On the convergence to local limit of nonlocal models with approximated interaction neighborhoods

TL;DR

This paper investigates how polygonal approximations of nonlocal interaction neighborhoods affect the convergence of solutions to their local limits, especially as the interaction horizon shrinks, revealing conditions for accurate convergence.

Contribution

It establishes the conditions under which polygonal approximations of nonlocal neighborhoods converge to the correct local limit, highlighting the importance of increasing polygon sides.

Findings

Convergence fails if the number of polygon sides is bounded.

Increasing polygon sides to infinity ensures convergence to the local limit.

Results guide computational approaches for nonlocal models.

Abstract

Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding solutions. While recent works have analyzed this issue for a fixed horizon parameter, the question remains open in the case of a small or vanishing horizon parameter, which happens often in many practical applications and has significant impact on the reliability and robustness of nonlocal modeling and simulations. In this work, we are interested in addressing this issue and establishing the convergence of the nonlocal solutions associated with polygonally approximated interaction neighborhoods to the local limit of the original nonlocal solutions. Our finding reveals that the new nonlocal solution does not converge to the correct local limit when the number of sides of polygons is uniformly bounded. On the other hand, if the number of sides tends to infinity, the desired convergence can be established. These results may be used to guide future computational studies of nonlocal models.