Goal-oriented adaptivity for GMsFEM

TL;DR

This paper introduces two goal-oriented adaptive strategies for the GMsFEM framework, improving the efficiency of error reduction in specific quantities of interest for multiscale problems.

Contribution

It develops and compares two novel goal-oriented adaptive error estimators within GMsFEM, enhancing targeted error reduction in multiscale finite element simulations.

Findings

The dual weighted residual method outperforms residual-based strategies in error reduction.

Both strategies effectively adaptively enrich the basis functions in high-contrast multiscale problems.

The proposed methods demonstrate improved efficiency in reducing goal functional errors.

Abstract

In this paper we develop two goal-oriented adaptive strategies for a posteriori error estimation within the generalized multiscale finite element framework. In this methodology, one seeks to determine the number of multiscale basis functions adaptively for each coarse region to efficiently reduce the error in the goal functional. Our first error estimator uses a residual based strategy where local indicators on each coarse neighborhood are the product of local indicators for the primal and dual problems, respectively. In the second approach, viewed as the multiscale extension of the dual weighted residual method (DWR), the error indicators are computed as the pairing of the local H^{-1} residual of the primal problem weighed by a projection into the primal space of the H_0^1 dual solution from an enriched space, over each coarse neighborhood. In both of these strategies, the goal-oriented indicators are then used in place of a standard residual-based indicator to mark coarse neighborhoods of the mesh for further enrichment in the form of additional multiscale basis functions. The method is demonstrated on high-contrast problems with heterogeneous multiscale coefficients, and is seen to outperform the standard residual based strategy with respect to efficient reduction of error in the goal function.