Math 275 Homework 1
Due: Wednesday, 9/28/05

Write a program to solve the Dirichlet problem

$$\begin{aligned}\Delta u&=4, & &\qquad\text{in}\quad \Omega \\ u &= x^2+y^2 & &\qquad\text{on}\quad \partial\Omega \end{aligned}$$

on each of the domains

using linear triangular elements on an unstructured grid. Compute the $H^1$ and $L^2$ norms of the error exactly for each of the meshes provided below, and make a log-log plot of the error vs. the mesh parameter. What do you think would have happened if you used quadratic elements instead of linear elements?

• meshes for unit circle
• meshes for unit square
• sparse Cholesky code and example (I wrote this code several years ago. It should be useful to students writing in C, C++, or Fortran. If you have another solver you prefer, feel free to use that instead -- but don't use dense linear algebra routines. If you're using Matlab, use the builtin sparse solvers.)
• vis.m (short visualization example in matlab to get you started).

Turn in a list of the exact errors you computed (two 12 digit numbers per grid), the log-log plots, and your discussion of what would have happened if you used quadratic elements instead of linear ones. E-mail me a zip or tar file of your code. If there are lots of files, include one file called "notes" which gives a short description of which files I should look at to understand what you did. Add enough comments to your code that you would know what you did if you looked at it in a year (but don't add excessive comments -- I'll probably understand what you were doing.)