  In this paper, we develop and analyze an adaptive multiscale approach for
heterogeneous problems in perforated domains. In many applications, these
problems have a multiscale nature arising because of the perforations, their
geometries, the sizes of the perforations, and configurations. In this paper,
we present a general offline/online procedure, which can adequately and
adaptively represent the local degrees of freedom and derive appropriate
coarse-grid equations. The main contributions of this paper are (1) the
rigorous analysis of the offline approach (2) the development of the online
procedures and their analysis (3) the development of adaptive strategies. We
present an online procedure, which allows adaptively incorporating global
information and is important for a fast convergence when combined with the
adaptivity. Our methodology allows adding and guides constructing new online
multiscale basis functions adaptively in appropriate regions. We present the
convergence analysis of the online adaptive enrichment algorithm for the Stokes
system. In particular, we show that the online procedure has a rapid
convergence with a rate related to the number of offline basis functions, and
one can obtain fast convergence by a sufficient number of offline basis
functions, which are computed in the offline stage. To illustrate the
performance of our method, we present numerical results with both small and
large perforations. We see that only a few (1 or 2) online iterations can
significantly improve the offline solution.
