The fast multipole method for the symmetric boundary integral formulation

A symmetric Galerkin boundary-element method is used for the solution of boundary-value problems with mixed boundary conditions of Dirichlet and Neumann type. As a model problem we consider the Laplace equation. When an iterative scheme is employed for solving the resulting linear system, the discrete boundary integral operators are realized by the fast multipole method. While the single-layer potential can be implemented straightforwardly as in the original algorithm for particle simulation, the double-layer potential and its adjoint operator are approximated by the application of normal derivatives to the multipole series for the kernel of the single-layer potential. The Galerkin discretization of the hypersingular integral operator is reduced to the single-layer potential via integration by parts. We finally present a corresponding stability and error analysis for these approximations by the fast multipole method of the boundary integral operators. It is shown that the use of the fast multipole method does not harm the optimal asymptotic convergence. The resulting linear system is solved by a GMRES scheme which is preconditioned by the use of hierarchical strategies as already employed in the fast multipole method. Our numerical examples are in agreement with the theoretical results.