Collocation with trigonometric polynomials for integral equations to the mixed boundary value problem

Abstract

We consider the direct boundary integral equation formulation for the mixed Dirichlet–Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and trigonometric polynomials. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. Special care is taken for handling the hypersingular operator as in Hartmann and Stephan (in: Dick, Kuo, Wozniakowski (eds) Festschrift for the 80th birthday of Ian Sloan, Springer, Berlin, 2018). With the indirect method used in Elschner et al. (Numer Math 76(3):355–381, 1997) this was avoided. Using Mellin transformation techniques a stability and solvability analysis of the transformed integral equations can be performed, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space.