Analysis of the Residual Type and the Recovery Type a Posteriori Error Estimators for a Consistent Atomistic-to-Continuum Coupling Method in 1D

TL;DR

This paper develops and compares residual and gradient recovery a posteriori error estimators for an atomistic-to-continuum coupling method in 1D, demonstrating their efficiency and optimal convergence through numerical experiments.

Contribution

It introduces a novel hybrid a posteriori error estimator combining residual and gradient recovery types for 1D atomistic-to-continuum models.

Findings

Proves the efficiency of the residual error estimator.

Establishes the equivalence between residual and gradient recovery estimators.

Demonstrates optimal convergence rates in numerical experiments.

Abstract

We consider the a posteriori error estimation for an atomistic-to-continuum cou- pling scheme for a generic one-dimensional many-body next-nearest-neighbour interaction model in 1D. We derive and rigorously prove the efficiency of the residual type estimator. We prove the equivalence between the residual type and the gradient recovery type estima- tor in the continuum region and propose a (novel) hybrid a a posteriori error estimator by combining the two types of estimators. Our numerical experiments illustrate the optimal con- vergence rate of the adaptive algorithms using these estimators whose efficiency factors are also presented.