Coursework
The course work consists
of a mixture of mathematical problem sets and computational projects.
The computational projects emphasize experimentation with mathematical
issues such as convergence, stability and accuracy. The writing of
mathematically correct and clear project reports is also emphasized
Prerequisites
Exposure to solutions of
the classic models in partial differential equations, elementary linear
algebra, ordinary differential equations, (as obtained in M531 for
example), and the ability to program in some language. The use of
MATLAB is strongly encouraged.
Textbooks
· Computational
Differential Equations, K. Eriksson, D. Estep, P. Hansbo, and C.
Johnson, Cambridge University Press, 1996
· Numerical
Solution of Partial Differential Equations: An Introduction, K. Morton
and D. Mayers, Cambridge University Press, 2-nd edition
Topics to be
Covered
1. Brief Introduction to Basic Numerical
Analysis
- Interpolation theory, Numerical quadrature, The need for numerical solutions of
differential equations
2.Elliptic Problems and the Finite Element
Method
- Models involving
conservation of heat, behavior of solutions
- Two-point boundary
value problems and the Laplace and Poisson equations
- The variational
formulation and the Galerkin finite element method
- Convergence in the
energy norm, a priori convergence, order of convergence
- Quadrature in the
finite element method and the finite difference method
- Brief overview of
complications that can occur in realistic models
3. (Very) Brief Introduction to Numerical
Linear Algebra
- Direct methods for
sparse and banded matrices , Basic
iterative methods
4. Parabolic Problems and the Method of Lines
- Explicit and implicit
method of lines methods using finite elements in space and finite
differences in time
- Numerical stability,
stiffness and dissipativity, convergence
5. Hyperbolic Problems and the Finite
Difference Method
- The transport
equation and wave equations, characteristics and the transport of
information, behavior of solutions
- Finite difference
schemes, consistency
- Stability,
dissipativity, dispersion, the CFL condition, convergence
6. Miscellaneous Topics as Time Permits : Error estimation, computational error
estimation and adaptive schemes, conservation laws