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

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Abstract
The Ritz-Galerkin solution of a linear integral or differential equation or set of equations leads to a set of linear algebraic equations, the structure of which depends on the type of expansion set used. For a finite-element expansion, the matrix involved is sparse, and reasonably efficient solution techniques are known. We study here the alternative case when a “global” expansion is chosen. Then the matrix involved is in general full, but has nonetheless a characteristic structure; we discuss the ways in which this structure can be used to yield efficient solution methods. Our main result is that a block iterative method can yield an arbitrarily high convergence rate; however, we also consider the stability of a direct solution of the equations.

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