Section: New Results

Asymptotics for a semi-linear convex problem with small inclusion

In [16], in collaboration with Lucas Chesnel (Inria Defi) and Sergei Nazarov (Saint-Petersbourg University), we recently investigated the asymptotics of the solution to a semi-linear problem in 2D with Dirichlet boundary condition. The partial differential operator under consideration was $-\Delta u+{\left(u\right)}^{2p+1}$ where $p$ is a positive integer. The computational domain is assumed to contain a small Dirichlet obstacle of size $\delta >0$. Using the method of matched asymptotic expansions, we compute an asymptotic expansion of the solution as $\delta$ tends to zero. Its relevance was 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 obtained numerical experiments to illustrate the analysis.