Bivariate Spline Method for Numerical Solution

We use the bivariate spline method to solve the steady state Navier-Stokes equations numerically. The bivariate spline we use in this paper is the space of splines of smoothness r and degree 3r over triangulated quadrangu-lations. The stream function formulation for the steady state Navier-Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth order equation, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in H 2 (() of the nonlinear fourth order problem and give an estimate of how fast the numerical solution converges to the weak solution. The Galerkin method with C 1 cubic splines is implemented in MATLAB. Our numerical experiments show that the method is eeective and eecient.