General description
Finite difference approximations to elliptic partial differential equations. Local truncation error and error analysis. Boundary conditions on a curved boundary. Variational formulation and the finite element method. Classical iterative methods for solving linear systems of algebraic equations: Jacobi, Gauss-Seidel, SOR. Fourier analysis of convergence.
Finite difference approximations to parabolic partial differential equations in one space variable. Local truncation error and error analysis. Explicit and implicit methods. Convergence and stability. The Thomas algorithm.

Aims
The aim of this course is to provide an introduction to numerical methods for solving partial differential equations of elliptic and parabolic type. Concepts such as consistency, convergence and stability of numerical methods will be discussed. Classical iterative methods for solving the systems of linear algebraic equations arising from the discretization of elliptic problems will be described and their convergence behaviour analysed.

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

• discretise an elliptic partial differential equation using finite difference and finite elements methods.
• perform an error analysis for the discrete approximation to elliptic equations.
• solve the discrete equations using classical iterative methods.
• analyse the convergence behaviour of these iterative methods.
• discretise parabolic partial differential equations in one space variable.
• perform an error analysis for the dicrete approximation to parabolic equations.