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Quadrature formulae

The quadrature formula is like the following:

\[\int_{D}{f(\boldx)} \approx \sum_{\ell=1}^{L}{\omega_\ell f(\boldxi_\ell)}\]

int1d

Quadrature formula on an edge.

Notations

\(|D|\) is the measure of the edge \(D\).

For a shake of simplicity, we denote:

\[f(\boldx) = g(t)\]

with \(0\leq t\leq 1\); \(\boldx=(1-t)\boldx_0+t\boldx_1\).

qf1pE

1
int1d(Th, qfe=qf1pE)( ... )

or

1
int1d(Th, qforder=2)( ... )

This quadrature formula is exact on \(\mathbb{P}_1\).

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

qf2pE

1
int1d(Th, qfe=qf2pE)( ... )

or

1
int1d(Th, qforder=3)( ... )

This quadrature formula is exact on \(\mathbb{P}_3\).

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

qf3pE

1
int1d(Th, qfe=qf3pE)( ... )

or

1
int1d(Th, qforder=6)( ... )

This quadrature formula is exact on \(\mathbb{P}_5\).

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

qf4pE

1
int1d(Th, qfe=qf4pE)( ... )

or

1
int1d(Th, qforder=8)( ... )

This quadrature formula is exact on \(\mathbb{P}_7\).

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

qf5pE

1
int1d(Th, qfe=qf5pE)( ... )

or

1
int1d(Th, qforder=10)( ... )

This quadrature formula is exact on \(\mathbb{P}_9\).

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

qf1pElump

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

This quadrature formula is exact on \(\mathbb{P}_2\).

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

int2d

Note

Complete formulas are no longer detailed

qf1pT

1
 int2d(Th, qfe=qf1pT)( ... )

or

1
 int2d(Th, qforder=2)( ... )

This quadrature formula is exact on \(\mathbb{P}_1\).

qf2pT

1
 int2d(Th, qfe=qf2pT)( ... )

or

1
 int2d(Th, qforder=3)( ... )

This quadrature formula is exact on \(\mathbb{P}_2\).

qf5pT

1
 int2d(Th, qfe=qf5pT)( ... )

or

1
 int2d(Th, qforder=6)( ... )

This quadrature formula is exact on \(\mathbb{P}_5\).

qf1pTlump

1
 int2d(Th, qfe=qf1pTlump)( ... )

This quadrature formula is exact on \(\mathbb{P}_1\).

qf2pT4P1

1
 int2d(Th, qfe=qf2pT4P1)( ... )

This quadrature formula is exact on \(\mathbb{P}_1\).

qf7pT

1
 int2d(Th, qfe=qf7pT)( ... )

or

1
 int2d(Th, qforder=8)( ... )

This quadrature formula is exact on \(\mathbb{P}_7\).

qf9pT

1
 int2d(Th, qfe=qf9pT)( ... )

or

1
 int2d(Th, qforder=10)( ... )

This quadrature formula is exact on \(\mathbb{P}_9\).

int3d

qfV1

1
 int3d(Th, qfe=qfV1)( ... )

or

1
 int3d(Th, qforder=2)( ... )

This quadrature formula is exact on \(\mathbb{P}_1\).

qfV2

1
 int3d(Th, qfe=qfV2)( ... )

or

1
 int3d(Th, qforder=3)( ... )

This quadrature formula is exact on \(\mathbb{P}_2\).

qfV5

1
 int3d(Th, qfe=qfV5)( ... )

or

1
 int3d(Th, qforder=6)( ... )

This quadrature formula is exact on \(\mathbb{P}_5\).

qfV1lump

1
 int3d(Th, qfe=qfV1lump)( ... )

This quadrature formula is exact on \(\mathbb{P}_1\).

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