A study for the Poisson equation with mixed boundary conditions on a cut rectangle

Abstract

In this thesis we study the mixed boundary value problem for the Poisson equation on a cut rectangle. We assign the zero Neumann on the cut portion of the boundary and the zero Dirichlet on the remaining boundary.

To analyze this problem we split the solution into the even and odd parts with respect to the axis y=0. The even part is then sufficiently smooth, that is, twice differentiable, but the odd part is not. The odd part must have the corner singularity expansion at the origin, that is, the corner singularity plus the regular part.

We give a formula for the coefficient of the corner singularity. We approximate the coefficient and the regular part by the finite element method and derive the error estimates showing convergence rates. We also confirm these by numerical experiments.