An asymptotically compatible coupling formulation for nonlocal interface problems with jumps

TL;DR

This paper develops a mathematically rigorous nonlocal interface formulation with jumps, introduces an asymptotically compatible finite element discretization, and demonstrates convergence to local solutions and robustness through numerical tests.

Contribution

It presents a novel asymptotically compatible finite element method for nonlocal interface problems with jumps, ensuring convergence to local limits and robustness.

Findings

Solutions converge to local counterparts as nonlocal horizon shrinks

Numerical tests confirm convergence to exact solutions

Method is robust and passes patch tests

Abstract

We introduce a mathematically rigorous formulation for a nonlocal interface problem with jumps and propose an asymptotically compatible finite element discretization for the weak form of the interface problem. After proving the well-posedness of the weak form, we demonstrate that solutions to the nonlocal interface problem converge to the corresponding local counterpart when the nonlocal data are appropriately prescribed. Several numerical tests in one and two dimensions show the applicability of our technique, its numerical convergence to exact nonlocal solutions, its convergence to the local limit when the horizons vanish, and its robustness with respect to the patch test.