About convergence of projection methods for solution of some Fredholm integral equations of the first kind

TL;DR

This paper investigates the convergence of projection methods like Galerkin and collocation for solving Fredholm integral equations of the first kind, providing conditions for convergence and error estimates in the context of potential theory.

Contribution

It establishes new convergence conditions for Galerkin and collocation methods applied to Fredholm equations of the first kind related to Laplace problems in 3D.

Findings

Conditions for convergence of Galerkin and collocation methods

Error estimates for approximate solutions

Application to Dirichlet problem for Laplace equation

Abstract

This article is dedicated to research of approximation properties of B-splines and Lagrangian finite elements in Hilbert spaces of functions defined on surfaces in three-dimensional space. Hereinafter the conditions are determined for convergence of Galerkin and collocation methods for solving Fredholm integral equations of the first kind for simple layer potential that is equivalent to Dirichlet problem for Laplace equation in R3. Estimation is determined for the error of approximate solution of this problem obtained using potential theory methods.