Certified Reduced Basis Methods for Parametrized Saddle Point Problems

We present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems. First, we develop new a posteriori error estimates that, unlike earlier approaches, provide upper bounds for the errors in the approximations of the primal variable and the Lagrange multiplier separately. The proposed method is an application of Brezzi's theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods. Second, based on an analysis of Brezzi's theory, we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost. Finally, we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and, compared to earlier approaches, computationally much more efficient. The method is applied to a Stokes flow problem in a two-dimensional channel with a parametrized rectangular obstacle. Numerical results demonstrate: (i) the need to appropriately enrich the approximation space for the primal variable; (ii) the significant effects of different enrichment strategies; (iii) the rapid convergence of (stable) reduced basis approximations; and (iv) the advantages of the proposed error bounds with respect to sharpness and computational cost.