Parallel numerical methods for incompressible flows

Abstract

The simulation of incompressible flows consists of an important part of computational fluid dynamics. Based on finite element discretization, this paper develops several parallel numerical methods for the incompressible flows governed by the Navier-Stokes (N-S) equations, which can be classified into two classes: one is based on two-grid discretization, where the fully nonlinear N-S equations are first solved on a coarse grid, and the correction is then calculated on a fine grid by solving a linear residual problem in a parallel manner; the other is based on a novel fully overlapping domain decomposition technique, where each processor computes a local finite element solution in its own sub-domain using a global multiscale mesh that is locally refined around its own sub-domain. These parallel numerical methods are easy to implement. They have low communication complexity, and can yield a finite element solution with the same order of convergence rate as the standard finite element methods. Theoretical analysis and numerical tests demonstrated the high efficiency of the studied methods.