Element-by-Element Parallel Computation of Incompressible Navier--Stokes Equations in Three Dimensions

Abstract

Development of a stable finite element model for solving steady incompressible viscous fluid flows in three dimensions is the main theme of the present study. For stability reasons, weighting functions are designed in favor of field variables on the upstream side. For accuracy reasons, it is required that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed. In the steady-state analysis of Navier--Stokes equations, we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the Ladyzhenskaya--Babuska--Brezzi (LBB) stability condition. To resolve difficulties arising from asymmetry and indefiniteness in the resulting large-size matrix equations, we abandon the elimination-like solution solver because the storage demand exceeds the ability of our hardware to solve for three-dimensional problems. A modern iteration solver, known as the biconjugate gradiant stabilized (BICGSTAB) solution solver, is thus implemented in an element-by-element fashion to effectively alleviate the problem. For performance reasons, the finite element code developed here should be implemented in a hardware environment which is suited to the use of an iterative solver. To this end, our analysis is implemented in shared memory parallel architectures, CRAY C-90 and J-90. We benchmark the parallel computing performance through a lid-driven cavity flow problem and a problem amenable to analytic solution.