Recently, Dahlke, Dahmen, Hochmuth and Schneider introduced a posteriori error estimates for Galerkin schemes for elliptic operator equations using stable multiscale bases. Based on this, they were able to construct adaptive space refinement strategies that could be proven to guarantee convergence.
Since their construction relies on positive definite operators, saddle point problems are ruled out. In this talk, we present some ideas and results for constructing convergent adaptive multiscale methods for saddle point problems. Moreover, we give a general criteria for ensuring the Ladyshenskaja-Babushka-Brezzi-condition for adaptively refined multiscale space. The general theory will be specified for a suitable multiscale wavelet discretization for the Stokes problem introduced by Dahmen, Kunoth and Urban.