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:
where 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:
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 and weights for .
The coordinates and related weights are:
|Number of points, p||Points, &xii||Weights, wi|