Aim:
The course is centred on the finite element technique. This is widely used in research and applied science and it is also an efficient method for solving practical problems of industry. The method is relevant generally to linear partial differential equations, and covers applications in the areas of heatflow, elasticity, acoustics, magnetism, etc. An alternative approach is the method of "finite differences", which is also covered in the course. Both methods are used to solve practical problems and the course involves workshop sessions to illustrate the theory, and to motivate with applied examples.The MATLAB toolbox on partial differential equations is suitable and available in the department.
Course Outline:
| An overview of partial differential equations, the situations they model and corresponding numerical methods. | 2 |
| The finite element method. Problems posed in variational form (rather than as the solution of a differential equation); discretizing the problem domain and the use of element shape functions. Element stiffness matrices developed into a global system. Initially the ideas are introduced in one dimension, and later extended into two in the context of steady-state heat transfer. | 18 |
| A more mathematical approach using trial and test functions. The Galerkin method. | 10 |
| General element computations using mappings from a master element. | 8 |
| The finite difference method: Finite differences approximating to derivatives. Elliptic equations, and parabolic equations with the explicit and Crank-Nicolson methods. | 10 |
Recommended Texts:
Additional Reading: