# 09-04【J.J.W. van der Vegt 】五教5206 研究生教育创新计划高水平学术前沿系列报告

Discontinuous Galerkin Finite Element Methods

Lecturer: Professor J.J.W. van der Vegt (University of Twente)

Classroom: USTC Teaching Building 5206

Zoom Link: Meeting ID: 850 2232 8209 Passcode: 209408

Time:

15:45-17:45, September 4, 2023 (Beijing time)

15:45-17:45, September 5, 2023 (Beijing time)

15:45-17:45, September 6, 2023 (Beijing time)

15:45-17:45, September 7, 2023 (Beijing time)

15:45-17:45, September 8, 2023 (Beijing time)

Discontinuous Galerkin (DG) finite element methods are nowadays one of the main numerical techniques to solve partial differential equations. The key feature of DG methods is that discontinuities are allowed in the test and trial spaces at element faces. This provides great flexibility to build higher order accurate, solution adaptive numerical discretizations, using local mesh refinement and the local adjustment of the polynomial order of the test and trial spaces. DG methods also allow for efficient parallel computing due to minimal element connectivity, and provide element wise conservative numerical discretizations, which is especially important for hyperbolic partial differential equations.

In these lectures we will discuss the basic principles of discontinuous Galerkin methods for several important classes of partial differential equations (hyperbolic, elliptic). Special attention will be given to the mathematical aspects of DG methods, such as stability, convergence, and accuracy by studying and analyzing several model problems in detail. If time permits also, the extension to space-time DG discretizations, which use discontinuous test and trial functions in space and in time, will be considered for the advection equation and the incompressible Navier-Stokes equations. As pre-existing knowledge for this course familiarity with standard conforming finite element methods and their analysis is assumed.

References:

1.D.A. Di Pietro, A. Ern, Mathematical aspects of discontinuous Galerkin methods, Springer 2012, ISBN 978-3-642-22979-4.

2.A. Ern, J.-L. Guermond, Theory and practice of finite elements, Sringer, 2004, ISBN 0-387-20574-8.

3.S. Rhebergen, B. Cockburn and J.J.W. van der Vegt, A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, Journal of Computational Physics, Vol. 233, pp. 339-358, 2013.

4.J. Sudirham, J.J.W. van der Vegt and R.M.J. van Damme, Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Applied Numerical Mathematics, Vol. 56, Issue 12, pp. 1491-1518, 2006.