  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.
