A GENERALIZED CONJUGATE GRADIENT METHOD FOR THE NUMERICAL SOLUTION OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

A generalized conjugate gradient method for solving sparse, symmetric, positive-definite systems of linear equations, principally those arising from the discretization of boundary value problems for elliptic partial differential equations is considered.

Abstract

This chapter presents a generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. The conjugate gradient method (CG) is used for solving the system of linear algebraic equations. This elegant method has as one of its important properties that in the absence of round-off error, the solution is obtained in at most n iteration steps. The initial interest and excitement in CG was dissipated, because in practice, the numerical properties of the algorithm differed from the theoretical ones, that is, even for small systems of equations, the algorithm did not necessarily terminate in n iterations. For large systems of equations arising from the discretization of two-dimensional elliptic partial differential equations, competing methods such as successive over-relaxation (SOR) required only O(√n) iterations to achieve a prescribed accuracy. The conjugate gradient method has a number of attractive properties when used as an iterative method: (1) it does not require an estimation of parameters, (2) it takes advantage of the distribution of the eigenvalues of the iteration operator, and (3) it requires fewer restrictions on the matrix A for optimal behavior than do such methods as SOR.