Colloquia
Chi-Wang Shu Abstract
MATH COLLOQUIUM
“Discontinuous Galerkin Finite Element Method: Survey and Recent Development”
Chi-Wang Shu
Division of Applied Mathematics
Brown University
Friday October 27, 2006
4:00PM in 117 Hayes-Healy Center
Abstract:
In this talk we will first give an introduction to the discontinuous Galerkin (DG) method, which is a finite element methods suitable for solving convection dominated partial differential equations (PDEs). This method has gained a lot of popularity in recent years because of its nice mathematical properties in stability and convergence, its flexibility to many different PDEs from applications, and its efficiency for adaptivity and parallel implementation. In the second part of the talk we will describe a recently developed DG method to solve the Hamilton-Jacobi (HJ) equations. Unlike the DG method previously available in the literature, which applies the discontinuous Galerkin framework on the conservation law system satisfied by the derivatives of the solution, this new method applies directly to the solution of the HJ equations. For the linear case, this method is equivalent to the traditional DG method for conservation laws with source terms. Thus, stability and error estimates are straightforward. For the nonlinear convex Hamiltonians, numerical experiments demonstrate that the method is stable and provides the optimal ($k+1$)-th order of accuracy for smooth solutions when using piecewise $k$-th degree polynomials. Singularities in derivatives can also be resolved sharply if the entropy condition is not violated. Special treatment is needed for the entropy violating cases. Both one and two dimensional numerical results are provided to demonstrate the good qualities of the scheme. The second part of the talk is joint work with Yingda Cheng.
