2.6. Numerical integration#
In the discrete equations of the finite element method, integrals appear. These need to be evualated
For example in the 1D Poisson equation it is possible to perform this analytically, as follows.
In practice, however, numerical integration is performed. The idea behind numerical integration is that an integral can be replaced by a weighted sum, as follows:
The function
The following two integration schemes are relevant for Finite Element Analysis:
Newton-Cotes
Gauss Integration
A Newton-Cotes scheme uses equally spaced integration points. In this scheme, with the appropriate set of weights,
Let’s consider a reference element defined from -1 to +1 in a local
For a O-th order polynomial (
) the position of the integration point is not important, as long as the weight is equal to the length of the domain, which in this case is 2.For a 1-st order polynomial (
), still we can be exact with one integration point if and only is the integration point is positioned at the centre of ξ-axis.For a 2-nd order polynomial (
), exact integration is possible with two integration points, located at and weight 1. In fact, this integration scheme is also exact for 3-rd order polynomials.
The following rule applies, regarding the order of the polynomial
This information can be summarised in the following manner.

Fig. 2.12 Gauss Integration points#
Number of points |
Position |
weight |
Polynomial order |
---|---|---|---|