Fast finite element solver for incompressible Navier-Stokes equation by parallel Gram-Schmidt process based GMRES and HSS

Abstract

After finite element discretization of incompressible Navier-Stokes equation, a sparse linear system is obtained in every iteration, and GMRES provides an efficient approach to solve this system. However, if the size of the original PDE model is large, the solution speed of the incompressible Navier-Stokes equation is still slow. Since the linear system from the incompressible Navier-Stokes equation is saddle point problem, we find that large portion of computational efforts for solving the linear system is occupied by the vector projection in GMRES. In this paper, by our parallel Gram-Schmidt process based GMRES and newly developed preconditioner Hermitian/Skew-Hermitian Separation (HSS), we develop a fast solver HSS-pGMRES for the saddle point problem from incompressible Navier-Stokes Equation. Theoretical analysis shows that, HSS-pGMRES decreases the computational complexity of finite element solver for incompressible Navier-Stokes equation from O(m²n) to O(mn), where m is the grid size. Computational experiments show that, the fast solver HSS-pGMRES significantly increases the solution speed for the saddle point problem of incompressible Navier-Stokes equation than the conventional solvers.