Algebraic Domain Decomposition

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Author(s) : Matthias Bollhöfer

Preprint series : PhD Thesis of the Department of Mathematics, Chemnitz University of Technology

MSC 2000

65Y05 Parallel computation
65F50 Sparse matrices
65F05 Direct methods for linear systems and matrix inversion
65F10 Iterative methods for linear systems

Abstract :
We discuss algebraic domain decomposition strategies for large sparse linear systems. This is done by use of the low rank modification formula due to Sherman, Morrison and Woodbury. Most part of this paper concentrates on the properties and treatment of the so-called coupling system, which arises from the application of the low rank modification formula. A strategy to improve the properties is presented and the close relations to algebraic multigrid methods are shown. Several splittings for large sparse linear systems are discussed, especially modified block diagonal splittings. An approach to improve the properties of the coupling system by the use of suitably modified block diagonal matrices is done and finally parallel aspects of implementation are pointed out. The results are illustrated by several examples.

Keywords : Domain decomposition, parallel computations, large sparse linear systems, iterative methods, direct methods