A numerical solution for the inhomogeneous Dirichlet boundary value problem on a non-convex polygon

Abstract

In this paper, we introduce an effective finite element scheme for the Poisson problem with inhomogeneous Dirichlet boundary condition on a non-convex polygon. Due to the corner singularities, the solution is composed of a singular part not belonging to the Sobolev space H2 and a smoother regular part. We first provide a generalized extraction formula for the coefficient of singular part, which is depending on the inhomogeneous boundary condition and the regular part. We then propose a stable finite element method for the regular part by the use of the derived formula. We show the H1 and L2 error estimates of the finite element solution for the regular part and the absolute error estimate of the approximation for the coefficient of singular part. Finally, we give some numerical examples to confirm the efficiency and reliability of the proposed method.