Dept. of Mathematical Sciences Fall 2004, UAF

Graduate Seminar in Finite Elements (Math 692)

Organizer/First Presenter: Ed Bueler
Tentative time & place:  Wednesday, 1pm-2:30pm, Chapman 107


I propose: A seminar in the basic mathematics of the finite element method for partial differential equations.  To control technical difficulties I propose to limit the seminar to two spatial variables and to primarily concern ourselves with linear elliptic and parabolic problems, but with an eye toward nonlinear problems.

To give structure to the seminar I propose to give at least the first six lectures from the small book:

Claes Johnson, Numerical Solution of PDEs by the Finite Element.Method, Cambridge University Press 1988.

(It is out of print.  There is a 1996 version, quite different.  I propose to start the course using photocopies of the 1988 text.)

The seminar will serve as a practical introduction to Sobolev spaces in the one and two variable cases and with many technical details suppressed.  The finite element method involves converting PDE into a bilinear form on such spaces (i.e. a translation to the "variational" form of the PDE).  One then approximates the solution Sobolev space by a finite-dimensional subspace of piecewise-polynomial functions (the space of "finite elements").

To make the mathematics meaningful we should use Matlab to do as-simple-as-possible examples of finite element computations.

Prerequisites:  Clearly one must have seen classical PDEs as taught in Math 421, and one must be comfortable with linear algebra.  Furthermore, either exposure to real analysis (Math 641) or methods of mathematical physics (Math 611) is strongly recommended because function spaces will be the object of interest from the beginning.
Students who want 2.0 credits should be prepared to do up to six homework assignments during the semester, using Matlab and certain FE tools, and give a one--hour talk. Such talks might be a:
    (i) lecture on a topic not covered by Bueler's lectures
    (ii) lecture on the content of a research paper, or
    (iii) description and analysis of a nontrivial computation performed by the student.
Graduate students in mathematics, engineering, and geosciences are especially encouraged to attend.

Contact Ed Bueler at elbueler@alaska.edu or x7693 or Chapman 301C
The first meeting on Wednesday, 8 September will include setting up a permanent time.