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 we use in this article is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. 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 H²(Ω) 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 C¹ cubic splines is implemented in MATLAB. Our numerical experiments show that the method is effective and efficient. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 147–183, 2000