Warning: Your browser doesn't support all of the features in this Web site. Please view our accessibility page for more details.

Search Web Site [?]WHAT IS THIS?
Performs a not very reliable search of the Web site and displays results in a new window.

This is the archived "Mathematics of the FEM" web page from 2011/2012.
If you are looking for the page for the current academic year, please see nuigalway.blackboard.com, or contact the module's lecturer.

Grad Course: Mathematics for the Finite Element Method

Course Information

Lecturer:	Dr Niall Madden
Lectures:	24 x 1 hour lectures in Semester 2, plus labs.
Schedule:	Lectures Wednesday 12-1 and Thursday 11-12 in C219 (Aras de Brun). Labs will be scheduled shortly.
Credit	Typically 5 ECTS for students in structured PhD programmes.
Content:
What you will learn:	Some fundamental concepts of boundary value problems, and their weak formulations. Key ideas in approximation of BVPs by finite differences and finite elements. The approximation of functions be piecewise polynomials; interpolation errors in various norms. Fundamentals of finite element methods. Best approximation and error analysis MATLAB implementation of FEMs for BVPs in 1 and 2 dimensions. Further topics (may be covered in class, labs, or part of a mini-project) Discontinuous Galerkin methods Solution of linear systems Adaptive meshing Exotic approximation spaces
Why take this course:	Many modelling problems lead to boundary value problems; obtaining useful information from the model requires that these problems be solve to "reasonable" accuracy. The mathematics of how this is done is rich and interesting. Furthermore, there are many black-box tools to help us find (approximate) solutions to BVPs, though they tend to be optimised for certain classes of problems. By understanding how these work, we are in a better position to chose which is best. By being able to implement our own versions, if even for simple test cases, gives us a way of verifying results generated by other methods.
Assessment:	Matlab programming projects and some written assignments.
Texts:	Brenner and Scott, The mathematical theory of finite element methods Gockenbach, Understanding and Implementing the Finite Element Method More to follow...
Useful references:	The Finite Element Primer by David Silvester (University of Manchester) forms an excellent starting point for acquainting yourself with core concepts. It is short (25 pages) and free. Chapter 14 of An Introduction to Numerical Analysis by Süli and Mayers contains the very basics of the Finite Element Method. Although it is intended that this course we extend well beyond its content, it is a good place to start. The set of lecture notes "Finite Element Methods for Partial Differential Equations" by Endre Süli has an accessible presentation of material related to this course. And is freely available. O. Axelsson, V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation. Full text access on campus through SIAM ebooks. Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems. Full access on campus through SIAM ebooks. The set of lecture notes "Finite Element Methods for Partial Differential Equations" by Endre Süli has an accessible presentation of material related to this course. And is freely available.
Projects:	Students taking this course for credit, as part of a structured PhD programme, are required to take part in assessment activities. Of these, the most important is a project that must be completed at the end of the semester. This will involve writing a technical report and producing a Matlab implementation of an FEM for a particular problem. Possible examples are given below. More will be added as I think of them. Google them to get more details. Discontinuous Galerkin Methods hp-FEM Adaptive meshing in 1 and 2 dimensions Stabilised methods for advection-diffusion problems
Lecture summaries:	Notes for the first two classes are extensive. The rest are very brief -- more like a check list of topics covered, than lecture notes. The Basics for FEM: model problem and weak form Discretisation More on weak forms and discretisation Error analysis Implementation (in Maple). Download the Maple implementation. Implementation (in Matlab). Download Solve1dFeLinear.m and Integrate.m 2D Implementation (in Matlab). Download FE_2D_V02.m Here is a slightly revised version of the above code. It is still quite inefficient, but this is deliberate, so as the make the operation of the code clear: Solve_2DFEV01.m and Test_2DFESolverV01.m A next version is much more efficient, though the operator is far less clear. It should work OK on a modest desktop computer taking N=2^10 (and so 10⁶ degrees of freedom). Solve_2DFEV02.m and Test_2DFESolverV02.m

School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway, University Road, Galway, Ireland.
Phone: +353 (0)91 492332 (direct) , +353 (0)91 524411 x2332 (switchboard)
Fax: +353 (0)91 494542 Email: Mary.Kelly@nuigalway.ie
This page was last updated Friday, September 04