On hp-Boundary Element Methods for the Laplace Operator in Two Dimensions. Implementation, hp-Adaptive Algorithms, and Data Compression.

Abstract

This work expands on three aspects of the hp-boundary element method (BEM) for the Laplace operator in two dimensions. First, we provide a new stable and efficient implementation of hp-BEM. Using Legendre functions 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 closely related to associated Legendre functions of the second kind. For the computation of these integrals we derive recurrence relations and discuss their accurate evaluation in detail. Our implementation of p- and hp-BEM is the first that produces accurate results even for large polynomial degrees (p>1000) while still being efficient. The next aspect considered is the construction of a posteriori error estimators for Symm"s integral equation. We generalize the idea of h-h/2 error estimators to obtain error estimators for hp-BEM. By a combination of theoretical considerations and numerical experiments, we show that weighted L^2 norms and projections have to be considered in order to obtain efficient and reliable estimators. Based on these estimators, we construct two hp-adaptive algorithms and prove their convergence. Numerical experiments show that both algorithms converge exponentially. Finally, we develop a new data compression scheme for the single layer Galerkin matrix arising in p- and hp-BEM. This novel approach uses different compression techniques for different sub-blocks of the Galerkin matrix, which results in an H-Matrix approximation. Complexity estimates show that the amount of storage needed by the Galerkin matrix and the effort for the matrix vector multiplication are of the order O(log(N)^a) with a less than or equal to 3.