Reduced Basis A Posteriori Error Bounds for the Stokes Equations in Parametrized Domains: A Penalty Approach

We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method is built upon a penalty formulation for saddle point problems and provides error bounds admitting affine geometric variations with relative ease. Essential features are: i) dimension reduction; ii) stable, good approximation of the pressure; iii) optimal and numerically stable approximations identified by an efficient Greedy sampling procedure; iv) certainty, through rigorous a posteriori bounds for the error in the reduced basis approximation; and v) efficiency, through an offline‐online computational strategy. The method is applied to a flow problem in a two‐dimensional channel with a parametrized rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter.