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A robust and parallel relaxation method based on algebraic splittings
Author(s) -
Ervin V. J.,
Layton William J.
Publication year - 1999
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199901)15:1<91::aid-num5>3.0.co;2-t
Subject(s) - discretization , robustness (evolution) , relaxation (psychology) , mathematics , algebraic number , linear system , finite element method , iterative method , mathematical optimization , mathematical analysis , psychology , social psychology , biochemistry , chemistry , physics , gene , thermodynamics
We propose and analyze a new relaxation scheme for the iterative solution of the linear system arising from the finite difference discretization of convection–diffusion problems. For problems that are convection dominated, the (nondimensionalized) diffusion parameter ϵ is usually several orders of magnitude smaller than computationally feasible mesh widths. Thus, it is of practical importance that approximation methods not degrade for small ϵ. We give a relaxation procedure that is proven to converge uniformly in ϵ to the solution of the linear algebraic system (i.e., “robustly”). The procedure requires, at each step, the solution of one 4 × 4 linear system per mesh cell. Each 4 × 4 system can be independently solved, and the result communicated to the neighboring mesh cells. Thus, on a mesh connected processor array, the communication requirements are four local communications per iteration per mesh cell. An example is given, which illustrates the robustness of the new relaxation scheme. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 91–110, 1999