Math 292A (Applied Mathematics)

Course Topics: Finite Element Methods for PDEs I
Instructor: Prof. Michael Holst
Term: Fall 1997

One-page course announcement and seven-page course outline.

HOMEWORK #1 can be found here (10-13-97).

DUE DATE: Homework #1 is do at 4pm on Wednesday 10-22-97.

NOTE: Except for those who came to see me Friday, no one was able to get the homework from the web page until today, due to my mistake. So, disregard the 10-17-97 due date printed on the homework itself; it is now due on 10-22-97 in class.

CLASS CANCELLED: Class on Monday (10-13-97) is cancelled; my apologies for those of you who walked over at 4:30pm only to see the announcement on the chalkboard. I was in a new faculty orientation meeting in which the chair of each department introduced the new faculty to the rest of the faculty. I had been told that it would be over by 4:30, but they hadn't even made it to the physical sciences by 4:25pm, so I had to call over and ask Tinh to cancel the class for me. Again, my apologies; I will lecture for two hours on Wednesday (or alternatively on Friday) if you have the stomach for this.

HOMEWORK #2 can be found here (10-26-97).

DUE DATE: Homework #2 is do at 4pm on Monday 11-3-97.

MATLAB FEM PACKAGE: Get this package from here. We will use this package to explore the finite element method in some homeworks. The package is mostly complete; part of your homework will be to finish a few missing pieces.

The package is a piecewise-linear Galerkin finite element code, for general second-order nonlinear elliptic equations on arbitrary polygonal domains in 2D. You provide the initial triangles representing the domain (which you can get from MATLAB's "delaunay" function if you like), and then you also specify the weak form of your problem (possibly nonlinear), as well as a linearization bilinear form. In the case of a linear problem like the Poisson equation, you simply specify the weak form of the Poisson equation in residual form, and then specify its linearization (just the weak form of the Laplace operator) as the associatied bilinear linearization form. (Refer to my class notes for the details.) The code then generates a sequence of Galerkin solutions to linearizations of the problem, and constructs the solution to the nonlinear problem with a Newton iteration. If your problem is linear, the Newton iteration terminates in one step.

Some of the missing pieces of this code include handling Robin boundary conditions, handling several unknowns per mesh point, using quadratic and higher-order basis functions, adding damping to the Newton iteration for global convergence properties, introducing inexactness in the Newton solve for efficiency, and using iterative (possibly multigrid) methods for the linear Jacobian solver. Homework #3 will involve picking one of these extensions and implementing it. Homework #4 will involve proving some things about the accuracy of the solutions that this code generates.

HOMEWORK #3 can be found here (11-12-97).

DUE DATE: Homework #3 is do at 4pm on Friday 12-5-97.