Frequency independent evaluation of highly oscillatory integrals

Abstract

Oscillatory integrals appear in many applications and therefore, several differ ent quadrature rules for their accurate and efficient evaluation have been proposed in the literature. The efficiency of these methods naturally depend on the degree of smoothness of the integrands. In this thesis, we present a survey for the paper [1] by Doḿınguez et al., which is recently exhibited in the literature, about composite algorithm which eases the con ditions of the Filon-Clenshaw-Curtis rule and enables the use of the algorithm with finitely many algebraic and logarithmic singularities or oscillators with a finite number of stationary points. Our survey includes the complete error analysis in [1], which is in the setting of appropriate Sobolev spaces of 2π-periodic functions. We support the results of this paper via providing some numerical experiments.