Posted: September 13th, 2017

Problem Solving by Computer Assignment 3

Paper, Order, or Assignment Requirements

1. Find a suitable freely available code (for example Distmesh) that creates a triangular finite element
mesh of domains in the plane including polygons. Get the code working and make sure you can
import the vertex coordinates and lists of vertices in each triangle in to MATLAB (or OCTAVE).
Include figures as well as code to show you can mesh a variety of shaped domains and that you
have some control over how fine the mesh is.
2. Write a MATLAB function to generate the cotangent Laplacian given the mesh data above.
Test this function writing a script that solves a Dirichlet problem for Laplace’s equation and a
non-trivial homogeneous Neumann (that is zero Neumann condition) Poisson’s equation, in both
cases compare with the known analytical solution and demonstrate numerically that using a finer
mesh gives a more accurate solution.
Use a suitable example to illustrate that the approximate solution of Laplace’s equation satisfies
the Maximum Principle (which you should look up).
3. The government of an island nation has a pollution problem. At one or more points in the interior
of the island a liquid is being injected in to the ground that contains a toxic chemical. The
concentration of the chemical in the soil can be ascertained by a test but this is time consuming
and fairly costly. They have found that the concentration at the coast is zero and it is hypothesised
that it satisfies Poisson’s equation with a delta function source at each point where the chemical
is being injected. The island is relatively flat without rivers and the soil fairly homogeneous.
Choose an irregular (and not necessarily convex) polygon as the coastline for the island (you can
measure a real island from a map or choose one with a realistic but interesting shape). Building
on your answer to 1 and 2 above produce contour plots of the concentration of the chemical from
one or more sources at different locations.
The government is interested in finding the sources of the pollution using a small number of
tests. Decide on a recommendation as as to how the government should go about doing these
tests. Explain your rational using whatever numerical or mathematical evidence you can, but then
summarise your recommendation in a succinct way that would be intelligible to environmental
scientists working for the government who lack any degree level mathematical knowledge.
Briefly, what ideas should be pursued to refine this model if it was found not to fit the data well.
You are not required to implement any of these ideas!