Introductory description

This module addresses the mathematical theory of discretization of partial differential equations (PDEs) which is one of the most important aspects of modern applied mathematics. Because of the ubiquitous nature of PDE based mathematical models in biology, finance, physics, advanced materials and engineering much of mathematical analysis is devoted to their study. The complexity of the models means that finding formulae for solutions is impossible in most practical situations. This leads to the subject of computational PDEs. On the other hand, the understanding of numerical solution requires advanced mathematical analysis. A paradigm for modern applied mathematics is the synergy between analysis, modelling and computation. This course is an introduction to the numerical analysis of PDEs which is designed to emphasise the interaction between mathematical theory and numerical methods.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Analysis and numerical analysis of two point boundary value problems.
Model finite difference methods and their analysis.
Variational formulation of elliptic PDEs; function spaces; Galerkin method; finite element method; examples of finite elements; error analysis.

Learning outcomes

By the end of the module, students should be able to:

• be able to discretise an elliptic partial differential equation using finite element and finite difference methods,
• carry out stability and error analysis for the discrete approximation to elliptic, parabolic and hyperbolic equations in certain domains.
• Students who have successfully taken this module should be aware of the issues around the discretization of several different types of pdes,
• have a knowledge of the finite element and finite difference methods that are used for discretizing,

Indicative reading list

Background reading:
Stig Larsson and Vidar Thomee, Partial differential equations with numerical methods, Springer Texts in Applied Mathematics Volume 45 (2005).
K W Morton and D F Mayers, Numerical solution of partial differential equations: an introduction Cambridge University Press Second edition (2005).

Subject specific skills

• Discretisation of partial differential equations: the students will learn typical problems that can be encountered when discretising PDEs and the most common ways of fixing or avoiding them.
• In particular, they will study convergence and stability properties of various numerical schemes (these are concepts that appear in numerical analysis in general and should be useful in the future).
• The students will also learn different forms of discretising PDEs in space and time, as well as how to decide which discretisation to use depending on the PDE they consider.

Transferable skills

• Problem solving and written communication
• Students will be provided Matlab and/or Python scripts to work on, which will give them the opportunity to improve their programming skills (Improving employability)
• Numerical solution of PDEs is a common required skill for applied mathematics researchers