# Numerical Integration

Numerical integration refers to all the procedures, algorithms and techniques in the numerical analysis to obtain an approximate solution to a definite integral.

That is, how to obtain a numerical value of:

$\int_{\lambda_a}^{\lambda_b}\! f(\lambda)\, d\lambda.$

where $\lambda \,$ can be a 1D, 2D or 3D domain.

For our interest in the Finite Element Method, the purpose is to describe how the element matrices can be integrated numerically.

## Gauss-Legendre Numerical Integration

To fix the most basic concepts on Numerical Integration, we will focus our description on a one dimensional integration using the Gauss-Legendre quadrature, that is, to solve:

$I=\int_{-1}^{+1} f(\xi) d\xi$

The Gauss-Legendre quadrature establish that the definite integral of a function can be approximate by using a weighted sum of function values at specified points within the domain of integration. An p-point Gaussian quadrature rule is constructed to yield an exact result for polynomials of degree 2p − 1 or less by a suitable choice of the points   $\xi_i \,$   and weights   $w_i \,$   for   $i = 1, \cdots p \,$.

$\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)$

The coordinates and related weights are:

Number of points, p Points, ±ξi Weights, wi
$1\,$ $0.0 \,$ $2.0\,$
$2\,$ $\pm\sqrt{1/3}$ $1.0\,$
$3\,$ $0.0 \,$ $\frac{8}{9}$
$\pm\sqrt{3/5}$ $\frac{5}{9}$
$4\,$ $\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}$ $\tfrac{18+\sqrt{30}}{36}$
$\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}$ $\tfrac{18-\sqrt{30}}{36}$
$5\,$ $0.0 \,$ $\frac{128}{225}$
$\pm\tfrac13\sqrt{5-2\sqrt{10/7}}$ $\tfrac{322+13\sqrt{70}}{900}$
$\pm\tfrac13\sqrt{5+2\sqrt{10/7}}$ $\tfrac{322-13\sqrt{70}}{900}$

or, using numerical values:

Number of points, p Points, ±ξi Weights, wi
1 0.0 2.0
2 0.5773502692 1.0
3 0.0 0.8888888889
0.774596697 0.5555555556
4 0.3399810436 0.6521451549
0.8611363116 0.3478548451
5 0.0 0.5688888889
0.5384693101 0.4786286705
0.9061798459 0.2369268851
6 0.2386191861 0.4679139346
0.6612093865 0.3607615730
0.9324695142 0.1713244924
7 0.0 0.4179591837
0.4058451514 0.3818300505
0.7415311856 0.2797053915
0.9491079123 0.1294849662
8 0.1834346425 0.3626837834
0.5255324099 0.3137066459
0.7966664774 0.2223810345
0.9602898565 0.1012285636

## References

1. Carlos A. Felippa, "A compendium of FEM integration formulas for symbolic work", Engineering Computations, Vol. 21 No. 8, 2004, pp. 867-890, (c) Emerald Group Publishing Limited [1]
2. Numerical Integration