ABSTRACT
This chapter covers the necessary mathematics for matrix solutions of the finite-element equations for triangular meshes. It reviews Gauss-Jordan elimination with pivoting, a standard method to invert matrices and to solve moderate sets of linear equations. With the speed and memory capacity of personal computers, it is practical to apply the method to about 500 equations. Many physical problems give rise to matrices where most of the elements equal zero. For example, the matrix that represents the difference equations for the one-dimensional Poisson equation has non-zero elements only on the matrix diagonal and adjacent rows. The chapter covers special fast methods to solve for tridiagonal matrices, and illustrates direct matrix solutions for one-dimensional electrostatics with space-charge. The example illustrates important points: setting up matrices, automatic implementation of Neumann boundary conditions, and handling constant potential points. The chapter describes solution techniques for tridiagonal block matrices. Critical issues are optimizing the use of random-access memory and minimizing scratch data storage on disk.
