An Adapted Boundary Element Method for the Dirichlet Problem in Polygonal Domains

The interior Dirichlet problem for Laplace’s equation is associated with the exterior Dirichlet problem obtained by taking the same boundary data. Then the solution may be expressed as the simple layer potential of the charge distribution q on the boundary $\Gamma $. q is the solution of a coercive variational problem on $\Gamma $ that can be solved numerically by a boundary element Galerkin method. Unfortunately the optimal order of convergence is not reached with an uniform mesh because of the singularities of q in the neighborhood of the vertices. Here it is proved that this optimal order can be restored by grading the mesh judiciously.