Banded systems

Whether the objective is to solve the Poisson equation using finite differences, finite elements or a spectral method, the outcome of discretization is a set of linear algebraic equations, e.g. (8.16) or (9.7). The solution of such equations ultimately constitutes the lion's share of computational expenses. This is true not just with regard to the Poisson equation or even elliptic PDEs since, as will become apparent in Chapter 16, the practical computation of parabolic PDEs also requires the solution of linear algebraic systems.

The systems (8.16) and (9.7) share two important characteristics. Our first observation is that in practical situations such systems are likely to be very large. Thus, five-point equations in an 81 × 81 grid result in 6400 equations. Even this might sound large to the uninitiated but it is, actually, relatively modest compared to what is encountered on a daily basis in real-life situations. Consider the equations of motion of fluids or solids, for example. The universe is three-dimensional and typical GFD (geophysical fluid dynamics) codes employ 14 variables – three each for position and velocity, one each for density, pressure, temperature and, say, the concentrations of five chemical elements. (If you think that 14 variables is excessive, you might be interested to learn that in combustion theory, say, even this is regarded as rather modest.) Altogether, and unless some convenient symmetries allow us to simplify the task in hand, we are solving equations in a three-dimensional parallelepiped.