Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

TL;DR

This paper analyzes the stability and error of the force-based quasicontinuum approximation, demonstrating its high accuracy and providing theoretical justification for its efficiency in coupling atomistic and continuum models.

Contribution

The paper proves the stability properties of the force-based quasicontinuum method and derives optimal error estimates, addressing the challenge of non-positivity of the linearized operator.

Findings

The linearized quasicontinuum operator is typically not positive definite.

No uniform inf-sup stability holds for discrete $W^{1,p}$-$W^{1,q}$ duality pairing with 1/p+1/q=1.

An inf-sup stability condition for discrete $W^{1,inity}$-$W^{1,1}$ pairing leads to optimal error estimates.

Abstract

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are an exciting development in computational physics. For example, the force-based quasicontinuum approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we analyze the stability of the force-based quasicontinuum approximation. We then use our stability result to obtain an optimal order error analysis of this coupling method that provides theoretical justification for the high accuracy of the force-based quasicontinuum approximation -- the computational efficiency of continuum modeling can be utilized without the loss of significant accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact (which we prove) that the linearized quasicontinuum operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the $W^{1,p}$-$W^{1,q}$ "duality pairing" with $1/p+1/q=1$, if $1 \leq p < \infty$. We must therefore derive an inf-sup stability condition for a discrete version of the $W^{1,\infty}$-$W^{1,1}$ "duality pairing" which then leads to optimal order error estimates in a discrete $W^{1,\infty}$-norm.