Analysis of an Energy-based Atomistic/Continuum Coupling Approximation of a Vacancy in the 2D Triangular Lattice

TL;DR

This paper provides a detailed error analysis of an energy-based atomistic/continuum coupling method for vacancy defects in 2D lattices, offering guidelines for optimal mesh design and demonstrating convergence through numerical tests.

Contribution

It offers the first comprehensive a priori error estimates for a practical energy-based atomistic/continuum coupling method in 2D with vacancies, including stability and consistency analysis.

Findings

Error estimates depend on mesh size and solution smoothness

Heuristics for optimal atomistic region and mesh selection

Numerical tests confirm analytical convergence predictions

Abstract

We present a comprehensive a priori error analysis of a practical energy based atomistic/continuum coupling method (Shapeev, arXiv:1010.0512) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects. The majority of the work is devoted to the analysis of consistency and stability of the method. These yield a priori error estimates in the H1-norm and the energy, which depend on the mesh size and the "smoothness" of the atomistic solution in the continuum region. Based on these error estimates, we present heuristics for an optimal choice of the atomistic region and the finite element mesh, which yields convergence rates in terms of the number of degrees of freedom. The analytical predictions are supported by extensive numerical tests.