Asymptotics of the Poisson Problem in Domains with Curved Rough Boundaries

Effective boundary conditions (wall laws) are commonly employed to approximate PDEs in domains with rough boundaries, but it is neither easy to design such laws nor to estimate the related approximation error. A two-scale asymptotic expansion based on a domain decomposition result is used here to mitigate such difficulties, and as an application we consider the Poisson equation. The proposed scheme considers rough curved boundaries and allows a complete asymptotic expansion for the solution, highlighting the influence of the boundary curvature. The derivation and estimation of high order effective conditions is a corollary of such development. Sharp estimates for first and second order wall law approximations are considered for different Sobolev norms and show superior convergence rates in the interior of the domain. A numerical test illustrates several of the results obtained here.