Very weak estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

TL;DR

This paper develops very weak estimates for a Poisson equation with rough boundaries and natural boundary conditions, improving previous results and validating them through numerical experiments on boundary layer approximations.

Contribution

It introduces fully developed very weak estimates in weighted Sobolev spaces for a Poisson problem with rough boundaries, extending prior a priori estimates and validating them numerically.

Findings

Optimal estimates in weighted Sobolev spaces are obtained.

Numerical validation confirms first-order accuracy for boundary layer approximations.

Results improve upon previous estimates in similar rough boundary problems.

Abstract

This work is a continuation of [E. Bonnetier, D.Bresch, V. Milisic, submitted]; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In [E. Bonnetier, D.Bresch, V. Milisic, submitted] we studied a priori estimates in this setting; here we fully develop very weak estimates a la Necas [J. Necas. Les m\'ethodes directes en th\'eorie des \'equations elliptiques] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [E. Bonnetier, D.Bresch, V. Milisic, submitted]. We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [W. J\"ager and A. Mikelic. J. Diff. Equa., N. Neus, M. Neus, A. Mikelic, Appl. Anal. 2006].