Four Variants of the Galerkin Method for Integral Equations of the Second Kind

Abstract

Four variants of the Galerkin method for integral equations of the form y = f+Ky, with K compact, are considered. They are the classical Galerkin method; the Kantorovich method (discussed recently by Schock), in which the inhomogeneous term f is treated exactly, and the Galerkin scheme applied to y−f; the iterated Galerkin method, obtained by substituting the Galerkin approximation into the right-hand side of the integral equation; and the similarly iterated Kantorovich method. The principal new result is that the iterated Kantorovich method converges faster than the Kantorovich method; and that the factor of improvement (the “superconvergence factor”) has the same theoretical bound as that already known for iteration of the Galerkin method. Thus in the circumstances in which the Kantorovich modification is indicated (i.e. when y−f is smoother than y), the iterated Kantorovich method will generally be the most accurate of the four methods considered.