Analysis of non-overlapping domain decomposition algorithms with inexact solves

Authors:
James H. Bramble, Joseph E. Pasciak and Apostol T. Vassilev

Journal:
Math. Comp. **67** (1998), 1-19

MSC (1991):
Primary 65N30, 65F10

DOI:
https://doi.org/10.1090/S0025-5718-98-00879-5

MathSciNet review:
1432125

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Abstract: In this paper we construct and analyze new non-overlapping domain decomposition preconditioners for the solution of second-order elliptic and parabolic boundary value problems. The preconditioners are developed using uniform preconditioners on the subdomains instead of exact solves. They exhibit the same asymptotic condition number growth as the corresponding preconditioners with exact subdomain solves and are much more efficient computationally. Moreover, this asymptotic condition number growth is bounded independently of jumps in the operator coefficients across subdomain boundaries. We also show that our preconditioners fit into the additive Schwarz framework with appropriately chosen subspace decompositions. Condition numbers associated with the new algorithms are computed numerically in several cases and compared with those of the corresponding algorithms in which exact subdomain solves are used.

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Additional Information

**James H. Bramble**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
bramble@math.tamu.edu

**Joseph E. Pasciak**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
pasciak@math.tamu.edu

**Apostol T. Vassilev**

Affiliation:
Schlumberger, 8311 N. FM 620 Rd., Austin, Texas 78726

Email:
vassilev@slb.com

DOI:
https://doi.org/10.1090/S0025-5718-98-00879-5

Received by editor(s):
February 21, 1996

Received by editor(s) in revised form:
September 6, 1996

Additional Notes:
This manuscript has been authored under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also supported in part under the National Science Foundation Grant No. DMS-9007185 and by the PICS ground water research initiative under contract AS-413-ASD.

Article copyright:
© Copyright 1998
American Mathematical Society