Parallel finite element multigrid solver for incompressible navier-stokes equations

Abstract

Fluid flows are generally modelled using the Navier-Stokes Equations (NSE). The model can also be applied in the simulation of various practical applications such as weather, ocean currents, air flow over aircrafts, etc. As analytical solutions to these equations are hard to come by, numerical techniques such as finite element methods (FEM), finite volume methods (FVM), finite difference methods (FDM), etc., are used to solve these problems. This involves solving a system of algebraic equations, where the size of the problem increases with the desired level of accuracy in the solution. FEM is the numerical approach adopted in this work. An efficient solution to these systems of equations requires the realization of robust iterative techniques. Initially, in this report, the saddle-point problem that arises in an NSE system is discussed. An iterative technique based on a Krylov subspace method, GMRES, is adopted. The coupling of this technique with a multigrid method is considered, in order to improve the rate of convergence. A study is performed, through a set of simulations, on the rate of convergence over the variants of multigrid methods considered. Based on this, the scalability of the chosen algorithms is observed through the proposed parallel algorithm (a flat MPI implementation). Ascertaining the complexity involved in the iterative method adopted, we finally consider the performance of the technique on two test problems, a steady-state problem and a time-dependent problem, by scaling their size.