Element-by-element parallel computation of incompressible Navier-Stokes equations in three dimensions

Development of a stable finite element model for solving steady incompressi ble viscous fluid flows in three dimensions is the main theme of the presen t study. For stability reasons, weighting functions are designed in favor o f field variables on the upstream side. For accuracy reasons, it is require d that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed. In the steady-state anal ysis of Navier-Stokes equations, we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the Ladyzhenskay a-Babuska-Brezzi (LBB) stability condition. To resolve difficulties arising from asymmetry and indefiniteness in the resulting large-size matrix equat ions, we abandon the elimination-like solution solver because the storage d emand exceeds the ability of our hardware to solve for three-dimensional pr oblems. A modern iteration solver, known as the biconjugate gradiant stabil ized (BICGSTAB) solution solver, is thus implemented in an element-by-eleme nt fashion to effectively alleviate the problem. For performance reasons, t he finite element code developed here should be implemented in a hardware e nvironment 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.