An Asymptotically Compatible Formulation for Local-to-Nonlocal Coupling Problems without Overlapping Regions

TL;DR

This paper introduces a new explicit local-to-nonlocal coupling method using Robin-type boundary conditions, ensuring asymptotic compatibility and convergence to the local model without overlapping regions.

Contribution

It develops a novel Robin-type boundary condition for local-to-nonlocal coupling that guarantees asymptotic compatibility and stability without overlapping domains.

Findings

The method achieves $O( ext{h})$ convergence rate as $ ext{h} o 0$.

Numerical experiments confirm the effectiveness of the optimal Robin coefficients.

The approach successfully couples heterogeneous local and nonlocal models in complex domains.

Abstract

In this paper we design and analyze an explicit partitioned procedure for a 2D dynamic local-to-nonlocal (LtN) coupling problem, based on a new nonlocal Robin-type transmission condition. The nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter $\delta$ characterizing the range of nonlocal interactions, and the local subproblem is described by the classical heat equation. We consider a heterogeneous system where the local and nonlocal subproblems present different physical properties, and employ no overlapping region between the two subdomains. We first propose a new generalization of classical local Neumann-type condition by converting the local flux to a correction term in the nonlocal model, and show that the proposed Neumann-type boundary formulation recovers the local case as $O(\delta^2)$ in the $L^{\infty}$ norm. We then extend the nonlocal Neumann-type boundary condition to a Robin-type boundary condition, and develop a local-to-nonlocal coupling formulation with Robin-Dirichlet transmission conditions. To stabilize the explicit coupling procedure and to achieve asymptotic compatibility, the choice of the coefficient in the Robin condition is obtained via amplification factor analysis for the discretized system with coarse grids. Employing a high-order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver, the selection of optimal Robin coefficients are verified with numerical experiments on heterogeneous and complicated domains. With the developed optimal coupling strategy, we numerically demonstrate the coupling framework's asymptotic convergence to the local limit with an $O(\delta)=O(h)$ rate, when there is a fixed ratio between the horizon size $\delta$ and the spatial discretization size $h$.