Instructor:
Dr. I. Oprea: http://www.math.colostate.edu/~juliana

OH: T2:10-3:00PM, W3:10-4:00PM and by appointment

This course covers the mathematical
analysis of finite element and finite difference methods for the
numerical solution of the classic linear partial differential equations
of science and engineering. The focus is on the mathematical issues
underlying accurate numerical solution of partial differential
equations
such as stability, convergence, and accuracy. Implementation issues as
far as they affect accuracy are also discussed. The design philosophy
of
the course is to cover the mathematical theory applied to the simplest
examples so as to minimize technical details, yet provide a strong
foundation that students can carry back to their particular fields.
OH: T2:10-3:00PM, W3:10-4:00PM and by appointment

__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, ISBN: 0521567386,
(paperback,
$55).

· Numerical
Solution of Partial Differential Equations: An Introduction, K. Morton
and D. Mayers, Cambridge University Press, 1993, ISBN: 0521429226,
(paperback, $30).

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