Commun. Comput. Phys., 9 (2011), pp. 587-606.


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

Roland Glowinski 1, Qiaolin He 2*

1 Department of Mathematics, University of Houston, Houston, TX 77204, USA; and Institute of Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
2 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; and Department of Mathematics, Sichuan University, Chengdu 610064, China.

Received 7 October 2009; Accepted (in revised version) 16 March 2010
Available online 17 September 2010
doi:10.4208/cicp.071009.160310s

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 $\Omega$ and $\omega$ be two bounded domains of $\mathbf{R}^{d}$ such that $\overline{\omega} \subset \Omega$. For a linear elliptic problem in $\Omega\setminus \overline{\omega}$ with Robin boundary condition on the boundary $\gamma$ of $\omega$, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full $\Omega$, followed by a well-chosen correction over $\omega$. 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.

AMS subject classifications: 65M85, 65N85, 76M10, 93E24

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Key words: Least-Square methods, fictitious domain methods, finite element methods, Robin boundary conditions.

*Corresponding author.
Email: roland@math.uh.edu (R. Glowinski), hqlaa@ust.hk (Q. He)
 

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