# Numerical integration (1)¶

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,

$\int_{-1}^{1} g(X)dX \approx \sum_{j=0}^M w_j g(\bar X_j),$

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,

$\int_{-1}^{1} g(X)dX \approx 2g(0),\quad \bar X_0=0,\ w_0=2,$

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,

(1)$\int_{-1}^{1} g(X)dX \approx g(-1) + g(1),\quad \bar X_0=-1,\ \bar X_1=1,\ w_0=w_1=1,$

and Simpson’s rule,

$\int_{-1}^{1} g(X)dX \approx \frac{1}{3}\left(g(-1) + 4g(0) + g(1)\right),$

where

$\bar X_0=-1,\ \bar X_1=0,\ \bar X_2=1,\ w_0=w_2=\frac{1}{3},\ w_1=\frac{4}{3}{\thinspace .}$

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

$M=1:\quad \bar X_0=-\frac{1}{\sqrt{3}},\ \bar X_1=\frac{1}{\sqrt{3}},\ w_0=w_1=1$
$M=2:\quad \bar X_0=-\sqrt{\frac{3}{{5}}},\ \bar X_0=0,\ \bar X_2= \sqrt{\frac{3}{{5}}},\ w_0=w_2=\frac{5}{9},\ w_1=\frac{8}{9}{\thinspace .}$

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.