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Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions

Authors: Qiya Hu and Jun Zou
Journal: Math. Comp. 73 (2004), 35-61
MSC (2000): Primary 65N30, 65N55
Published electronically: August 19, 2003
MathSciNet review: 2034110
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Abstract: This paper is concerned with the saddle-point problems arising from edge element discretizations of Maxwell’s equations in a general three dimensional nonconvex polyhedral domain. A new augmented technique is first introduced to transform the problems into equivalent augmented saddle-point systems so that they can be solved by some existing preconditioned iterative methods. Then some substructuring preconditioners are proposed, with very simple coarse solvers, for the augmented saddle-point systems. With the preconditioners, the condition numbers of the preconditioned systems are nearly optimal; namely, they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size.

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Additional Information

Qiya Hu
Affiliation: Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China

Jun Zou
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
ORCID: 0000-0002-4809-7724

Keywords: Maxwell’s equations, Nédélec finite elements, nonoverlapping domain decomposition, condition numbers
Received by editor(s): February 21, 2002
Received by editor(s) in revised form: July 15, 2002
Published electronically: August 19, 2003
Additional Notes: The work of the first author was supported by Special Funds for Major State Basic Research Projects of China G1999032804.
The work of the second author was partially supported by Hong Kong RGC Grants CUHK4048/02P and CUHK4292/00P
Article copyright: © Copyright 2003 American Mathematical Society