A Crank-Nicolson Scheme for the Dirichlet-to-Neumann Semigroup

Abstract

The aim of this work is to study a semidiscrete Crank-Nicolson type scheme in order to approximate numerically the Dirichlet-to-Neumann semigroup. We construct an approximating family of operators for the Dirichlet-to-Neumann semigroup, which satisfies the assumptions of Chernoff’s product formula, and consequently the Crank-Nicolson scheme converges to the exact solution. Finally, we write a $P_{\mathrm{1}}$ finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.