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Quadrature formulae

\newcommand{\boldx}{\mathbf{x}} \newcommand{\boldxi}{\boldsymbol{\xi}}

The quadrature formula is like the following:

int1d#

Quadrature formula on an edge.

Notations#

|D| is the measure of the edge D.

For a shake of simplicity, we denote: with 0\leq t\leq 1; \boldx=(1-t)\boldx_0+t\boldx_1.

qf1pE#

1
int1d(Th, qfe=qf1pE)( ... )
or
1
int1d(Th, qfe=qforder=2)( ... )

This quadrature formula is exact on \P_1.

\int_{D}{f(\boldx)} \approx |D|g\left(\frac{1}{2}\right)

qf2pE#

1
int1d(Th, qfe=qf2pE)( ... )
or
1
int1d(Th, qfe=qforder=3)( ... )

This quadrature formula is exact on \P_3.

\int_{D}{f(\boldx)} \approx \frac{|D|}{2}\left( g\left( \frac{1+\sqrt{1/3}}{2} \right) + g\left( \frac{1-\sqrt{1/3}}{2} \right) \right)

qf3pE#

1
int1d(Th, qfe=qf3pE)( ... )
or
1
int1d(Th, qfe=qforder=6)( ... )

This quadrature formula is exact on \P_5.

\int_{D}{f(\boldx)} \approx \frac{|D|}{18}\left( 5g\left( \frac{1+\sqrt{3/5}}{2} \right) + 8g\left( \frac{1}{2} \right) + 5g\left( \frac{1-\sqrt{3/5}}{2} \right) \right)

qf4pE#

1
int1d(Th, qfe=qf4pE)( ... )
or
1
int1d(Th, qfe=qforder=8)( ... )

This quadrature formula is exact on \P_7.

\int_{D}{f(\boldx)} \approx \frac{|D|}{72}\left( (18-\sqrt{30})g\left( \frac{1-\frac{\sqrt{525+70\sqrt{30}}}{35}}{2} \right) + (18-\sqrt{30})g\left( \frac{1+\frac{\sqrt{525+70\sqrt{30}}}{35}}{2} \right) + (18+\sqrt{30})g\left( \frac{1-\frac{\sqrt{525-70\sqrt{30}}}{35}}{2} \right) + (18+\sqrt{30})g\left( \frac{1+\frac{\sqrt{525-70\sqrt{30}}}{35}}{2} \right) \right)

qf5pE#

1
int1d(Th, qfe=qf5pE)( ... )
or
1
int1d(Th, qfe=qforder=10)( ... )

This quadrature formula is exact on \P_9.

\int_{D}{f(\boldx)} \approx |D|\left( \frac{(332-13\sqrt{70})}{1800}g\left( \frac{1-\frac{\sqrt{245+14\sqrt{70}}}{21}}{2} \right) + \frac{(332-13\sqrt{70})}{1800}g\left( \frac{1+\frac{\sqrt{245+14\sqrt{70}}}{21}}{2} \right) + \frac{64}{225}g\left( \frac{1}{2} \right) + \frac{(332+13\sqrt{70})}{1800}g\left( \frac{1-\frac{\sqrt{245-14\sqrt{70}}}{21}}{2} \right) + \frac{(332+13\sqrt{70})}{1800}g\left( \frac{1+\frac{\sqrt{245-14\sqrt{70}}}{21}}{2} \right) \right)

qf1pElump#

1
int1d(Th, qfe=qf1pElump)( ... )

This quadrature formula is exact on \P_2.

\int_{D}{f(\boldx)} \approx \frac{|D|}{2}\left( g\left( 0 \right) + g\left( 1 \right) \right)

int2d#

qf1pT#

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qf2pT#

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qf5pT#

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qf1pTlump#

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qf2pT4P1#

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qf7pT#

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qf9pT#

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int3d#

qfV1#

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qfV2#

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qfV5#

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qfV1lump#

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