Reduced Basis A Posteriori Error Bounds for the Instationary Stokes Equations

Abstract: We present reduced basis approximations and rigorous a posteriori error bounds for the instationary Stokes equations. We shall discuss both a method based on the standard formulation as well as a method based on a penalty approach, which combine techniques developed in our previous work on parametrized saddle point problems with current reduced basis techniques for parabolic problems. The analysis then shows how time integration affects the development of reduced basis a posteriori error bounds as well as the construction of computationally efficient reduced basis approximation spaces. To demonstrate their performance in practice, the methods are applied to a Stokes flow in a two-dimensional microchannel with a parametrized rectangular obstacle; evolution in time is induced by a time-dependent velocity profile on the inflow boundary. Numerical results illustrate (i) the rapid convergence of reduced basis approximations, (ii) the performance of a posteriori error bounds with respect to sharpness, and (iii) computational efficiency.