Volume 10, Issue 3
A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions

Qiaolin He & Xiaomin Lv

Adv. Appl. Math. Mech., 10 (2018), pp. 634-651.

Published online: 2018-10

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  • Abstract

In this article, we discuss a modified least--squares/fictitious domain method for the solution of linear elliptic boundary  value  problems  with the third type of boundary conditions (Robin boundary conditions).  Let $\Omega$ and $\omega$ be two bounded domains of $\mathbb{R}^{d}$ such that  $\overline{\omega} \subset \Omega$.  For a linear  elliptic problem in $\Omega\setminus \overline{\omega}$ with Robin boundary conditions on the boundary $\gamma$ of $\omega$,  we accelerate the original least--squares/fictitious domain method in Glowinski & He [1] and present a modified least--squares formulation.  This method is still a virtual control type and relies on a least-squares formulation, which makes the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results show that our method costs much less iterations and the optimal order of convergence is obtained.

  • Keywords

Least–squares methods, fictitious domain methods, finite element methods, Robin boundary conditions.

  • AMS Subject Headings

65M85, 65N85, 76M10, 93E24

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-634, author = {}, title = {A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {3}, pages = {634--651}, abstract = {In this article, we discuss a modified least--squares/fictitious domain method for the solution of linear elliptic boundary  value  problems  with the third type of boundary conditions (Robin boundary conditions).  Let $\Omega$ and $\omega$ be two bounded domains of $\mathbb{R}^{d}$ such that  $\overline{\omega} \subset \Omega$.  For a linear  elliptic problem in $\Omega\setminus \overline{\omega}$ with Robin boundary conditions on the boundary $\gamma$ of $\omega$,  we accelerate the original least--squares/fictitious domain method in Glowinski & He [1] and present a modified least--squares formulation.  This method is still a virtual control type and relies on a least-squares formulation, which makes the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results show that our method costs much less iterations and the optimal order of convergence is obtained.}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0193}, url = {http://global-sci.org/intro/article_detail/aamm/12228.html} }
TY - JOUR T1 - A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 634 EP - 651 PY - 2018 DA - 2018/10 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0193 UR - https://global-sci.org/intro/aamm/12228.html KW - Least–squares methods, fictitious domain methods, finite element methods, Robin boundary conditions. AB - In this article, we discuss a modified least--squares/fictitious domain method for the solution of linear elliptic boundary  value  problems  with the third type of boundary conditions (Robin boundary conditions).  Let $\Omega$ and $\omega$ be two bounded domains of $\mathbb{R}^{d}$ such that  $\overline{\omega} \subset \Omega$.  For a linear  elliptic problem in $\Omega\setminus \overline{\omega}$ with Robin boundary conditions on the boundary $\gamma$ of $\omega$,  we accelerate the original least--squares/fictitious domain method in Glowinski & He [1] and present a modified least--squares formulation.  This method is still a virtual control type and relies on a least-squares formulation, which makes the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results show that our method costs much less iterations and the optimal order of convergence is obtained.
Qiaolin He & Xiaomin Lv. (2020). A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions. Advances in Applied Mathematics and Mechanics. 10 (3). 634-651. doi:10.4208/aamm.OA-2017-0193
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