A Fat Boundary Method for the Poisson Problem in a Domain with Holes

AbstractWe consider the Poisson equation with Dirichlet boundary conditions, in a domain Ω\ $$\bar B$$ , where Ω⊂ ℝn, and B is a collection of smooth open subsets (typically balls). The objective is to split the initial problem into two parts: a problem set in the whole domain Ω, for which fast solvers can be used, and local subproblems set in narrow domains around the connected components of B, which can be solved in a fully parallel way. We shall present here a method based on a multi-domain formulation of the initial problem, which leads to a fixed point algorithm. The convergence of the algorithm is established, under some conditions on a relaxation parameter θ. The dependence of the convergence interval for θ upon the geometry is investigated. Some 2D computations based on a finite element discretization of both global and local problems are presented.

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