Math 621 - Numerical Methods for Partial Differential Equations

Spring 2013 - Matthias K. Gobbert

Detailed Schedule - Last Updated January 30, 2013

This schedule is designed to give you an overview of the material to be covered and is tentative in nature. This is a living document and will be updated throughout the semester.
The entries in the column References point to a number of standard references in the field. See my webpage on recommended literature for the complete citations.
The color coding classifies the topic of the material, with green and blue colors for material on the finite difference and the finite element methods, respectively, the actual contents of the course. But to apply either of these methods, we need tools including linear solvers, non-linear solvers, and ODE solvers, color-coded in red. The orange color indicates software workshops on coordinated material in Matlab and COMSOL Multiphysics, offered by the Center for Interdisciplinary Research and Consulting (www.umbc.edu/circ).

Class	Date	Main Topic	References
1	Tu 01/29	Overview: Math 621 in a nutshell	Gobbert (SISC)
	We 01/30	12:00-01:00, ENGR 122: CIRC Software workshop: Advanced MATLAB Programming
2	Th 01/31	Prototype problems for elliptic, parabolic, and hyperbolic PDEs	Evans (Ch. 1 and 2)
3, #1	Tu 02/05	The finite difference method for elliptic problems	Gobbert (Notes)
4	Th 02/07	Linear solvers and Matlab programming	Watkins (Ch. 8)
5, #2	Tu 02/12	The method of lines for a parabolic prototype problem using FD	Gobbert (SISC)
	We 02/13	12:00-01:00, ENGR 122: CIRC Software workshop: Intermediate MATLAB Programming
6	Th 02/14	Implementation issues and higher dimensional problems	Gobbert (Notes)
7	Tu 02/19	Review of basic numerical methods for ODEs and their convergence theory	Iserles (Part I)
	We 02/20	12:00-01:00, ENGR 122: CIRC Software workshop: An Introduction to MATLAB Programming
8	Th 02/21	Linear stability theory for ODE methods and the concept of stiffness	Iserles (Part I)
9, #3	Tu 02/26	Numerical methods for stiff systems of ODEs	Shampine & Reichelt
10	Th 02/28	The non-linear solver inside implicit ODE methods	Gobbert (slides)
11	Tu 03/05	Theory of the finite difference method for elliptic problems	Gobbert (Notes)
12	Th 03/07	Idea of the finite element method for elliptic problems
13, #4	Tu 03/12	Weak derivatives and Sobolev spaces	Braess (Sec. II.1)
	We 03/13	12:00-01:00, ENGR 122: CIRC Software workshop: COMSOL Multiphysics
14	Th 03/14	The weak form of elliptic PDEs	Braess (Sec. II.2)
	Tu 03/19	Spring Break
	Th 03/21	Spring Break
15	Tu 03/26	Report on project background; project proposal due
16	Th 03/28	Galerkin orthogonality and Céa's lemma, standard finite elements	Braess (Sec. II.4, II.5)
17, #5	Tu 04/02	Approximation theory and convergence of the FEM in the H¹- and L²-norms	Braess (Sec. II.6-II.7)
18	Th 04/04	Introduction to COMSOL Multiphysics: GUI	Trott & Gobbert (HPCF-2010-8)
19	Tu 04/09	Demonstration of the CAD facilities of COMSOL Multiphysics
20	Th 04/11	Analysis of FEM convergence in COMSOL Multiphysics	Trott & Gobbert (HPCF-2010-8)
21, #6	Tu 04/16	The finite element method for parabolic problems	Gobbert (SISC)
22	Th 04/18	Theory of the finite element method for parabolic problems	Thomée (Ch. 1)
23	Tu 04/23	Parallel computing for non-linear reaction-diffusion equations	Gobbert (SISC)
24	Th 04/25	Introduction to hyperbolic partial differential equations	Strikwerda (Sec. 1.1-1.2)
25, #7	Tu 04/30	Finite differences for the scalar transport equation	Strikwerda (Sec. 1.3-1.4)
26	Th 05/02	Theory for FD for the scalar transport equation	Strikwerda (Sec. 1.5-1.6)
27	Tu 05/07	Update on project work; project report due
28	Th 05/09	Finite elements for hyperbolic conservation laws	Baumann & Oden
29, #8	Tu 05/14	Other numerical methods (finite volume method) for PDEs and review	Schäfer et al.
	Fr 05/17	01:00 p.m. Project Presentations	Note date and time!