Stable multiscale discretizations for saddle point problems and preconditioning

Stability for discretizations of saddle point problems is typically the result of satisfying the discrete Babuska-Brezzi condition. As a consequence a number of natural discretizations are ruled out and some effort is required to provide stable ones. Therefore ideas for circumventing the Babuska-Brezzi condition are interesting. Here an ansatz presented in a series of papers by Hughes et al. is described and investigated in the framework of multiscale discretizations. In particular discretizations for appending boundary conditions by Lagrange multipliers and the stationary Stokes problem are considered. Sufficient conditions for their stability are given and diagonal preconditioners which give uniformly bounded condition numbers are proposed.