A Reduced Basis Method for steady Stokes flow
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In light of the increasing demand for many-query and real-time PDE solutions, reduced basis methods stand as a promising technique for developing solvers with the desired performance. In this thesis, we present the theory to implement both a finite element method and a proper orthogonal decomposition (POD) Galerkin reduced basis method. Numerical analysis is done for the parametrized steady Stokes equations. For flow around a NACA airfoil, we find that only a small number of reduced basis functions Nu and Np for velocity and pressure, respectively, are needed to obtain a reduced solution of sufficient accuracy. Reduced basis methods based on inf-sup stable finite element solutions do not generally inherit the inf-sup stability of the underlying finite element method. For the reduced basis method, we find a region of different values Nu and Np, where the reduced solutions are stable. The reduced basis method gives a reduction in degrees of freedom of 1500:1 and 360:1 for the reduced velocity and the reduced pressure, respectively. This reduction gives an attractive speedup of order O(10 N5) compared to the finite element method. In order to expand the stable region of different values Nu and Np, a reduced basis method with supremizer stabilization is implemented. Enriching the reduced velocity space by supremizers do in fact expand the stable region, but it introduces additional degrees of freedom to the performance-critical online stage, slowing it down.