Fast divergence-conforming reduced basis methods for stationary and transient flow problems

Reduced basis methods (RB methods or RBMs) form one of the most promising techniques to deliver numerical solutions of parametrized PDEs in real-time with reasonable accuracy [1]. For the Navier-Stokes equation, RBMs based on stable velocity-pressure spaces do not generally inherit the stability of the high-fdelity method. Common techniques for working around this problem (e.g. [2]) have the effect of deteriorating the performance of the RBM in the performance-critical online stage. We show how divergence-free reduced formulations eliminates this problem, producing RBMs that are faster by an order of magnitude or more in the online stage. This is most easily achieved using divergence-conforming compatible B-spline bases, using a transformation that can maintain the divergence-free property under variable geometries. See [3] for more details. We also demonstrate the flexibility of RBMs for non-stationary flow problems using a problem with two stages: an initial, finite transient stage where the flow pattern settles from the initial data, followed by a terminal and infinite oscillatory stage characterized by vortex shedding. We show how an RBM whose data is only sourced from the terminal stage nevertheless can produce solutions that pass through the initial stage without critical problems (e.g. crashing, diverging or blowing up).