Bivariate Spline Method for Numerical Solution of Navier-Stokes Equations in Stream Function Formulation

We use the bivariate spline functions to solve the 2D NavierStokes equations numerically. The bivariate spline space we use in this paper is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier-Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth order equation, Crank-Nicholson's method is applied to discretize the time variable, 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(Ω) of the nonlinear fourth order problem corresponding to the steady-state Navier-Stokes equations. We also show the existence and uniqueness of the weak solution in L₂(0^,T;H ²(Ω)) of the nonlinear fourth order problem associated with the time-dependent Navier-Stokes equations. We 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 bivariate spline method is effective and efficient.