Math 621 - Numerical Methods for Partial Differential Equations

Spring 2015 - Matthias K. Gobbert

Detailed Schedule - Last Updated March 10, 2015

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 numbers #1, #2, etc. in the Class column indicated that the homework with that number is due at the beginning of class that day.
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 01/27	Overview: Math 621 in a nutshell	Huang et al. (2015)
2	Th 01/29	Prototype problems for elliptic, parabolic, and hyperbolic PDEs	Evans (Ch. 1 and 2)
3, #1	Tu 02/03	Classification of PDEs	Evans (Ch. 1 and 2)
	We 02/04	12:00-01:00, ENGR 122: CIRC Software workshop: Advanced MATLAB Programming	CIRC Tutorial
4	Th 02/05	The finite difference method for elliptic problems	Gobbert (Notes)
5	Tu 02/10	Overview of the conjugate gradient method	Watkins (Ch. 8)
6, #2	Th 02/12	Preconditioning for iterative methods	Watkins (Ch. 8)
7	Fr 03/13	The method of lines for a parabolic prototype problem using FD	Huang et al. (2015)
8	Tu 02/17	Implementation issues and higher dimensional problems	Gobbert (Notes)
	We 02/18	12:00-01:00, ENGR 122: CIRC Software workshop: Intermediate MATLAB Programming	CIRC Tutorial
9	Th 02/19	Derivation of BDFk and NDFk, Newton method, linear solver	Gobbert (slides)
10, #3	Tu 02/24	Numerical methods for stiff systems of ODEs	Shampine & Reichelt
11	Th 02/26	Implementation issues of MOL-FD for non-linear systems of PDEs	Gobbert (slides)
12, #4	Tu 03/03	Idea of the finite element method for elliptic problems	Braess (Sec. II.7)
	We 03/04	12:00-01:00, ENGR 122: CIRC Software workshop: COMSOL Multiphysics	CIRC Tutorial
13	Th 03/05	Snow cancellation
14	Tu 03/10	Weak derivatives and Sobolev spaces	Braess (Sec. II.1)
15, #5	Th 03/12	The weak form of elliptic PDEs	Braess (Sec. II.2)
	Tu 03/17	Spring Break
	Th 03/19	Spring Break
16, #6	Tu 03/24	The weak form of more general elliptic PDEs	Braess (Sec. II.3)
	We 03/25	12:00-01:00, ENGR 122: CIRC Software workshop: Octave	CIRC Tutorial
17	Th 03/26	Report on project background; project proposal due
18, #7	Tu 03/31	Abstract formulation of weak with bilinear form	Braess (Sec. II.2)
	Th 04/02	Class cancelled; make-up was 03/13/15
19	Tu 04/07	Galerkin orthogonality and Céa's lemma	Braess (Sec. II.4)
20, #8	Th 04/09	Standard finite elements	Braess (Sec. II.5)
21	Tu 04/14	Approximation theory for the FEM	Braess (Sec. II.6)
22	Th 04/16	Convergence of the FEM in the H¹- and L²-norms	Braess (Sec. II.7)
23	Tu 04/21	Theory of the finite element method for parabolic problems	Thomée (Ch. 1)
24, #9	Th 04/23	Introduction to hyperbolic partial differential equations	Strikwerda (Sec. 1.1-1.2)
25	Tu 04/28	Finite differences for the scalar transport equation	Strikwerda (Sec. 1.3-1.4)
26	Th 04/30	Theory for FD for the scalar transport equation	Strikwerda (Sec. 1.5-1.6)
27	Tu 05/05	Update on project work; project report due
28, #10	Th 05/07	Finite elements for hyperbolic conservation laws	Baumann & Oden
29	Tu 05/12	Other numerical methods (finite volume method) for PDEs and review	Huang et al. (2015)
	Tu 05/19	01:00 p.m. Project Presentations	Note date and time!