Project 14 consideres the Poisson equation, a simple second order partial differential equation which shows up across many fields of science and engineering.
For example, Poisson's equation describes the relationship between the charge density and the electostatic potential field; or between mass density and the gravitational potential; or between the steady state temperature of a material with given heat sources.
The particular Poisson equation we will look at will be defined in a square, and has the form
- uxx - uyy = lambda * u
where u(x,y) is our unknown function, and lambda is an unknown
scalar.
In order to study a Poisson equation computationally, we will start by using a finite difference approach, as discussed in project 13. This will give us a vector U of unknown values, and a matrix A containing coefficients used to approximate the derivatives. Our partial differential equation now becomes
A * U = lambda * U
and now it's clear that lambda must be an eigenvalue of A, and U must
be an eigenvector.
In this project, we will look at the relationship between the approximate solutions we compute, and the exact solutions we are trying to model. Usually these exact solutions are unknown, but in this project we will know the solution to the original partial differential equation, and so we can study how well we approximate the true solution, and how much better our approximations become as we use more points in our mesh.
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