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

We consider the Poisson equation with Dirichlet boundary conditions, in a domain , where ⁿ , 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 , 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 . The dependence of the convergence interval for upon the geometry is investigated. Some 2D computations based on a finite element discretization of both global and local problems are presented.