The Numerical Solution of Sets of Linear Equations Arising from Ritz-Galerkin Methods

The Ritz-Galerkin solution of a linear integral or differentialequation or set of equations leads to a set of linear algebraicequations, the structure of which depends on the type of expansionset used. For a finite-element expansion, the matrix involvedis sparse, and reasonably efficient solution techniques areknown. We study here the alternative case when a "global" expansionis chosen. Then the matrix involved is in general full, buthas nonetheless a characteristic structure; we discuss the waysin which this structure can be used to yield efficient solutionmethods. Our main result is that a block iterative method canyield an arbitrarily high convergence rate; however, we alsoconsider the stability of a direct solution of the equations.