Special boundary approximation methods for laplace equation problems with boundary singularities— Applications to the motz problem

We investigate the convergence of special boundary approximation methods (BAMs) used for the solution of Laplace problems with a boundary singularity. In these methods, the solution is approximated in terms of the leading terms of the asymptotic solution around the singularity. Since the approximation of the solution satisfies identically the governing equation and the boundary conditions along the segments causing the singularity, only the boundary conditions along the rest of the boundary need to be enforced. Four methods of imposing the essential boundary conditions are considered: the penalty, hybrid, and penalty/hybrid BAMs and the BAM with Lagrange multipliers. A priori error analyses and numerical experiments are carried out for the case of the Motz problem, and comparisons between all methods are made.