A posteriori error estimation and adaptive strategy for the control of MsFEM computations

TL;DR

This paper develops a reliable a posteriori error estimation method for MsFEM, enabling adaptive control of accuracy and computational cost through indicators that guide refinement, compatible with offline/online strategies.

Contribution

It introduces a fully computable error estimate for MsFEM based on dual analysis and CRE, with adaptive indicators for improved accuracy-cost trade-offs.

Findings

Effective error control in MsFEM simulations demonstrated

Adaptive strategy improves computational efficiency

Compatible with offline/online MsFEM framework

Abstract

We introduce quantitative and robust tools to control the numerical accuracy in simulations performed using the Multiscale Finite Element Method (MsFEM). First, we propose a guaranteed and fully computable a posteriori error estimate for the global error measured in the energy norm. It is based on dual analysis and the Constitutive Relation Error (CRE) concept, with recovery of equilibrated fluxes from the approximate MsFEM solution. Second, the estimate is split into several indicators, associated to the various MsFEM error sources, in order to drive an adaptive procedure. The overall strategy thus enables to automatically identify an appropriate trade-off between accuracy and computational cost in the MsFEM numerical simulations. Furthermore, the strategy is compatible with the offline/online paradigm of MsFEM. The performances of our approach are demonstrated on several numerical experiments.