Reduced-Basis Approximation and A Posteriori Error Estimation for Saddle-Point Problems

The Stokes problem is a saddle point problem. A stable discretization enables applying classical results from the Brezzi and Babuška Theories in order to compute a posteriori upper bounds for the approximation error. The error can be the norm of the difference between the exact solution and the Galerkin Finite Elements discretization or in the case of this work between the FE and the Reduced Basis approximation. The RB approximation allows a quick computation of the problem solution accordingly to the variation of geometric parameters. A correct implementation of the RB method requires a verification of the stability: in particular the role of the supremizer operator is analyzed. A rigorous error bound is required for enhancing the application of the RB method and the motivation for a research of a Brezzi-based error bound relies on the possibility of efficiently computing lower or upper bounds for the involved constants. The classical Babuška approach for variational problems provides the error bound for the velocity and the pressure jointly, whereas the Brezzi approach deals with the two quantities alone. This does not necessarily mean a sharper and improved bound. Numerical tests show that for reasonable (O(1)) values of the ratio of the dual norms of the momentum and continuity equations residuals, the Brezzi error is sharper (by a factor 10) than the Babuška error only for the velocity. In a parametrized domain, e.g. a channel with a rectangular obstacle, the behaviour of the Brezzi error bound gets worse as the parameters assume values which are far from the reference ones.