A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions

A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions

Year:    2018

Author:    Qiaolin He, Xiaomin Lv

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 3 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2017-0193

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 3 : pp. 634–651

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    18

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

Author Details

Qiaolin He

Xiaomin Lv