Computing the Dirichlet-to-Neumann map via an integral equation with the adjoint generalized Neumann kernel

Abstract

A new numerical method for computing the Dirichlet-to-Neumann map for Laplace’s equation in simply and multiply connected smooth domains is introduced. This method is based on an integral equation with the adjoint generalized Neumann kernel. Contrary to the classical approach which requires numerical differentiation in a post-processing step, our method allows computing the Dirichlet-to-Neumann map directly without the need of numerical differentiation in post-processing. The results of our numerical experiments demonstrate that the proposed method gives better accuracy and is more efficient than the classical approach for large problems with unbounded multiply connected domains.

Keywords

• Riemann–Hilbert problem
• Laplace’s equation
• Dirichlet-to-Neumann map
• Integral equation
• Adjoint generalized Neumann kernel