Laplace's equation and the Dirichlet-Neumann map: a new mode for Mikhlin's method

Mikhlin's method for solving Laplace's equation in domains exterior to a number of closed contours is discussed with particular emphasis on the Dirichlet-Neumann map. In the literature there already exist two computational modes for Mikhlin's method. Here a new mode is presented. The new mode is at least as stable as the previous modes. Furthermore, its computational complexity in the number of closed contours is better. As a result, highly accurate solutions in domains exterior to tens of thousands of closed contours can be obtained on a simple workstation.