Content:

1. Review analysis for PDEs
2. Review numerical discretisation
3. Finite element theory for linear problems
4. Concepts for discretises problems
5. Semilinear monotone equations
6. Obstacle problems

7. Time dependent problems

Aims:

The goal of this course is to introduce some fundamental concepts, methods and theory associated with the numerical analysis of partial differential equations and in particular nonlinear equations.

Finite element theory will provide the core machinery for devising methods and their analysis. Abstract notions of stability, consistency and convergence will be introduced as applied to convergence of minimisers, approximation of equilibrium points and the solution of nonlinear discrete problems

Objectives:

By the end of the module the student should be able to:

- Recognise the nature of the problem to be approximated

- Recognise where the Galerkin method is appropriate for use

- Formulate discrete approximations of nonlinear PDEs

- Obtain stability estimates in a variety of settings

- Apply elementary iterative methods for fixed point equations and optimisation.

Books:

• Soren Bartels Numerical methods for nonlinear PDEs Springer Series in Computtaional Mathematics Vol 47 (2015)
• S Larrsson and V. Theme

PDEs and numerical methods Springer Texts in Applied Maths Vol 45 (2005)