The Petrov--Galerkin and Iterated Petrov--Galerkin Methods for Second-Kind Integral Equations

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Abstract
We develop in this paper a theoretical framework for the analysis of convergence for the Petrov-Galerkin method and superconvergence for the iterated Petrov--Galerkin method for Fredholm integral equations of the second kind. As important approaches to the analysis, we introduce notions of the generalized best approximation and the regular pair of trial space sequence and test space sequence. In Hilbert spaces, we characterize the regular pair in terms of the angle of two space sequences or the generalized best approximation projections. Several specific constructions of the Petrov--Galerkin elements for equations of both one dimension and two dimensions are presented and the convergence of the Petrov--Galerkin method and the iterated Petrov--Galerkin method using these elements is proved.