Joseph Kovac

The finite difference method represents a highly straightforward and logical approach to approximating continuous problems using discrete methods. At its heart is a simple idea: substitute finite, discrete differences for derivatives in some way appropriate for a given problem, make the time and space steps the right size, run the difference method, and get an approximation of the answer. Many of these finite difference methods can ultimately be written in a matrix form, with a finite difference matrix multiplying a vector of unknowns to equal a known quantity or source term. In this paper, we will be examining the problem Au=f, where A represents a finite difference matrix operating on u, a vector of unknowns, and f represents a time-independent vector of source terms. While this is a general problem, we will specifically examine the case where A is the finite difference approximation to the centered second derivative. We will examine solutions arising when f is zero (Laplace’s equation) and when it is nonzero (Poisson’s equation). The discussion would be quite straightforward if we wanted it to be; to find u, we would simply need to multiply both sides of the equation by A, explicitly finding u= Af. While straightforward, this method becomes highly impractical as the mesh becomes fine and A becomes large, requiring inversion of an impractically large matrix. This is especially true for the 2D and 3D finite difference matrices, whose dimensions grow as the square and cube of the length of one edge of the square grid. It is for this reason that relaxation methods became both popular and necessary. Many times in engineering applications, getting the exact answer is not necessary; getting the answer right to within a certain percentage of the actual answer is often good enough. To this end, relaxation methods allow us to take steps toward the right answer. The advantage here is that we can take a few iterations toward the answer, see if the answer is good enough, and if it is not, iterate until it is. Oftentimes, using such an approach, getting an answer “good enough” could be done with orders of magnitude less time and computational energy than with an exact method. However, relaxation methods are not without their tradeoffs. As will be shown, the error between the actual answer and the last iteration’s answer ultimately will decay to zero. However, not all frequency components of the error will get to zero at the same rate. Some error modes will get there faster than others. What we seek is to make all the error components get to zero as fast as possible by compensating for this difference in decay rates. This is the essence of multi-grid; multi-grid seeks to allow the error modes of the solution to decay as quickly as possible by changing the resolution of the grid to let the error decay properties of the grid be an advantage rather than a liability.