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Journal of Applied Mathematics
Volume 2015 (2015), Article ID 429641, 5 pages
http://dx.doi.org/10.1155/2015/429641
Research Article

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

1Department of Mathematics, Faculty of Science, Beirut Arab University, P.O. Box 11-5020, Beirut, Lebanon
2Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, 57 045 Metz, France

Received 4 May 2015; Accepted 28 June 2015

Academic Editor: M. Montaz Ali

Copyright © 2015 Rola Ali Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.