An Optimal Order Error Analysis of the One-Dimensional Quasicontinuum Approximation

TL;DR

This paper provides an analysis of the optimal-order convergence rates for quasicontinuum approximations in one-dimensional models, focusing on error estimates and interface coupling effects.

Contribution

It introduces a simplified model problem to analyze the convergence and error estimates of energy-based and quasi-nonlocal quasicontinuum methods.

Findings

Optimal-order error estimates for all strains up to fracture strain.

Explicit treatment of coupling error at atomistic-continuum interface.

Analysis based on stability and error due to atomistic and continuum schemes.

Abstract

We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimal-order convergence rate in the continuum limit for both the energy-based quasicontinuum approximation and the quasi-nonlocal quasicontinuum approximation. The optimal-order error estimates for the quasi-nonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture. The analysis is based on an explicit treatment of the coupling error at the atomistic to continuum interface, combined with an analysis of the error due to atomistic and continuum schemes using the stability of the quasicontinuum approximation.