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Reduced-basis methods (RB methods or RBMs) form one of the most promisingtechniques to deliver numerical solutions of parametrized PDEs in real-timeperformance with reasonable accuracy. For incompressible flow problems, RBMsbased on LBB stable velocity-pressure spaces do not generally inherit thestability of the underlying high-fidelity model and, instead, additionalstabilization techniques must be introduced. One way of bypassing the loss ofLBB stability in the RBM is to inflate the velocity space with supremizermodes. This however deteriorates the performance of the RBM in theperformance-critical online stage, as additional DOFs must be introduced toretain stability, while these DOFs do not effectively contribute to accuracy ofthe RB approximation. In this work we consider a velocity-only RBapproximation, exploiting a solenoidal velocity basis. The solenoidal reducedbasis emerges directly from the high-fidelity velocity solutions in the offlinestage. By means of Piola transforms, the solenoidality of the velocity space isretained under geometric transformations, making the proposed RB methodsuitable also for the investigation of geometric parameters. To ensure exactsolenoidality of the high-fidelity velocity solutions that constitute the RB,we consider approximations based on divergence-conforming compatible B-splines.We show that the velocity-only RB method leads to a significant improvement incomputational efficiency in the online stage, and that the pressure solutioncan be recovered a posteriori at negligible extra cost. We illustrate thesolenoidal RB approach by modeling steady two-dimensional Navier-Stokes flowaround a NACA0015 airfoil at various angles of attack.

Fast divergence-conforming reduced basis methods for steady Navier-Stokes flow