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Trapezoidal Rule for a Rectangle

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The weights of the trapezoidal rule with step size $ h=(b-a)/n$ are equal to $ h$ at inner knots and equal to $ h/2$ at the endpoints $ a$ and $ b$ of the integration interval. Hence, for a rectangle

$\displaystyle [a_1,b_1] \times [a_2,b_2] \,, $

we obtain three different weights, as is illustrated in the figure.

\includegraphics[width=0.5\linewidth]{Bild_Gewichte_Trapezregel}

More generally, for an $ m$-dimensional rectangle with step sizes $ h_\nu$ with respect to the $ \nu$-th coordinate,

$\displaystyle w_{k_1,\ldots,k_m} = h^m 2^{-\alpha_1} \cdots 2^{-\alpha_m} \,, $

where $ \alpha_\nu=0$ ($ =1$) for an inner (a boundary) index of the $ \nu$-th coordinate.

As in the univariate case, the error of the multivariate trapezoidal rule has a quadratic expansion, and, consequently, Romberg extrapolation is applicable.

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  automatisch erstellt am 17.  1. 2017