学术报告七:Ming-Jun Lai:Numerical Solution of PDE based on Bivariate Spline Functions


报告题目:Numerical Solution of PDE based on Bivariate Spline Functions






报告人:  Ming-Jun Lai,


工作单位:Department of Mathematics, University of Georgia, Athens, GA, U.S.A.


报告人简介:Ming-Jun Lai is a full professor of Department of Mathematics, University of Georgia, Athens. He received his B. S. in Jan. 1982 from Hangzhou University, China and his Ph. D. in Aug. 1989 from Texas A&M University. His Ph. D. advisor is Professor Charles K. Chui. After his Ph. D., he became an instructor at University of Utah, Salt Lake City during 1989-1992.He started as an assistant professor at University of Georgia in 1992-1995.Then he was promoted to associate professor in 1995.He has been a full professor since 2000 at University of Georgia. He has several specialized research areas: Approximation Theory, Compressed Sensing, Mathematical Image Analysis, Multivariate Splines, Numerical Analysis, Numerical Solution of Partial Differential Equations, Wavelet and Frame Analysis. He published a monograph Spline Functions over Triangulations, 585+pages by Cambridge University Press, Cambridge, U.K. 2007 coauthored with L. L. Schumaker.


报告简介: 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.