A Posteriori Error Estimate and Adaptive Mesh Refinement Algorithm for Atomistic/Continuum Coupling with Finite Range Interactions in Two Dimensions

TL;DR

This paper develops residual-based a posteriori error estimates and an adaptive mesh refinement algorithm for atomistic/continuum coupling with finite range interactions in 2D, improving computational efficiency and accuracy.

Contribution

It introduces a new explicitly computable stress tensor formula and applies the GRAC coupling scheme with optimal discretization, enhancing adaptive refinement methods.

Findings

Numerical results confirm the effectiveness of the adaptive mesh refinement.

The error estimates are consistent with optimal a priori error bounds.

The proposed method improves computational accuracy for atomistic/continuum models.

Abstract

In this paper, we develop the residual based a posteriori error estimates and the corresponding adaptive mesh refinement algorithm for atomistic/continuum (a/c) coupling with finite range interactions in two dimensions. We have systematically derived a new explicitly computable stress tensor formula for finite range interactions. In particular, we use the geometric reconstruction based consistent atomistic/continuum (GRAC) coupling scheme, which is optimal if the continuum model is discretized by $P^1$ finite elements. The numerical results of the adaptive mesh refinement algorithm is consistent with the optimal a priori error estimates.