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The construction of local reduced-order models via multiscale basis functionshas been an area of active research. In this paper, we propose onlinemultiscale basis functions which are constructed using the offline space andthe current residual. Online multiscale basis functions are constructedadaptively in some selected regions based on our error indicators. We derive anerror estimator which shows that one needs to have an offline space withcertain properties to guarantee that additional online multiscale basisfunction will decrease the error. This error decrease is independent ofphysical parameters, such as the contrast and multiple scales in the problem.The offline spaces are constructed using Generalized Multiscale Finite ElementMethods (GMsFEM). We show that if one chooses a sufficient number of offlinebasis functions, one can guarantee that additional online multiscale basisfunctions will reduce the error independent of contrast. We note that theconstruction of online basis functions is motivated by the fact that theoffline space construction does not take into account distant effects. Usingthe residual information, we can incorporate the distant information providedthe offline approximation satisfies certain properties. In the paper,theoretical and numerical results are presented. Our numerical results showthat if the offline space is sufficiently large (in terms of the dimension)such that the coarse space contains all multiscale spectral basis functionsthat correspond to small eigenvalues, then the error reduction by adding onlinemultiscale basis function is independent of the contrast. We discuss variousways computing online multiscale basis functions which include a use of smalldimensional offline spaces.

Residual-driven online Generalized Multiscale Finite Element Methods