M676: Introduction to Numerical Methods for Partial Differential Equations
Spring 2007  Reference Number: 354249;
Instructor: Dr. I. Oprea
Note: the course is officially scheduled from 11 to 11:50AM; however, the proposed running time/classroom is 2-3:15PM on MW, Engineering B2

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.

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
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.
· 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