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A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

Published online by Cambridge University Press:  20 August 2015

Roland Glowinski  and
Qiaolin He
Roland Glowinski*
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA Institute of Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Qiaolin He*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Department of Mathematics, Sichuan University, Chengdu 610064, China
*
Email:roland@math.uh.edu
Corresponding author.Email:hqlaa@ust.hk
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Abstract

In this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of Rd such that ω̅⊂Ω. For a linear elliptic problem in Ω\ω̅ with Robin boundary condition on the boundary ϒ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.

Keywords

65M8565N8576M1093E24Least-Square methodsfictitious domain methodsfinite element methodsRobin boundary conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 587 - 606
Copyright
Copyright © Global Science Press Limited 2011

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