 Department of Mathematics 

Next: MA4021 Wavelets and Signal Processing
Up: ModuleGuide0304
Previous: MA4001 Dynamical Systems
MA4011 Finite Element Methods for Partial Differential Equations
Credits: 20 
Convenor: Dr. Ciprian Coman 
Semester: 2 (weeks 15 to 26) 
Prerequisites: 

desirable: MA2001(=MC224), MA2121(=MC243) 
Assessment: 
coursework: 40% 
2 hour exam: 60% 
Lectures: 
36 
Problem Classes: 
10 
Tutorials: 
none 
Private Study: 
104 
Labs: 
none 
Seminars: 
none 
Project: 
none 
Other: 
none 
Surgeries: 
none 
Total: 
150 
Subject Knowledge
Aims
The aim of this course is to introduce the basic theory of finite
element methods for the numerical approximation of partial
differential equations and provide a good basis
for those students who wish to pursue the study of more
advanced topics.
Learning Outcomes
Students should understand the mathematical framework of finite
element methods. In particular they should be able to calculate weak derivatives
for simple functions, formulate the weak form for a large class of elliptic
partial differential equations and understand the role played by boundary conditions
in obtaining the weak forms. They should be able to construct various approximations of
these weak problems and be able to characterise the error bounds involved.
Students should also gain some measure of familiarity with the software package MATLAB
and its use in evaluating numerical algorithms.
Methods
Class sessions, computer labs and problem classes.
Assessment
The coursework will consist of regularly assigned exercise sheets, including computer
assignments. A substantial individual project will be required and the student will
have to present the project orally to the class and the instructor.
The June examination will have 4 questions, it being (theoretically) possible to obtain full
marks by answering 3 of them.
Subject Skills
Aims
To foster and develop the ability to synthesise and build upon knowledge
acquired in the previous years.
The ability to apply the techniques developed in this course to
a large variety of partial differential equations.
Learning Outcomes
Students will gain an appreciation of
mathematical modelling and solving practical problems and in this process
they will learn how to design an implement complex solutions for specific
problems, and be able to present this information effectively in either written
or oral form. A special emphasis will be placed on simplicity combined with
depth of understanding rather than on sophistication.
Methods
Class sessions, computer lab sessions and problem classes.
Assessment
Marked problem sheets, lab reports and class tests.
Explanation of Prerequisites
A good understanding of linear space theory and
the calculus of functions of several variables.
Some knowledge of MATLAB is desirable but it is
not essential.
Course Description
Partial differential equations arise in the mathematical modelling of
many physical, chemical and biological phenomena. Indeed, they play
a crucial role in many diverse subject areas, such as fluid dynamics,
electromagnetism, materials science, astrophysics and financial
modelling, for example. Typically, the equations under
consideration are so complicated that their solution may not be
determined by purely analytical techniques; instead one has to resort
to computing numerical approximations to the unknown analytical
solution.
This course is devoted to a particular class of numerical
techniques for determining the approximate solution of partial
differential equations: finite element methods. Here,
we will provide an introduction to their mathematical theory,
with special emphasis on theoretical and practical issues such as accuracy,
reliability and implementation.
Syllabus
Review of the linear space theory. Hilbert space theory (subspaces
and projections). Riesz representation theorem; Lax Milgram lemma.
Basic facts about function spaces; spaces and Sobolev spaces.
The theory of weak solutions to elliptic boundary value problems.
Finite element methods for elliptic partial differential equations;
Galerkin orthogonality; Cèa's lemma; piecewise polynomial approximation
in Sobolev spaces; optimal error bounds in the energy norm;
AubinNitsche duality argument; variational crimes.
A posteriori error analysis by duality.
Computer implementation of FEM algorithms for simple problems.
Reading list
Recommended:
K. Eriksson, D. Estep, P. Hansbo and C. Johnson,
Computational Partial Differential Equations,
Cambridge University Press, 1996.
C. Johnson,
Numerical Solution of Partial Differential Equations by the
Finite Element Method,
Cambridge University Press, 1990.
W. Cheney, D. Kincaid,
Numerical Mathematics and Computing (fifth edition),
Brooks Cole, 2003.
P.G. Ciarlet,
The Finite Element Method for Elliptic Problems,
NorthHolland, 1978.
S. Brenner and R. Scott,
The Mathematical Theory of Finite Element Methods,
SpringerVerlag, 1994.
Resources
Problem sheets, handouts, computer labs and lecture rooms.
Module Evaluation
module questionnaires, module review, year review.
Next: MA4021 Wavelets and Signal Processing
Up: ModuleGuide0304
Previous: MA4001 Dynamical Systems
Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 20040221
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.