Analysis of patch-test consistent atomistic-to-continuum coupling with higher-order finite elements

TL;DR

This paper develops a higher-order finite element atomistic-to-continuum coupling scheme that is proven to be (quasi-)optimal in energy-norm error estimates, though higher-order discretization does not improve convergence rates.

Contribution

It introduces a patch test consistent a/c scheme using higher-order finite elements and provides sharp energy-norm error estimates demonstrating its optimality.

Findings

The scheme is (quasi-)optimal among energy-based sharp-interface a/c methods.

Employing higher-order finite elements does not improve convergence rates.

The method maintains accuracy while using higher-order discretizations.

Abstract

We formulate a patch test consistent atomistic-to-continuum coupling (a/c) scheme that employs a second-order (potentially higher-order) finite element method in the material bulk. We prove a sharp error estimate in the energy-norm, which demonstrates that this scheme is (quasi-)optimal amongst energy-based sharp-interface a/c schemes that employ the Cauchy--Born continuum model. Our analysis also shows that employing a higher-order continuum discretization does not yield qualitative improvements to the rate of convergence.