An Adaptive Variable Order Quadrature Strategy

TL;DR

This paper introduces an adaptive quadrature method that dynamically adjusts both the subdivision of the integration domain and the number of quadrature points locally, inspired by hp-version finite element techniques, to improve accuracy and efficiency.

Contribution

It presents a novel adaptive quadrature strategy combining local domain subdivision and variable quadrature points, inspired by hp finite element methods, for enhanced numerical integration.

Findings

Achieves high accuracy with fewer function evaluations.

Effectively handles functions with varying smoothness.

Demonstrates superior convergence rates compared to traditional methods.

Abstract

In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this way we aim to account for local smoothness properties of the function to be integrated as effectively as possible, and thereby achieve highly accurate results in a very efficient manner. Indeed, this idea originates from so-called hp-version finite element methods which are known to deliver high-order convergence rates, even for nonsmooth functions.