A Note on the Convergence of Symmetric Triangle Quadrature Rules

TL;DR

This paper investigates symmetric polynomial quadrature rules for triangles, revealing that when the polynomial degree is even, the convergence rate exceeds traditional expectations, leading to potential computational savings in finite-element methods.

Contribution

It demonstrates that symmetric triangle quadrature rules with even polynomial degrees achieve higher convergence rates, reducing the number of points needed for accurate integration.

Findings

Quadrature error for even degree polynomials is lower than expected.

Fewer quadrature points are needed for a given accuracy.

Numerical examples confirm faster convergence rates.

Abstract

Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of points can exactly integrate, smooth integrands are generally not polynomials of finite degree. Therefore, for such integrands, one needs to balance integration accuracy and computational cost. A natural approach to this balance is to choose the number of points such that the convergence rate with respect to the mesh size $h$ matches that of the other properties of the scheme, such as the planar or curved triangles that approximate the geometry or the basis functions that approximate the solution. In general, it is expected that a quadrature rule capable of integrating polynomials up to degree $d$ yields an integration error that is $\mathcal{O}(h^p)$, where $p=d+1$. However, as we describe in this paper, for symmetric triangle quadrature rules, when $d$ is even, $p=d+2$; therefore, for a $p^\text{th}$-order-accurate quadrature rule, fewer quadrature points are necessary, reducing the time required for matrix assembly in finite-element-type problems. This reduction in cost is modest for local differential operators that yield sparse matrices but appreciable for global integral operators that yield dense matrices. In this paper, we briefly summarize the details of symmetric triangle quadrature rules, discuss error implications for quadrature rules for one dimension and triangles, and we provide numerical examples that support our observation that polynomials that exactly integrate even maximum degrees converge faster than the conventional expectation for sequences of regular meshes.