Bivariate splines for fluid flows

We discuss numerical approximations of the 2D steady-state Navier-Stokes equations in stream function formulation using bivariate splines of arbitrary degree d and arbitrary smoothness r with r<d. We derive the discrete Navier-Stokes equations in terms of B-coefficients of bivariate splines over a triangulation, with curved boundary edges, of any given domain. Smoothness conditions and boundary conditions are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and our numerical experiments show that our method is effective. Numerical simulations of several fluid flows will be included to demonstrate the effectiveness of the bivariate spline method.