The Discrete Galerkin Method for Integral Equations

A general theory is given for discretized versions of the Galerkin method for solving Fredholm integral equations of the second kind.The discretized Galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the Galerkin method.The theoretical framework that is given parallels that of the regular Galerkin method, including the error analysis of the superconvergence of the iterated Galerkin and discrete Galerkin solutions.In some cases, the iterated discrete Galerkin solution is shown to coincide with the Nystrm solution with the same numerical integration method.The paper concludes with applications to finite element Galerkin methods.1. Introduction.Consider the numerical solution of the Fredholm integral equation of the second kind, (1.1)x(s)-f K(s,t)x(t)da(t)=y(s), s e D. JDIn this paper we will define and analyze the discrete Galerkin method for the numerical solution of (1.1).The Galerkin method is a well-known procedure for the approximate solution of this equation (e.g., see 5, p. 62); and the discrete Galerkin method results when the integrations of the Galerkin method are evaluated numerically.Before giving a more precise definition of the discrete Galerkin method, we review results for the Galerkin method.In Eq. (1.1), the region D is to be a closed subset of Rm, some m > 1; and the dimension of D can be less than m, for example, if D is a surface in R3.For the discrete Galerkin method, we will assume that K(s, t) is continuous for s, t e D, although that is not necessary for the discussion of Galerkin's method given below.The equation (1.1) is written symbolically as (1.2) (I-Jf)x=y, with the integral operator assumed to be a compact operator from L2(D) to L2(D) and from L(D) to C(D).Further, it is assumed that (1.1) is uniquely solvable in SC for all y e 3C, for both 3C= C(D) and SC= L2(D).Additional assumptions on D and K(s, t) will be given as they are needed in the applications presented later in the paper.Generally y e C(D), and this will imply x e C(D).