A Crank-Nicolson scheme for the Dirichlet-to-Neumann semigroup. (English) Zbl 1435.65185

Ali Ahmad, Rola; El Arwadi, Toufic; Chrayteh, Houssam; Sac-Epée, Jean-Marc

J. Appl. Math. 2015, Article ID 429641, 5 p. (2015).

Summary: 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 \(P_1\) finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.

MSC:

65N06	Finite difference methods for boundary value problems involving PDEs
65J05	General theory of numerical analysis in abstract spaces
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

FreeFem++

References:

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