Efficient and accurate implementation of hp-BEM for the Laplace operator in 2D

We discuss the accurate and efficient implementation of hp-BEM for the Laplace operator in two dimensions. Using Legendre polynomials and their antiderivatives as local bases for the discrete ansatz spaces, we are able to reduce both the evaluation of potentials and the computation of Galerkin entries to the evaluation of basic integrals. For the computation of these integrals we derive recurrence relations and discuss their accurate evaluation. Our implementation of p- and hp  -BEM produces accurate results even for large polynomial degrees (p>1000p>1000) while still being efficient. While this work only treats Symm's integral equation for the Laplace operator in 2D, our approach can be used to solve Symm's, hypersingular and mixed integral equations for Laplace, Lamé and Stokes problems in two dimensions.