Continuum limit for Laplace and Elliptic operators on lattices

TL;DR

This paper investigates the continuum limits of Laplace and elliptic operators on various lattices, demonstrating convergence to continuous elliptic operators in the Euclidean space, including special cases like the hexagonal lattice.

Contribution

It extends the theory of continuum limits to general lattices and specifically analyzes the hexagonal lattice, showing convergence to continuous operators in the limit.

Findings

Laplace operators on general lattices converge to elliptic operators.

Hexagonal lattice Laplace operator converges to the continuous Laplace operator.

Discrete operators on square lattices with variable coefficients also converge in the continuum limit.

Abstract

Continuum limits of Laplace operators on general lattices are considered, and it is shown that these operators converge to elliptic operators on the Euclidean space in the sense of the generalized norm resolvent convergence. We then study operators on the hexagonal lattice, which does not apply the above general theory, but we can show its Laplace operator converges to the continuous Laplace operator in the continuum limit. We also study discrete operators on the square lattice corresponding to second order strictly elliptic operators with variable coefficients, and prove the generalized norm resolvent convergence in the continuum limit.