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), 935-949

MSC (2000):
Primary 65N30, 65N55, 65F10, 78M10

Published electronically:
March 1, 2000

MathSciNet review:
1710632

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 low-order 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 York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****2.**Ana Alonso and Alberto Valli,*An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations*, Math. Comp.**68**(1999), no. 226, 607–631. MR**1609607**, 10.1090/S0025-5718-99-01013-3**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**, 10.1090/S0025-5718-97-00826-0**4.**-,*Multigrid in H(div) and H(curl)*, Numer. Math. to appear.**5.**-,*Multigrid preconditioning in on nonconvex polygons*, Comput. Appl. Math.**17**(1998), 303-315. 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**, 10.1090/S0025-5718-1986-0829613-0**7.**Franco Brezzi and Michel Fortin,*Mixed and hybrid finite element methods*, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR**1115205****8.**Robert Dautray and Jacques-Louis Lions,*Mathematical analysis and numerical methods for science and technology. Vol. 2*, Springer-Verlag, 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****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**, 10.1137/0731086**10.**Vivette Girault and Pierre-Arnaud Raviart,*Finite element methods for Navier-Stokes equations*, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR**851383****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****12.**R. Hiptmair,*Multigrid method for Maxwell’s equations*, SIAM J. Numer. Anal.**36**(1999), no. 1, 204–225. MR**1654571**, 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, Springer-Verlag, Berlin, 2000, pp. 181-208.**14.**J.-C. Nédélec,*Mixed finite elements in 𝑅³*, Numer. Math.**35**(1980), no. 3, 315–341. MR**592160**, 10.1007/BF01396415**15.**Ch. G. Makridakis and P. Monk,*Time-discrete 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****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, Springer-Verlag, New York-Heidelberg, 1969. MR**0253638****17.**Alfio Quarteroni and Alberto Valli,*Numerical approximation of partial differential equations*, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR**1299729****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****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.**-,*Neumann-Neumann 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****24.**Barbara I. Wohlmuth,*Adaptive Multilevel-Finite-Elemente 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 Raviart-Thomas vector fields in three dimensions*, SIAM J. Numer. Anal., to appear.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65N30,
65N55,
65F10,
78M10

Retrieve articles in all journals with MSC (2000): 65N30, 65N55, 65F10, 78M10

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, D-86 159 Augsburg, Germany

Email:
wohlmuth@math.uni-augsburg.de

DOI:
http://dx.doi.org/10.1090/S0025-5718-00-01244-8

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 NSF-CCR-9732208 and NSF-ECS-9527169, and in part by the U.S. Department of Energy under Contract DE-FG02-92ER25127.

The work of the second author was supported in part by the National Science Foundation under Grants NSF-CCR-9732208 and NSF-ECS-9527169, and in part by the U.S. Department of Energy under Contract DE-FG02-92ER25127.

The work of the third author was supported in part by the Deutsche Forschungsgemeinschaft.

Article copyright:
© Copyright 2000
American Mathematical Society