Preprint 86/2005

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

Anja Schlömerkemper

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Submission date: 08. Oct. 2005
Pages: 30
published in: Asymptotic analysis, 52 (2007) 1/2, p. 95-115 
MSC-Numbers: 40A30, 31C20, 78A30, 78M35
Keywords and phrases: approximation of surface integrals, lattice-to-continuum theories, convergence of multidimensional singular series, magnetostatics
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Let formula10 be a lattice in formula12, formula14, and let formula16 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 formula18. 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 formula14 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 [10] from d=3 to formula14 and to a more general geometric setting, which is no longer restricted to nested sets.

18.10.2019, 02:12