Workshop on Numerical Methods and Analysis in CFD - Abstract
Novo, Julia
We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations. Our aim is to get error bounds with constants independent of inverse powers of the viscosity parameter. This type of error bounds are called robust. In the case of small viscosity coefficients and coarse grids, only robust estimates provide useful information about the behavior of a numerical method on coarse grids. To this end, we compute the snapshots with a full order stabilized method (FOM). We also add stabilization to the POD method. We study a case in which non inf-sup stable elements are used for the FOM and a case in which inf-sup stable elements are used. In the last case to approximate the pressure we use a supremizer pressure recovery method.
We show that in case we have some a priori information about the velocity, a POD data assimilation algorithm converges to the true solution exponentially fast improving the accuracy of the standard POD method.
In practical simulations one can apply some given software to compute the snapshots. It could then be the case that a different discretization for the nonlinear term is used in the FOM and the POD methods. We analyze the influence of using different discretizations for the nonlinear term. Finally, we also analyze the influence of including snapshots that approach the velocity time derivative. We study the differences between projecting onto L2 and H1 and prove pointwise in time error bounds in both cases.