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