Small obstacle asymptotics for a 2D semi-linear convex problem

Lucas Chesnel 1, 2 Xavier Claeys 3 Sergei Nazarov 4, 5
2 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
3 ALPINES - Algorithms and parallel tools for integrated numerical simulations
LJLL - Laboratoire Jacques-Louis Lions, INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris
Abstract : We study a 2D semi-linear equation in a domain with a small Dirichlet obstacle of size δ. Using the method of matched asymptotic expansions, we compute an asymptotic expansion of the solution as δ tends to zero. Its relevance is justified by proving a rigorous error estimate. Then we construct an approximate model, based on an equation set in the limit domain without the small obstacle, which provides a good approximation of the far field of the solution of the original problem. The interest of this approximate model lies in the fact that it leads to a variational formulation which is very simple to discretize. We present numerical experiments to illustrate the analysis.
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Lucas Chesnel, Xavier Claeys, Sergei Nazarov. Small obstacle asymptotics for a 2D semi-linear convex problem. Applicable Analysis, Taylor & Francis, 2017, pp.20. ⟨http://www.tandfonline.com/action/showCitFormats?doi=10.1080%2F00036811.2017.1295449⟩. ⟨10.1080/00036811.2017.1295449⟩. ⟨hal-01427617⟩

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