LARGE-SCALE COMPUTATION OF INCOMPRESSIBLE VISCOUS-FLOW BY LEAST-SQUARES FINITE-ELEMENT METHOD

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.