Quadrature formulae

The quadrature formula is like the following:

\int_{D} f (x) \approx \sum_{ℓ = 1}^{L} ω_{ℓ} f (ξ_{ℓ})

int1d

Quadrature formula on an edge.

Notations

$| D |$ is the measure of the edge $D$ .

For a shake of simplicity, we denote:

f (x) = g (t)

with $0 \leq t \leq 1$ ; $x = (1 - t) x_{0} + t x_{1}$ .

qf1pE

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

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

This quadrature formula is exact on $P_{1}$ .

\int_{D} f (x) \approx | D | g (\frac{1}{2})

qf2pE

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

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

This quadrature formula is exact on $P_{3}$ .

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

qf3pE (default)

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

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

This quadrature formula is the default one and be exact on $P_{5}$ .

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

qf4pE

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

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

This quadrature formula is exact on $P_{7}$ .

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

qf5pE

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

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

This quadrature formula is exact on $P_{9}$ .

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

qf1pElump

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

This quadrature formula is exact on $P_{1}$ .

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

int2d

Note

Complete formulas are no longer detailed

qf1pT

 int2d(Th, qft=qf1pT)( ... )

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

This quadrature formula is exact on $P_{1}$ .

qf2pT

 int2d(Th, qft=qf2pT)( ... )

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

This quadrature formula is exact on $P_{2}$ .

qf5pT (default)

 int2d(Th, qft=qf5pT)( ... )

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

This quadrature formula is the default and be exact on $P_{5}$ .

qf1pTlump

 int2d(Th, qft=qf1pTlump)( ... )

This quadrature formula is exact on $P_{1}$ .

qf2pT4P1

 int2d(Th, qft=qf2pT4P1)( ... )

This quadrature formula is exact on $P_{1}$ .

qf7pT

 int2d(Th, qft=qf7pT)( ... )

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

This quadrature formula is exact on $P_{7}$ .

qf9pT

 int2d(Th, qft=qf9pT)( ... )

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

This quadrature formula is exact on $P_{9}$ .

int3d

qfV1

 int3d(Th, qfV=qfV1)( ... )

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

This quadrature formula is exact on $P_{1}$ .

qfV2

 int3d(Th, qfV=qfV2)( ... )

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

This quadrature formula is exact on $P_{2}$ .

qfV5 (default)

 int3d(Th, qfV=qfV5)( ... )

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

This quadrature formula is the default one and be exact on $P_{5}$ .

qfV1lump

 int3d(Th, qfV=qfV1lump)( ... )

This quadrature formula is exact on $P_{1}$ .

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Quadrature formulae

int1d

Notations

qf1pE

qf2pE

qf3pE (default)

qf4pE

qf5pE

qf1pElump

int2d

qf1pT

qf2pT

qf5pT (default)

qf1pTlump

qf2pT4P1

qf7pT

qf9pT

int3d

qfV1

qfV2

qfV5 (default)

qfV1lump

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