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