Schwarz preconditioned CG algorithm for the mortar finite element

Authors:
Talal Rahman
Affiliations:
Dept. Comp. Engg., Bergen University College, Bergen, Norway 5020 and Dept. Comp. Sci., Stamford University Bangladesh, Dhaka, Bangladesh 1209
Venue:
Numerical Algorithms
Year:
2011

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Abstract

We propose a simple and effective hybrid (multiplicative) Schwarz precondtioner for solving systems of algebraic equations resulting from the mortar finite element discretization of second order elliptic problems on nonmatching meshes. The preconditioner is embedded in a variant of the classical preconditioned conjugate gradient (PCG) for an effective implementation reducing the cost of computing the matrix-vector multiplication in each iteration of the PCG. In fact, it serves as a framework for effective implementation of a class of hybrid Schwarz preconditioners. The preconditioners of this class are based on solving a sequence of non-overlapping local subproblems exactly, and the coarse problems either exactly or inexactly (approximately). The classical PCG algorithm is reformulated in order to make reuse of the results of matrix-vector multiplications that are already available from the preconditioning step resulting in an algorithm which is cost effective. An analysis of the proposed preconditioner, with numerical results, showing scalability with respect to the number of subdomains, and a convergence which is independent of the jumps of the coefficients are given.