There is no pointwise consistent quasicontinuum energy

TL;DR

This paper proves fundamental limitations on the pointwise consistency of finite-range quasicontinuum coupling methods, establishing that achieving o(1)-consistency at the interface is impossible, and provides convergence bounds in 1D.

Contribution

It demonstrates a theoretical impossibility result for finite-range quasicontinuum couplings and derives convergence bounds in discrete norms.

Findings

Pointwise o(1)-consistency cannot be achieved with finite-range coupling.

Provides an upper bound on convergence order in 1D discrete norms.

Shows fundamental limitations in quasicontinuum energy coupling methods.

Abstract

Much work has gone into the construction of quasicontinuum energies that reduce the coupling error along the interface between atomistic and continuum regions. The largest consistency errors are typically pointwise $O(\frac{1}{\eps})$ errors, and in some cases this has been reduced to pointwise O(1) errors. In this paper we show that one cannot create a coupling method using a finite-range coupling interface that has o(1)-consistency in the interface, and we use this to give an upper bound on the order of convergence in discrete $w^{1,p}$-norms in 1D.