A priori and a posteriori $W^{1,\infty}$ error analysis of a QC method for complex lattices

TL;DR

This paper develops and proves new a priori and a posteriori error estimates in the $W^{1,inity}$ norm for a multiscale computational method applied to multilattice equilibria, simplifying analysis through an equivalent coarse-grained formulation.

Contribution

It introduces a novel formulation of the coarse-grained problem that simplifies the derivation of error bounds for a multiscale method in one dimension.

Findings

Error estimates are validated through numerical experiments.

The formulation simplifies the derivation of both a priori and a posteriori bounds.

The analysis applies to complex multilattice structures under external forces.

Abstract

In this paper we prove a priori and a posteriori error estimates for a multiscale numerical method for computing equilibria of multilattices under an external force. The error estimates are derived in a $W^{1,\infty}$ norm in one space dimension. One of the features of our analysis is that we establish an equivalent way of formulating the coarse-grained problem which greatly simplifies derivation of the error bounds (both, a priori and a posteriori). We illustrate our error estimates with numerical experiments.