Seminar in Numerical Analysis: Marc Dambrine (Université de Pau)

From asymptotic analysis for simulation to generalized Ventcel's boundary value problem

I am interested in the influence of small geometrical perturbations on the solution of elliptic problems. The cases of a single inclusion or several well-separated inclusions have been deeply studied.  I will first recall here the techniques to construct an asymptotics expansion in that case. Then I will consider moderately close inclusions, i.e. the distance between the inclusions tends to zero more slowly than their characteristic size and provide a complete asymptotic description of the solution of Laplace equation. I will also present numerical simulations based on the multiscale superposition method derived from the first order expansion.

I will explain how some mathematical questions about the loss of coercivity arise from the computation of the profiles appearing in the expansion. Ventcel boundary conditions are second order differential conditions that appears  when looking for a transparent boundary condition for an exterior boundary value problem in planar linear elasticity. The goal is to bound the infinite domain by a large “box” to make numerical approximations possible. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open.  I will present, in the generic case,  existence and uniqueness result of the solution for the Ventcel boundary value problem without the sign condition. Then, I will consider perforated geometries and give conditions to remove the genericity restriction.

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