SPECTRAL ELEMENT APPROXIMATION OF FUNCTIONAL INTEGRAL EQUATIONS
Vol. 8(2) July 2020, No. 16, pp. 172-187
J. S. AZEVEDO, S. P. OLIVEIRA AND A. M. ROCHA
We discuss the existence, uniqueness, and numerical approximation of solutions to a class of nonlinear Fredholm-type integral equations of the second kind. The analysis is performed in the space of square integrable functions, providing the functional setting to study Galerkin approximations. In particular, we choose the spectral element method with Gauss-LobattoLegendre collocation points, which leads to an explicit fixed-point problem that is solved with the Picard iterative method. We prove an optimal convergence rate for the proposed iterative method, which is also superconvergent at the quadrature nodes. The theoretical error bounds are verified through numerical experiments as well.