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, 93E24Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 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) |