Bivariate Spline Method for Numerical Solution of Time Evolution Navier-Stokes Equations over Polygons in Stream Function Formulation

We use a bivariate spline method to solve the time evolution Navier-Stokes equations numerically. The bivariate splines we use in this article are in the spline space 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 discretize the space variables of the 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 L₂(0, T; H²(Ω)) ∩ L _∞(0, T; H¹(Ω)) of the 2D nonlinear fourth-order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier-Stokes equations numerically. Our numerical experiments show that the method is effective and efficient.