Finite Element Methods for Eigenvalue Problems
August 11 - August 15, 2024
Lecturer: Daniele Boffi (King Abdullah University of Science and Technology)
Classroom: USTC Teaching Building
Abstract
Eigenvalue problems are important in many applications areas, such as electromagnetics, structural mechanics, quantum theory, and control. Important examples are the Laplace, Maxwell and Schrodinger eigenvalue problems. Finite element methods provide a well-established numerical method to solve eigenvalue problems and are supported by an extensive mathematical theory. In this short course, we will discuss several finite element methods for the computation of eigenvalue problems. Special attention will be given to the theoretical analysis of finite element discretizations for eigenvalue problems. In particular, the convergence theories from Descloux, Nassif and Rapaz, and from Babuska and Osborn will be discussed and used to analyze the convergence of finite element discretizations for eigenvalue problems.
Reference: D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), pp. 1–120.
Discontinuous Galerkin Finite Element Methods
August 17 - August 21, 2024
Lecturer: Professor J.J.W. van der Vegt (University of Twente)
Classroom: USTC Teaching Building
Abstract
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: D.A. Di Pietro, A. Ern, Mathematical aspects of discontinuous Galerkin methods, Springer 2012, ISBN 978-3-642-22979-4.
Introduction to Multigrid Methods
August 23 - August 27, 2024
Classroom: USTC Teaching Building
Lecturer: Professor J.J.W. van der Vegt (University of Twente)
Abstract
Multigrid methods provide very efficient iterative methods for the solution of large systems of (non)linear algebraic equations that result for instance from the discretization of partial differential equations. In a multigrid method several coarsened approximations of the algebraic system and well-designed smoothers are used to accelerate the convergence of the iterative method. This can result in very efficient iterative methods, but if one wants to develop new multigrid algorithms or understand the performance of existing algorithms, then multilevel analysis is indispensible. In this class an outline of basic multigrid and iterative methods will be given and mathematical techniques to understand and predict their performance will be discussed. No prior knowledge of multigrid or iterative methods will be required. After this class you should be able to use basic iterative and multigrid methods, analyze and (approximately) predict multigrid performance using multilevel analysis and apply these techniques to improve and test multigrid algorithms. The main applications will be from numerical discretizations of partial differential equations.
Reference: U. Trottenberg, C.W. Oosterlee, A. Schüller, Multigrid, Academic Press, ISBN 0-12-701070-X, 2001.
