Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds

TL;DR

This paper introduces a robust adaptive numerical integration method for functions with sharp gradients and cusps in n-dimensional parallelepipeds, demonstrating its effectiveness in applications like modeling sharp gradients and solving Coulomb problems in crystalline materials.

Contribution

It presents a novel adaptive integration algorithm that is independent of the location of sharp features and achieves optimal convergence rates in high-dimensional settings.

Findings

Achieves convergence rate of n+1 for $C^0$ functions in $R^n$

Effectively integrates regularized Heaviside functions with sharp gradients

Enables efficient finite element solutions for Coulomb problems with cusps

Abstract

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.