An iterative and multigrid solution of boundary integral equations
The author considers the mixed boundary value problem (1) \(\Delta u = 0\) in \(\Omega \subset \mathbb{R}^2\), \(u = u_0\) on \(\Gamma_1\), \(\partial u/\partial n = \nu_0\) on \(\Gamma_2\) for the two-dimensional Laplace equation as some model problem. Instead of applying the boundary integral equation method (BIEM) to (1) directly, the author defines a sequence of pure Dirichlet and Neumann type subproblems in such a way that it remains consistent with the original problem (1). Now, the subproblems can be transformed to integral equations by BIEM in a standard way and can be discretized by the boundary element method. The author proves that the sequence of functions produced by the alternate solution of the Dirichlet and Neumann type subproblems converges rapidly to the solution of (1). The same is true for the discrete counterparts. The discrete subproblems can be solved very efficiently by multigrid techniques. The results of the numerical experiments reported confirm the efficiency of the proposed method.
scientific article; zbMATH DE number 779206
scientific article