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Multi-parameterized Schwarz alternating methods for elliptic boundary value problems

Author

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  • Kim, S.-B.
  • Hadjidimos, A.
  • Houstis, E.N.
  • Rice, J.R.

Abstract

The convergence rate of a numerical procedure based on Schwarz Alternating Method (SAM) for solving elliptic boundary value problems (BVPs) depends on the selection of the so-called interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the weighted mixed interface conditions (g(u) = ωu + (1 − ω)ϖuϖn), controlled by the parameter ω, can optimize SAMs convergence rate. In this paper, we present a matrix formulation of this method based on finite difference approximation of the BVP, review its known computational behavior in terms of the parameter α = /gf(ω, h), where h is the discretization parameter and /gf is a derivable relation, and obtain analytically explicit and implicit expressions for the optimum α. Moreover, we consider a parameterized SAM where the parameter ω or α is assumed to be different in each overlapping area. For this SAM and the one-dimensional (1-D) elliptic model BVPs, we determine analytically the optimal values of αi. Furthermore, we extend some of these results to two-dimensional (2-D) elliptic problems.

Suggested Citation

  • Kim, S.-B. & Hadjidimos, A. & Houstis, E.N. & Rice, J.R., 1996. "Multi-parameterized Schwarz alternating methods for elliptic boundary value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 42(1), pages 47-76.
    • Handle: RePEc:eee:matcom:v:42:y:1996:i:1:p:47-76
    DOI: 10.1016/0378-4754(95)00111-5
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    Cited by:

    1. Hadjidimos, A. & Noutsos, D. & Tzoumas, M., 2000. "Nonoverlapping domain decomposition:," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 51(6), pages 597-625.

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