Reduced basis method for multi-parameter dependent steady Navier-Stokes equations: applications to natural convection in a cavity

This work focuses on the approximation of parametric steady Navier-- Stokes equations by the reduced basis method. For a particular instance of the parameters under consideration, we are able to solve the underlying partial differential equations, compute an output, and give sharp error bounds. The computations are split into an offline part, where the value of the parameters is not yet identified, but only within a range of interest, and an online part, where the problem is solved for an instance of the parameters. The offline part is expensive and is used to build a reduced basis and prepare all the ingredients -- mainly matrix-vector and scalar products, but also eigenvalue computations -- necessary for the online part, which is fast. We provide a model problem -- describing natural convection phenomena in a laterally heated cavity -- characterized by three parameters: Grashof and Prandtl numbers and the aspect ratio of the cavity. We show the feasibility and efficiency of the a posteriori error estimation by the natural norm approach considering several test cases by varying two different parameters. The gain in terms of CPU time with respect to a parallel finite element approximation is of three magnitude orders with an acceptable -- indeed less than 0.1% -- error on the selected outputs.