General description
Boundary value problems, in ordinary and partial differential equations, occur naturally in science and engineering, eg clamped beam problems, slow viscous flow, and elasticity. Over the centuries many famous mathematicians have been challenged by such problems and have produced elegant classical solution methods. Today it is possible to marry some of these classical discoveries with modern computational methods, to enable the solution of contemporary problems.

Aims
To teach students how to solve linear boundary problems using modern analytic and computational methods.

Learning outcomes
On completion of this module, a student should be able to:

• discretize elliptic boundary value problems in an efficient way;
• derive accurate numerical solutions of elliptic boundary value problems;
• explain and use spectral methods and spectral element methods.

Syllabus
1. TWO POINT BOUNDARY VALUE PROBLEMS: Variational and weak formulations.
2. GALERKIN AND PSEUDOSPECTRAL GALERKIN METHODS: Pseudospectral Galerkin and collocation methods.
3. ERROR ESTIMATE AND CONVERGENCE RATES FOR FINITE DIMENSIONAL APPROXIMATIONS
4. ELLIPTIC BOUNDARY VALUE PROBLEMS IN THE PLANE: Approximation in Tensor Product Spaces of Polynomials
5. INTRODUCTION TO ELEMENT METHODS.