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MC450 Finite Element Methods for Partial Differential Equations

MC450 Finite Element Methods for Partial Differential Equations

Credits: 20 Convenor: Dr P. Houston Semester: 2

Prerequisites: essential: MC224,MC243 desirable:
Assessment: Coursework: 10% Three hour exam: 90%

Lectures: 36 Problem Classes: 10
Tutorials: 0 Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

The essential prerequisites for this course are a knowledge of Hilbert space theory and the calculus of functions of several variables.

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, material 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, efficiency and adaptivity.

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.

Objectives

To learn about the mathematical theory of finite element methods for partial differential equations.

To understand the theory of weak solutions to elliptic partial differential equations.

To provide analytical techniques for deriving optimal a priori and a posteriori error bounds.

To understand how to efficiently implement finite element methods.

Transferable Skills

The ability to apply the techniques developed in this course to a large variety of partial differential equations.

The importance of partial differential equations in all areas of science and engineering makes this course an essential prerequisite for any student wishing to pursue a career in applied mathematics.

Syllabus

Elements of function spaces; theory of weak solutions to elliptic boundary value problems; Lax Milgram theorem.

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; Aubin-Nitsche duality argument; variational crimes.

A posteriori error analysis by duality; design of reliable and efficient adaptive finite element algorithms.

Finite element methods for parabolic partial differential equations; forward and backward Euler timestepping schemes; stability; error analysis.

Finite element methods for hyperbolic partial differential equations; Petrov-Galerkin schemes; streamline-diffusion stabilisation; error analysis.

Reading list

Essential:

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994.

Recommended:

P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1990.

Background:

K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Partial Differential Equations, Cambridge University Press, 1996.

Details of Assessment

The final assessment of this module will consist of 10% coursework and 90% from a three hour examination during the Summer exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 6 questions with full marks on the paper obtainable from 4 complete answers.


Next: MC451 Dynamical Systems Up: Year 4 Previous: MC446 Algebraic Topology

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