Michael Higley and Dr. Peter Bates, Mathematics
Laplace’s equation is a differential equations that describes many physical properties in steady state systems. It can be used to describe heat distributions, displacements in elastic media, or electrostatic fields. My interest is in solving an electrostatics problem where the domain—the area in which the equation has physical meaning—is irregular enough that the only way to solve the problem is to use some sort of numerical approximation.
To do this the domain is divided into pieces, and a solution is sought to a much simpler equation that mimics Laplace’s equation on each small piece. Using more pieces provides a clearer picture of the solution in the same way that having more pixels per square inch on a computer monitor can provide a sharper image.
Unfortunately, dividing the domain into more pieces creates a very large amount of data that needs to be stored and manipulated. If the domain is split into N pieces in each direction, that makes a total of N3 pieces. Because each piece is represented in an equation that contains variables representing each piece of the domain, this gives N6 pieces of information that must be stored and manipulated. For example, if the domain were split into 100 pieces in each direction, there would be a total of 1,000,000,000,000 pieces of information to keep track of and manipulate. For a computer to store that many decimal numbers would require at least four thousand gigabytes of memory.
Because most of these numbers (all but 7N3) are zero, I had hoped to be able to create a program that could ignore enough of the zeroes so that a computer could solve the equations using Gaussian or Gauss-Jordan elimination such as students learn in any college algebra class. Unfortunately, I discovered that there are large blocks of numbers (size N4) that unavoidably fill in while finding a solution. There are about N such blocks, leaving N5 pieces of information to keep track of. In the previous example, this would reduce the required storage space from four thousand gigabytes to forty gigabytes—a minor improvement, but certainly not sufficient to make it feasible on a personal computer.
It was suggested to me that the problem would be more readily solved using an iterative method like successive over-relaxation. As I read about this method, I discovered that it had been created for the very reason that I had failed in solving the problem: even though matrices may start out sparse (containing mostly zeroes) the zeroes fill in rapidly and unavoidably, leaving too much information to keep track of. (1) Some modification of the problem was necessary to make it solvable using these techniques, but I have accomplished that and am examining the electrostatics problem using successive over-relaxation.
- Strang, Gilbert. 1986. pp. 367-369. In: Introduction to Applied Mathematics. Wellesley- Cambridge Press, Wellesley.