# Online basis construction for goal-oriented adaptivity in the Generalized Multiscale Finite Element Method

## TL;DR

This paper introduces an online enrichment framework for goal-oriented adaptivity in the generalized multiscale finite element method, enhancing the approximation of quantities of interest in heterogeneous flow problems.

## Contribution

It proposes residual-based primal-dual online basis functions and three enrichment strategies, advancing adaptive multiscale methods for complex heterogeneous media.

## Key findings

- The methods effectively improve accuracy in high-contrast heterogeneous problems.
- Numerical experiments demonstrate the efficiency of the proposed online enrichment strategies.
- The primal-dual approaches outperform standard methods in computational experiments.

## Abstract

In this research, we develop an online enrichment framework for goal-oriented adaptivity within the generalized multiscale finite element method for flow problems in heterogeneous media. The method for approximating the quantity of interest involves construction of residual-based primal and dual basis functions used to enrich the multiscale space at each stage of the adaptive algorithm. Three different online enrichment strategies based on the primal-dual online basis construction are proposed: standard, primal-dual combined and primal-dual product based. Numerical experiments are performed to illustrate the efficiency of the proposed methods for high-contrast heterogeneous problems.