Oscillation Preserving Galerkin Methods for Fredholm Integral Equations of the Second Kind with Oscillatory Kernels

Abstract

Solutions of Fredholm integral equations of the second kind with oscillatory kernels likely exhibit oscillation. Standard numerical methods applied to solving equations of this type have poor numerical performance due to the influence of the highly rapid oscillation in the solutions. Understanding of the oscillation of the solutions is still inadequate in the literature and thus it requires further investigation. For this purpose, we introduce a notion to describe the degree of oscillation of an oscillatory function based on the dependence of its norm in a certain function space on the wavenumber. Based on this new notion, we construct structured oscillatory spaces with oscillatory structures. The structured spaces with a specific oscillatory structure can capture the oscillatory components of the solutions of Fredholm integral equations with oscillatory kernels. We then further propose oscillation preserving Galerkin methods for solving the equations by incorporating the standard approximation subspace of spline functions with a finite number of oscillatory functions which capture the oscillation of the exact solutions of the integral equations. We prove that the proposed methods have the optimal convergence order uniformly with respect to the wavenumber and they are numerically stable. A numerical example is presented to confirm the theoretical estimates.