Continuum limit of discrete Sommerfeld problems on square lattice

TL;DR

This paper proves that solutions to discrete Sommerfeld diffraction problems on a square lattice converge to the classical continuum solutions as the lattice spacing approaches zero, under certain conditions.

Contribution

It establishes the convergence of discrete lattice solutions to the continuum Sommerfeld problem for both Dirichlet and Neumann boundary conditions.

Findings

Discrete solutions converge to continuum solutions in Sobolev space

Convergence proven for positive imaginary part of wavenumber

Results apply to both boundary condition types

Abstract

A low frequency approximation of the discrete Sommerfeld diffraction problems, involving the scattering of a time harmonic lattice wave incident on square lattice by a discrete Dirichlet or a discrete Neumann half-plane, is investigated. It is established that the exact solution of the discrete model converges to the solution of the continuum model, i.e. the continuous Sommerfeld problem, in certain discrete Sobolev space defined by W. Hackbusch. The proof of convergence has been provided for both types of boundary conditions when the imaginary part of incident wavenumber is positive.