An Extrapolated Crank Nicholson VMS-POD Method for Darcy Brinkman Equations

Abstract

This study presents a variational multiscale (VMS) reduced order model (ROM) based on proper orthogonal decomposition (POD) for the Darcy Brinkman equations. The idea of POD consists of extracting the dominant features of a data set, which are naturally assumed to represent Galerkin finite element solution of a partial differential equation. In this way, POD reduces the complexity of systems. Despite the widespread use of POD, it can perform quite poorly without any numerical stabilization. In this work, to obtain POD solution of the Darcy Brinkman system with double convection, we consider a projection-based VMS method as a numerical stabilization. The proposed scheme uses VMS type stabilization in POD for the velocity, temperature and concentration. For the fully discretization of the system, the finite element method is utilized for space variables and Crank Nicholson time discretization for time variables. The nonlinear term is treated with an extrapolated Crank Nicholson method. The numerical analysis for the VMS-POD is carried out and optimal error estimates are proved for each fluid variable. Numerical studies are performed to verify the theoretical findings.