The Interface Probing Technique in Domain Decomposition

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Abstract
The interface probing technique, which was developed and used by Chan and Resasco and Keyes and Gropp, is an algebraic technique for constructing interface preconditioners in domain decomposition algorithms. The basic technique is to approximate interface matrices by matrices having a specified sparsity pattern. The construction involves only matrix-vector products, and thus the interface matrix need not be known explicitly. A special feature is that the approximations adapt to the variations in the coefficients of the equations and the aspect ratios of the subdomains. This preconditioner can then be used in conjunction with many standard iterative methods, such as conjugate gradient methods. In this paper, some old results are summarized and new ones are presented, both algebraic and analytic, about the interface probing technique and its applications to interface operators. Comparisons are made with some optimal preconditioners.