$$ \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\tp}{\thinspace .} \newcommand{\x}{\boldsymbol{x}} \newcommand{\X}{\boldsymbol{X}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\w}{\boldsymbol{w}} \newcommand{\e}{\boldsymbol{e}} \newcommand{\f}{\boldsymbol{f}} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\Iz}{\mathcal{I}_z} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}} \newcommand{\sequencej}[1]{\left\{ {#1}_j \right\}_{j\in\If}} \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\psib}{\boldsymbol{\psi}} \newcommand{\xno}[1]{x_{#1}} \newcommand{\Xno}[1]{X_{(#1)}} \newcommand{\xdno}[1]{\boldsymbol{x}_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\Real}{\mathbb{R}} \newcommand{\Integer}{\mathbb{Z}} $$

Numerical integration

Finite element codes usually apply numerical approximations to integrals. Since the integrands in the coefficient matrix often are (lower-order) polynomials, integration rules that can integrate polynomials exactly are popular.

The numerical integration rules can be expressed in a common form, $$ \begin{equation} \int_{-1}^{1} g(X)\dX \approx \sum_{j=0}^M w_j g(\bar X_j), \tag{107} \end{equation} $$ where \( \bar X_j \) are integration points and \( w_j \) are integration weights, \( j=0,\ldots,M \). Different rules correspond to different choices of points and weights.

The very simplest method is the Midpoint rule, $$ \begin{equation} \int_{-1}^{1} g(X)\dX \approx 2g(0),\quad \bar X_0=0,\ w_0=2, \tag{108} \end{equation} $$ which integrates linear functions exactly.

Newton-Cotes rules

The Newton-Cotes rules are based on a fixed uniform distribution of the integration points. The first two formulas in this family are the well-known Trapezoidal rule, $$ \begin{equation} \int_{-1}^{1} g(X)\dX \approx g(-1) + g(1),\quad \bar X_0=-1,\ \bar X_1=1,\ w_0=w_1=1, \tag{109} \end{equation} $$ and Simpson's rule, $$ \begin{equation} \int_{-1}^{1} g(X)\dX \approx \frac{1}{3}\left(g(-1) + 4g(0) + g(1)\right), \tag{110} \end{equation} $$ where $$ \begin{equation} \bar X_0=-1,\ \bar X_1=0,\ \bar X_2=1,\ w_0=w_2=\frac{1}{3},\ w_1=\frac{4}{3}\tp \tag{111} \end{equation} $$ Newton-Cotes rules up to five points is supported in the module file numint.py.

For higher accuracy one can divide the reference cell into a set of subintervals and use the rules above on each subinterval. This approach results in composite rules, well-known from basic introductions to numerical integration of \( \int_{a}^{b}f(x)\dx \).

Gauss-Legendre rules with optimized points

More accurate rules, for a given \( M \), arise if the location of the integration points are optimized for polynomial integrands. The Gauss-Legendre rules (also known as Gauss-Legendre quadrature or Gaussian quadrature) constitute one such class of integration methods. Two widely applied Gauss-Legendre rules in this family have the choice $$ \begin{align} M=1&:\quad \bar X_0=-\frac{1}{\sqrt{3}},\ \bar X_1=\frac{1}{\sqrt{3}},\ w_0=w_1=1 \tag{112}\\ M=2&:\quad \bar X_0=-\sqrt{\frac{3}{{5}}},\ \bar X_0=0,\ \bar X_2= \sqrt{\frac{3}{{5}}},\ w_0=w_2=\frac{5}{9},\ w_1=\frac{8}{9}\tp \tag{113} \end{align} $$ These rules integrate 3rd and 5th degree polynomials exactly. In general, an \( M \)-point Gauss-Legendre rule integrates a polynomial of degree \( 2M+1 \) exactly. The code numint.py contains a large collection of Gauss-Legendre rules.