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 where is a positive integer.
The computational domain is assumed to contain a small Dirichlet
obstacle of size . Using the method of matched
asymptotic expansions, we compute an asymptotic expansion of the
solution as 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.