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Interactions between moderately close inclusions for the Laplace equation
 Math. Models Meth. in Appl. Sc
, 2009
"... The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been deeply studied for a single inclusion or well separated inclusions. We investigate the case where the distance between the holes tends to zero but remains large with ..."
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The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been deeply studied for a single inclusion or well separated inclusions. We investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω0. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion. 1
A numerical approach for the Poisson equation in a planar domain with a small inclusion
"... Abstract. We consider the Poisson equation in a domain with a small hole of size δ. We present a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as δ goes to zero without meshing the small hole. We prove the stability of t ..."
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Abstract. We consider the Poisson equation in a domain with a small hole of size δ. We present a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as δ goes to zero without meshing the small hole. We prove the stability of the scheme and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique. Key words. Small hole, asymptotic analysis, singular perturbation, finite element method. 1