A modified FC-Gram approximation algorithm with provable error bounds

TL;DR

This paper introduces a modified FC-Gram approximation algorithm that uses explicit Hermite polynomial extensions, enabling provable error bounds and improved computational efficiency, with convergence rates confirmed by numerical experiments.

Contribution

The paper proposes a new modified FC-Gram algorithm with explicit Hermite extensions, allowing for theoretical error bounds and easier extension length adjustments.

Findings

The modified algorithm achieves convergence rates consistent with theoretical predictions.

Explicit Hermite extensions eliminate the need for precomputed extension data.

Numerical experiments confirm the practical effectiveness of the proposed method.

Abstract

The FC-Gram trigonometric polynomial approximation of a non-periodic function that interpolates the function on equispaced grids was introduced in 2010 by Bruno and Lyon [J. Comput. Phys, 229(6):2009-2033, 2010]. Since then, the approximation algorithm and its further refinements have been used extensively in numerical solutions of various PDE-based problems, and it has had impressive success in handling challenging configurations. While much computational evidence exists in the literature confirming the rapid convergence of FC-Gram approximations, a theoretical convergence analysis has remained open. In this paper, we study a modified FC-Gram algorithm where the implicit least-squares-based periodic extensions of the Gram polynomials are replaced with an explicit extension utilizing two-point Hermite polynomials. This modification brings in two significant advantages - (i) as the extensions are known explicitly, the need to use computationally expensive precomputed extension data is eliminated, which, in turn, facilitates seamlessly changing the extension length, and (ii) allows for establishing provable error bounds for the modified approximations. We show that the numerical convergence rates are consistent with those predicted by the theory through a variety of computational experiments.