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 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, using iterative (possibly multigrid) methods for the linear Jacobian solver, and finally, extensions to (nonlinear) parabolic equations. Our homeworks this quarter will involve implementing some of these extensions, and proving some things about the extensions.

DUE DATE: Homework #1 is do at 4pm on Friday 2-20-98.

DUE DATE: Homework #2 is do at 4pm on Friday 3-20-98.