Open Access

ARTICLE

On the Multigrid Method Based on Finite Difference Approximate Inverses

by Christos K. Filelis-Papadopoulos¹, George A. Gravvanis¹

1 Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, GR 67100 Xanthi, Greece

Computer Modeling in Engineering & Sciences 2013, 90(3), 233-253. https://doi.org/10.3970/cmes.2013.090.232

Abstract

During the last decades, multigrid methods have been extensively used in order to solve large scale linear systems derived from the discretization of partial differential equations using the finite difference method. Approximate Inverses in conjunction with Richardon’s iterative method could be used as smoothers in the multigrid method. Thus, a new class of smoothers based on approximate inverses could be derived. Effectiveness of explicit approximate inverses relies in the fact that they are close approximants to the inverse of the coefficient matrix and are fast to compute in parallel. Furthermore, the class of finite difference approximate inverses proposed in conjunction with the explicit preconditioned Richardson method present improved results against the classic smoothers such as Jacobi and Gauss – Seidel method. Moreover, a dynamic relaxation scheme is proposed based on the Dynamic Over / Under Relaxation (DOUR) algorithm. Furthermore, results for the multigrid preconditioned Bi-CGSTAB based on approximate inverse smoothing and a dynamic relaxation technique are presented for a class of model problems.

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