# Goal-oriented error estimation and adaptivity in MsFEM computations

## TL;DR

This paper presents a goal-oriented adaptive error estimation strategy for MsFEM that combines Constitutive Relation Error with an adjoint problem to improve accuracy and efficiency in multiscale simulations.

## Contribution

It introduces a novel goal-oriented approach coupling CRE with adjoint problems for adaptive MsFEM error control and proposes a local enrichment technique for better bounds.

## Key findings

- Error estimates are highly accurate and reliable.
- The adaptive procedure significantly reduces computational costs.
- The method performs well across various numerical test cases.

## Abstract

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.