Multigrid and Multipole Techniques in the Boundary Integral Equation Methods

Summary

Because of the relatively bad properties of the boundary element matrices (they are generally neither self-adjoint nor sparse) the computational cost of the Boundary Integral Equation Method is often unnecessarily high. Moreover, in case of mixed boundary conditions, the corresponding boundary integral equation is not of the second kind, so that the traditional well-known iterative methods can hardly be applied. In this paper we present a special iterative method which converts the original mixed boundary value problem to a sequence of pure Dirichlet and pure Neumann subproblems converging rapidly to the solution of the original problem. In the solution of these subproblems, standard multi-grid tools can be used, so that a significant reduction of the computational cost can be achieved. We also derive a multipole-based technique to evaluate the appearing boundary integrals in an economic way, which can further reduce the overall computational cost.