Content: 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.

Topics in this module include:

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

Aims: The aim of this module is to provide an introduction to the analysis and design of numerical methods for solving partial differential equations of elliptic, hyperbolic and parabolic type.

Objectives: Students who have successfully taken this module should be able to:

• Become aware of the issues around the discretization of several different types of PDEs
• Gain knowledge of the finite element and finite difference methods that are used for discretizing
• 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

Books:
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).