A finite element method using singular functions for Poisson equations: Mixed boundary conditions

In [Z. Cai, S. Kim, A finite element method using singular functions for the Poisson equation: corner singularities, SIAM J. Numer. Anal. 39 (2001) 286–299], we proposed a new finite element method to compute singular solutions of Poisson equations on a polygonal domain with re-entrant angles. Singularities are eliminated and only the regular part of the solution that is in H2 is computed. The stress intensity factor and the solution can be computed as a post-processing step. This method is extended to problems with crack singularities and to a higher-order method for smooth data in [Z. Cai, S. Kim, G. Woo, A finite element method using singular functions for the Poisson equation: crack singularities, Numer. Linear Algebra Appl. 9 (2002) 445–455]. In this paper, we study the Poisson equation with mixed boundary conditions. Examples with various singular points and numerical results are presented.