# 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.