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IMA Journal of Numerical Analysis Advance Access published online on April 2, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drm011
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A Lions non-overlapping domain decomposition method for domains with an arbitrary interface

S. H. Lui{dagger}

Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2

{dagger} Email: luish{at}cc.umanitoba.ca

Received on 24 August 2005. Accepted for publication 5 November 2006.


  Abstract

Lions’ non-overlapping domain decomposition method for the solution of elliptic partial differential equations has been analysed extensively by many authors. There have been numerous works on the convergence of the iterative method as well as variations of it. In the present work, we analyse several formulations of Lions’ method. For two of these, we show that the spectral radius of the operator in the fixed-point iteration for the interface boundary function is bounded above by 1 – O(h1/2) when the optimal value (O(h–1/2)) of the parameter in the Robin boundary condition along the artificial interface is used. While this result is already known for rectangular domains with a straight interface, our analysis is valid for essentially arbitrary geometry. The method of Guo and Hou, which is Lions’ method with relaxation, has a simple proof of convergence using our method of analysis. Similarly, we provide a simple proof of convergence of the scheme of Deng.

Key Words: domain decomposition; Lions' non-overlapping method; convergence acceleration; Poincare–Steklov operator; optimized Schwarz methods


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