Finite element solution of Navier-Stokes equations using Krylov Subspace methods

Abstract

Paper presented to the 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Florida, 14-16 July 2014.

This paper presents an analysis of different Krylov subspace methods used to solve non-symmetric, non-linear matrix equations obtained after finite element discretization of Navier-Stokes equations. Mixed velocity-pressure formulation, also known as the primitive variable formulation, which consists of two momentum equations and a zero-velocity-divergence constraint representing mass conservation is applied (2D problem). Matrix equations obtained are solved using following Krylov subspace methods: Least Squares Conjugate Gradient, Bi-Conjugate Gradient, Conjugate Gradient Squared, Bi-Conjugate Gradient Stabilized and Bi-Conjugate Gradient Stabilized (ell). Also, a comparison between these iterative methods and direct Gaussian elimination was made. Findings presented in this paper show that Least Square Conjugate Gradient method with its stability, which has been abandoned by many authors as the slowest, has became very fast when the 'element-by-element' method is applied. Lid-driven cavity is chosen to be the test case, and results obtained for two different Reynolds numbers; Re = 400 and Re = 1000, and for two discretization schemes (10x10 and 48x48; uniform and non-uniform) are compared with the results presented in literature.