A boundary element method for Laplace's equation without numerical integrations

A boundary element method (BEM), without any numerical integration, is presented here for the treatment of boundary value problems in Laplace's equation on a plane domain with a polygonal boundary. We use the double-layer potential to approximate the solution. The exact forms of the double-layer potentials for the first- and second-order spline density are to be derived here. These potentials can be differentiated directly. Then, the collocation method is applied to mixed boundary value problems. The collocation equations of the first-order scheme for Dirichlet's problems on a convex domain are shown to be well-conditioned; i.e., the condition number of the resulting matrix is bounded by a constant independent of the number of the elements. The second-order method can be applied to mixed boundary problems on interior and exterior regions. Two mixed boundary value problems with singular solutions are solved by the second-order scheme. Numerical examples show that the scheme is capable of dealing with singularities.