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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 (default)

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


or

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


This quadrature formula is the default one and be 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 $$\mathbb{P}_1$$.

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

## int2d

Note

Complete formulas are no longer detailed

### qf1pT

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


or

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


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

### qf2pT

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


or

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


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

### qf5pT (default)

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


or

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


This quadrature formula is the default and be exact on $$\mathbb{P}_5$$.

### qf1pTlump

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


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

### qf2pT4P1

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


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

### qf7pT

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


or

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


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

### qf9pT

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


or

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


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

## int3d

### qfV1

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


or

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


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

### qfV2

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


or

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


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

### qfV5 (default)

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


or

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


This quadrature formula is the default one and be exact on $$\mathbb{P}_5$$.

### qfV1lump

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


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