Bn. Jiang et al., LARGE-SCALE COMPUTATION OF INCOMPRESSIBLE VISCOUS-FLOW BY LEAST-SQUARES FINITE-ELEMENT METHOD, Computer methods in applied mechanics and engineering, 114(3-4), 1994, pp. 213-231
The least-square finite element method (LSFEM) based on the velocity-p
ressure-vorticity formulation is applied to large-scale/three-dimensio
nal steady incompressible Navier-Stokes problems. This method can acco
mmodate equal-order interpolations, and results in a symmetric, positi
ve definite algebraic system which can be solved effectively by simple
iterative methods. The first-order velocity-Bernoulli pressure-vortic
ity formulation for incompressible viscous flows is also tested. The f
irst-order velocity-pressure-stress formulation is not elliptic in the
ordinary sense. so we do not recommend its use for Newtonian flows. F
or three-dimensional flows, a compatibility equation, i.e., zero diver
gence of vorticity vector, is included to make the first-order system
elliptic. As a by-product of proving the ellipticity of first-order sy
stems, a rigorous mathematical technique has been developed to justify
the number of permissible boundary conditions for the Navier-Stokes e
quations. The simple substitution or Newton's method is employed to li
nearize the partial differential equations, the LSFEM is used to obtai
n discretized equations, and the system of algebraic equations is solv
ed using the Jacobi preconditioned conjugate gradient method which avo
ids formation of either element or global matrices (matrix-free) to ac
hieve high efficiency. To show the validity of this method for large-s
cale computation, we give numerical results for the 2D driven cavity p
roblem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3
D cavity is calculated at Re = 100, 400, and 1000 with 50 x 52 x 50 tr
ilinear elements. The Taylor-Gortler-like vortices are observed for Re
= 1000.