Dirichlet-to-Neumann semigroup acts as a magnifying glass

The first aim of this paper is to illustrate numerically that the Dirichlet-to-Neumann semigroup represented by P. Lax acts as a magnifying glass. In this perspective, we used the finite element method for the discretization of the correspondent boundary dynamical system using the implicit and explicit Euler schemes. We prove by using the Chernoff’s Theorem that the implicit and explicit Euler methods converge to the exact solution and we use the (P1)-finite elements to illustrate this convergence through a FreeFem++ implementation which provides a movie available online. In the Dirichlet-to-Neumann semigroup represented by P. Lax the conductivity γ is the identity matrix In, but for an other conductivity γ, the authors of [3] supplied an estimation of the operator norm of the difference between the Dirichlet-toNeumann operator Λγ and Λ1, when γ = βIn and β = 1 near the boundary ∂Ω (see Lemma 2.1). We will use this result to estimate the accuracy between the correspondent Dirichlet-to-Neumann semigroup and the Lax semigroup, for f ∈ H(∂Ω).