Volume 9, Issue 3
A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

Roland Glowinski & Qiaolin He

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

Published online: 2011-03

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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 Rsuch that ω⊂Ω. For a linear elliptic problem in Ω\ω with Robin boundary condition on the boundary γ 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.

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@Article{CiCP-9-587, author = {}, title = {A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions}, journal = {Communications in Computational Physics}, year = {2011}, volume = {9}, number = {3}, pages = {587--606}, 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 Rsuch that ω⊂Ω. For a linear elliptic problem in Ω\ω with Robin boundary condition on the boundary γ 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.071009.160310s}, url = {http://global-sci.org/intro/article_detail/cicp/7512.html} }
TY - JOUR T1 - A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions JO - Communications in Computational Physics VL - 3 SP - 587 EP - 606 PY - 2011 DA - 2011/03 SN - 9 DO - http://doi.org/10.4208/cicp.071009.160310s UR - https://global-sci.org/intro/article_detail/cicp/7512.html KW - AB -

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 Rsuch that ω⊂Ω. For a linear elliptic problem in Ω\ω with Robin boundary condition on the boundary γ 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.

Roland Glowinski & Qiaolin He. (2020). A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions. Communications in Computational Physics. 9 (3). 587-606. doi:10.4208/cicp.071009.160310s
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