The Discrete Galerkin Method for Integral Equations

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Abstract
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 Nyström solution with the same numerical integration method. The paper concludes with applications to finite element Galerkin methods.