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 discretizing 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 \(\gamma \) is the identity matrix \(I_n\) , but for a different conductivity \(\gamma \) , the authors of Cornean et al. (J Inverse Ill-posed Prob 12:111–134, 2006) supplied an estimation of the operator norm of the difference between the Dirichlet-to-Neumann operator \(\Lambda _\gamma \) and \(\Lambda _1\) , when \(\gamma =\beta I_n\) and \(\beta =1\) near the boundary \(\partial \Omega \) (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 \(f\in H^{1/2}(\partial \Omega )\) .