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Gauss quadrature

Element integrals are rarely evaluated analytically – instead numerical integration at a few points is used. Gauss quadrature chooses the point locations and weights so that low-degree polynomials are integrated exactly with very few points, which makes FEM computations fast.

\int_{-1}^1 g(\xi)\,d\xi \approx \sum_i w_ig(\xi_i)

numerical integration at Gauss points

Symbols

$\xi_i$	Gauss point
$w_i$	weight
$n$	number of points

Example

1 point ( $\xi=0$ , weight 2) for $g(\xi)=3\xi^2+2\xi+4$ :

gives $2\cdot 4 = 8$ , while the exact value is $3\cdot\tfrac23+8=10$ – the $\xi^2$ term is lost.

n

Gauss points integrate polynomials up to degree

2n-1

exactly.

Practise weak form and shape functions for free →

← Weak form · The CST element →

Part of Finite Element Method: Weak form and shape functions.