Lattice approximation of a surface integral and convergence of a singular lattice sum

Abstract

Let ℒ be a lattice in $\mathbb{R}^{d}$ , d≥2, and let $A\subset\mathbb{R}^{d}$ be a Lipschitz domain which satisfies some additional weak technical regularity assumption. In the first part of the paper we consider certain lattice sums over points which are close to $\curpartial A$ . The main result is that these lattice sums approximate corresponding surface integrals for small lattice spacing. This is not obvious since the thickness of the domain of summation is comparable to the scale of the lattice.

In the second part of the paper we study a specific singular lattice sum in d≥2 and prove that this lattice sum converges as the lattice spacing tends to zero. This lattice sum and its convergence are of interest in lattice-to-continuum approximations in electromagnetic theories—as is the above approximation of surface integrals by lattice sums.

This work generalizes previous results (Schlömerkemper, Arch. Rational Mech. Anal. 176 (2005), 227–269) from d=3 to d≥2 and to a more general geometric setting, which is no longer restricted to nested sets.