Non-linear Petrov–Galerkin methods for reduced order modelling of the Navier–Stokes equations using a mixed finite element pair

Abstract

A new nonlinear Petrov–Galerkin approach has been developed for proper orthogonal decomposition (POD) reduced order modelling (ROM) of the Navier–Stokes equations. The new method is based on the use of the cosine rule between the advection direction in Cartesian space–time and the direction of the gradient of the solution. A finite element pair, P1DGP2, which has good balance preserving properties is used here, consisting of a mix of discontinuous (for velocity components) and continuous (for pressure) basis functions. The contribution of the present paper lies in applying this new non-linear Petrov–Galerkin method to the reduced order Navier–Stokes equations, and thus improving the stability of ROM results without tuning parameters. The results of numerical tests are presented for a wind driven 2D gyre and the flow past a cylinder, which are simulated using the unstructured mesh finite element CFD model in order to illustrate the numerical performance of the method. The numerical results obtained show that the newly proposed POD Petrov–Galerkin method can provide more accurate and stable results than the POD Bubnov–Galerkin method.