A New Fictitious Domain Method for Elliptic Problems with the Third Type Boundary Conditions (Advances in Applied Mathematics and Mechanics, 2018)
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 $Ω$ and $ω$ be two bounded domains of $\mathbb{R}^{d}$ such that $\overline{ω} \subset Ω$. For a linear elliptic problem in $Ω\setminus \overline{ω}$ with Robin boundary conditions on the boundary $γ$ of $ω$, 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.