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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
DOI: https://doi.org/10.1090/S0025-5718-00-01244-8
Published electronically: March 1, 2000
MathSciNet review: 1710632
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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 $H^1$, 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 $H(\mathrm{curl};\Omega)$ in two dimensions. Results of numerical experiments are also provided.


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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, D-86 159 Augsburg, Germany
Email: wohlmuth@math.uni-augsburg.de

DOI: https://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

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