TY - JOUR T1 - A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions AU - He , Qiaolin AU - Lv , Xiaomin JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 634 EP - 651 PY - 2018 DA - 2018/10 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2017-0193 UR - https://global-sci.org/intro/article_detail/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.