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 aP1finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.
Highlights
Let Ω be a bounded smooth domain Ω ⊂ Rn and let γ(x) = [γi,j(x)]ni,j=1 be a real-valued matrix function, which is known as the electrical conductivity matrix
We construct an approximating family of operators for the Dirichlet-to-Neumann semigroup, which satisfies the assumptions of Chernoff ’s product formula, and the Crank-Nicolson scheme converges to the exact solution
In the last section of this paper, we present the resolution of (CNS) in the particular case where γ is the unit matrix and Ω is the open unit disk
Summary
The aim of this work is to study a semidiscrete Crank-Nicolson type scheme in order to approximate numerically the Dirichletto-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 the Crank-Nicolson scheme converges to the exact solution. We write a P1 finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation
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