Bivariate spline method for numerical solution of steady state Navier-Stokes equations over polygons in stream function formulation

We use the bivariate spline finite elements to numerically solve the steady state Navier-Stokes equations. The bivariate spline finite element space w e use in this article is the space of splines of smoothness tau and degree 3 tau over triangulated quadrangulations. The stream function formulation f or the steady state Navier-Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth-order equation, and Newton's i terative method is then used to solve the resulting nonlinear system. We sh ow the existence and uniqueness of the weak solution in H-2(Omega) Of the n onlinear fourth-order problem and give an estimate of how fast the numerica l solution converges to the weak solution. The Galerkin method with C-1 cub ic splines is implemented in MATLAB. Our numerical experiments show that th e method is effective and efficient. (C) 2000 John Wiley & Sons, Inc.