Pasciak, and A.
H. Schatz, An iterative method for elliptic
problems on regions partitioned into substructures, Math. Comp. 46 (1986), no. 174, 361–369. MR 829613
(88a:65123), http://dx.doi.org/10.1090/S00255718198608296130
 7.
Franco
Brezzi and Michel
Fortin, Mixed and hybrid finite element methods, Springer
Series in Computational Mathematics, vol. 15, SpringerVerlag, New
York, 1991. MR
1115205 (92d:65187)
 8.
Robert
Dautray and JacquesLouis
Lions, Mathematical analysis and numerical methods for science and
technology. Vol. 2, SpringerVerlag, Berlin, 1988. Functional and
variational methods; With the collaboration of Michel Artola, Marc Authier,
Philippe Bénilan, Michel Cessenat, Jean Michel Combes,
Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude
Zuily; Translated from the French by Ian N. Sneddon. MR 969367
(89m:00001)
 9.
Maksymilian
Dryja, Barry
F. Smith, and Olof
B. Widlund, Schwarz analysis of iterative substructuring algorithms
for elliptic problems in three dimensions, SIAM J. Numer. Anal.
31 (1994), no. 6, 1662–1694. MR 1302680
(95m:65211), http://dx.doi.org/10.1137/0731086
 10.
Vivette
Girault and PierreArnaud
Raviart, Finite element methods for NavierStokes equations,
Springer Series in Computational Mathematics, vol. 5, SpringerVerlag,
Berlin, 1986. Theory and algorithms. MR 851383
(88b:65129)
 11.
R.
Hiptmair, Multigrid method for
𝐇(𝐝𝐢𝐯) in three dimensions, Electron.
Trans. Numer. Anal. 6 (1997), no. Dec.,
133–152. Special issue on multilevel methods (Copper Mountain, CO,
1997). MR
1615161 (99c:65232)
 12.
R.
Hiptmair, Multigrid method for Maxwell’s equations, SIAM
J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571
(99j:65229), http://dx.doi.org/10.1137/S0036142997326203
 13.
Ralf Hiptmair and Andrea Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, Parallel solution of PDEs, IMA Volumes in Mathematics and its Applications, SpringerVerlag, Berlin, 2000, pp. 181208.
 14.
J.C.
Nédélec, Mixed finite elements in
𝑅³, Numer. Math. 35 (1980), no. 3,
315–341. MR
592160 (81k:65125), http://dx.doi.org/10.1007/BF01396415
 15.
Ch.
G. Makridakis and P.
Monk, Timediscrete finite element schemes for Maxwell’s
equations, RAIRO Modél. Math. Anal. Numér.
29 (1995), no. 2, 171–197 (English, with
English and French summaries). MR 1332480
(96i:78002)
 16.
Claus
Müller, Foundations of the mathematical theory of
electromagnetic waves, Revised and enlarged translation from the
German. Die Grundlehren der mathematischen Wissenschaften, Band 155,
SpringerVerlag, New YorkHeidelberg, 1969. MR 0253638
(40 #6852)
 17.
Alfio
Quarteroni and Alberto
Valli, Numerical approximation of partial differential
equations, Springer Series in Computational Mathematics, vol. 23,
SpringerVerlag, Berlin, 1994. MR 1299729
(95i:65005)
 18.
Barry
F. Smith, Petter
E. Bjørstad, and William
D. Gropp, Domain decomposition, Cambridge University Press,
Cambridge, 1996. Parallel multilevel methods for elliptic partial
differential equations. MR 1410757
(98g:65003)
 19.
Andrea Toselli, Domain decomposition methods for vector field problems, Ph.D. thesis, Courant Institute of Mathematical Sciences, 1999, Technical Report 785, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University.
 20.
, Overlapping Schwarz methods for Maxwell's equations in three dimensions, Numer. Math. (2000), To appear.
 21.
, NeumannNeumann methods for vector field problems, Tech. Report 786, Department of Computer Science, Courant Institute, June 1999, Submitted to Electron. Trans. Numer. Anal.
 22.
Andrea Toselli and Axel Klawonn, A FETI domain decomposition method for Maxwell's equations with discontinuous coefficients in two dimensions, Tech. report 788, Department of Computer Science, Courant Institute, September 1999.
 23.
Olof
B. Widlund, Iterative substructuring methods: algorithms and theory
for elliptic problems in the plane, First International Symposium on
Domain Decomposition Methods for Partial Differential Equations (Paris,
1987) SIAM, Philadelphia, PA, 1988, pp. 113–128. MR 972514
(90c:65138)
 24.
Barbara I. Wohlmuth, Adaptive MultilevelFiniteElemente Methoden zur Lösung elliptischer Randwertprobleme, Ph.D. thesis, TU München, 1995.
 25.
Barbara I. Wohlmuth, Andrea Toselli, and Olof B. Widlund, Iterative substructuring method for RaviartThomas vector fields in three dimensions, SIAM J. Numer. Anal., to appear.
 1.
 Robert A. Adams, Sobolev spaces, Academic Press New York, 1975. MR 56:9247
 2.
 Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for lowfrequency timeharmonic Maxwell equations, Math. Comp. 68 (1998), no. 226, 607631. MR 99i:78002
 3.
 Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Preconditioning in H(div) and applications, Math. Comp. 66 (1997), 957984. MR 97i:65177
 4.
 , Multigrid in H(div) and H(curl), Numer. Math. to appear.
 5.
 , Multigrid preconditioning in on nonconvex polygons, Comput. Appl. Math. 17 (1998), 303315. CMP 99:12
 6.
 James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), no. 173, 361369. MR 88a:65123
 7.
 Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, SpringerVerlag, New York, 1991. MR 92d:65187
 8.
 Robert Dautray and JaquesLouis Lions, Mathematical analysis and numerical methods for science and technology, SpringerVerlag, New York, 1988.MR 89m:00001
 9.
 Maksymilian Dryja, Barry F. Smith, and Olof B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal. 31 (1994), no. 6, 16621694. MR 95m:65211
 10.
 Vivette Girault and PierreArnaud Raviart, Finite element methods for NavierStokes equations, SpringerVerlag, New York, 1986. MR 88b:65129
 11.
 Ralf Hiptmair, Multigrid method for in three dimensions, Electron. Trans. Numer. Anal. 6 (1997), 133152. MR 99c:65232
 12.
 , Multigrid method for Maxwell's equations, SIAM J. Numer. Anal. 36 (1998), 204225. MR 99j:65229
 13.
 Ralf Hiptmair and Andrea Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, Parallel solution of PDEs, IMA Volumes in Mathematics and its Applications, SpringerVerlag, Berlin, 2000, pp. 181208.
 14.
 JeanClaude Nédélec, Mixed finite elements in , Numer. Math. 35 (1980), 315341.MR 81k:65125
 15.
 Charalambos G. Makridakis and Peter Monk, Timediscrete finite element schemes for Maxwell's equations, RAIRO 29 (1995), 171197.MR 96i:78002
 16.
 Claus Müller, Foundations of the mathematical theory of electromagnetic waves, SpringerVerlag, Berlin, 1969. MR 40:6852
 17.
 Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, SpringerVerlag, Berlin, 1994. MR 95i:65005
 18.
 Barry F. Smith, Petter E. Bjørstad, and William D. Gropp, Domain decomposition: Parallel multilevel methods for elliptic partial differential equations, Cambridge University Press, 1996. MR 98g:65003
 19.
 Andrea Toselli, Domain decomposition methods for vector field problems, Ph.D. thesis, Courant Institute of Mathematical Sciences, 1999, Technical Report 785, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University.
 20.
 , Overlapping Schwarz methods for Maxwell's equations in three dimensions, Numer. Math. (2000), To appear.
 21.
 , NeumannNeumann methods for vector field problems, Tech. Report 786, Department of Computer Science, Courant Institute, June 1999, Submitted to Electron. Trans. Numer. Anal.
 22.
 Andrea Toselli and Axel Klawonn, A FETI domain decomposition method for Maxwell's equations with discontinuous coefficients in two dimensions, Tech. report 788, Department of Computer Science, Courant Institute, September 1999.
 23.
 Olof B. Widlund, Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Philadelphia, PA) (Roland Glowinski, Gene H. Golub, Gérard A. Meurant, and Jacques Périaux, eds.), SIAM, 1988. MR 90c:65138
 24.
 Barbara I. Wohlmuth, Adaptive MultilevelFiniteElemente Methoden zur Lösung elliptischer Randwertprobleme, Ph.D. thesis, TU München, 1995.
 25.
 Barbara I. Wohlmuth, Andrea Toselli, and Olof B. Widlund, Iterative substructuring method for RaviartThomas vector fields in three dimensions, SIAM J. Numer. Anal., to appear.
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Additional Information
Andrea Toselli
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012
Email:
toselli@cims.nyu.edu
Olof B. Widlund
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012
Email:
widlund@cs.nyu.edu
Barbara I. Wohlmuth
Affiliation:
Math. Institut, Universität Augsburg, Universitätsstr. 14, D86 159 Augsburg, Germany
Email:
wohlmuth@math.uniaugsburg.de
DOI:
http://dx.doi.org/10.1090/S0025571800012448
PII:
S 00255718(00)012448
Keywords:
Maxwell's equations,
N\'ed\'elec finite elements,
domain decomposition,
iterative substructuring methods
Received by editor(s):
August 14, 1998
Received by editor(s) in revised form:
September 7, 1999
Published electronically:
March 1, 2000
Additional Notes:
The work of the first author was supported in part by the National Science Foundation under Grants NSFCCR9732208 and NSFECS9527169, and in part by the U.S. Department of Energy under Contract DEFG0292ER25127.
The work of the second author was supported in part by the National Science Foundation under Grants NSFCCR9732208 and NSFECS9527169, and in part by the U.S. Department of Energy under Contract DEFG0292ER25127.
The work of the third author was supported in part by the Deutsche Forschungsgemeinschaft.
Article copyright:
© Copyright 2000
American Mathematical Society
