Mathematics 462 - Fitted Numerical Methods for Singular Perturbatiion Problems

Lecturer: Professor John J H Miller

Date: 1997-98

Groups: Optional JS and SS Mathematics, SS Two-subject moderatorship

Prerequisites: 261

Duration: 21 weeks.

Lectures per week: 3 (including 1 tutorial)

Assessment:

Examinations: One 3-hour examination

An introduction to numerical methods for solving singular perturbation problems with special emphasis on the methods of G. I. Shishkin.

• Motivation for the study of singular perturbation problems
• Simple examples of singular perturbation problems
• Numerical methods for singular perturbation problems
• Simple fitted operator methods in one dimension
• Simple fitted mesh methods in one dimension
• Convergence of fitted mesh finite difference methods for linear reaction-diffusion problems in one dimension
• Properties of upwind finite difference operators on piecewise uniform fitted meshes
• Convergence of fitted mesh finite difference methods for linear convection-diffusion problems in one dimension
• Fitted mesh finite element methods for linear convection-diffusion problems in one dimension
• Convergence of Schwarz iterative methods for fitted mesh methods in one dimension
• Linear convection-diffusion problems in two dimensions and their numerical solution
• Bounds on the derivatives of solutions of linear convection-diffusion problems in two dimensions with regular boundary layers
• Convergence of fitted mesh finite difference methods for linear convection-diffusion problems in two dimensions with regular boundary layers
• Limitations of fitted operator methods on uniform rectangular meshes for problems with parabolic boundary layers