# Numerical Integration

(Difference between revisions)
 Revision as of 11:12, 3 November 2009 (view source)JMora (Talk | contribs) (→Gauss-Legendre Numerical Integration)← Older edit Revision as of 11:17, 3 November 2009 (view source)JMora (Talk | contribs) (→Gauss-Legendre Numerical Integration)Newer edit → Line 16: Line 16: :$I=\int_{-1}^{+1} f(\xi) d\xi$ :$I=\int_{-1}^{+1} f(\xi) d\xi$ + + + 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 $\xi \,$ and weights $wi \,$ for $i = 1,...,n \,$. + + + :$\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)$ == References == == References ==

## Revision as of 11:17, 3 November 2009

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 $\xi \,$ and weights $wi \,$ for $i = 1,...,n \,$. $\int_{-1}^{+1} f(\xi)\,d\xi \approx \sum_{i=1}^p w_i f(\xi_i)$