Large-scale computation of incompressible viscous flow by least-squares finite element method

Abstract

The least-square finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in a symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli pressure-vorticity formulation for incompressible viscous flows is also tested. The first-order velocity-pressure-stress formulation is not elliptic in the ordinary sense, so we do not recommend its use for Newtonian flows. For three-dimensional flows, a compatibility equation, i.e., zero divergence of vorticity vector, is included to make the first-order system elliptic. As a by-product of proving the ellipticity of first-order systems, a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the Navier-Stokes equations. The simple substitution or Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this method for large-scale computation, we give numerical results for the 2D driven cavity problem at Re = 10000 with 408 × 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1000 with 50 × 52 × 50 trilinear elements. The Taylor-Görtler-like vortices are observed for Re = 1000.