Numerical Solution of PDE based on Bivariate Spline Functions
摘要：Bivariate spline functions are piecewise polynomial functions over triangulation. I shall explain a constrained minimization approach to use bivariate splines for numerical solution to partial differential equations. Three different PDEs will be used to demonstrate the effectiveness and efficiency of bivariate spline methods. (1) second order elliptic PDE in non-divergence form, (2) Navier-Stokes equations in stream function formulation, and (3) Helmholtz equation with large wave number. Mainly, I will explain the usefulness of smooth constraints. For example, bivariate splines enable us to use the stream function formulation which leads to numerical solution of one stream function instead of two components of velocity and one pressure function. For another example, when using the potential function formulation of Maxwell equations, we need to solve Helmholtz equation. As the electric and/or magnetic fields are very smooth, smooth spline functions are good choices to approximate these fields. Our numerical results show that we are able to solve Helmholtz equation with wave number 500 or more over my laptop computer. Some theoretical study on the existence, uniqueness and stability of spline solutions will also be explained.