Finite Element Methods for PDEs

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Description

The aim of this module is to present aspects of the theory of finite element methods (conforming, non-conforming and discontinuous Galerkin methods) approximating solutions to elliptic, parabolic and hyperbolic PDE problems. Finite Element Methods (FEM, for short) are abundant in modern numerical approximation of PDEs and also offer a highly developed theory for error analysis, but also for practical error estimation. The module will cover the basic theory for elliptic problems (best approximation, Galerkin orthogonality, a priori error analysis, a posteriori error analysis), before moving to parabolic problems whereby the corresponding results will be presented. Finally, we will give a relatively detailed account of a non-conforming class of finite elements, the discontinuous Galerkin methods, which have been rather popular in recent years for the numerical approximation of hyperbolic problems. The course will contain both a theoretical and a computational component.

Assessment: We will follow the standard mode of assesment with two take home Assignments/problem sheets, one at the end of Lecture 5 and one at the end of Lecture 10. Students taking the assesment will have two weeks to submit their work. Some questions will involve programming.