basix::quadrature Namespace Reference

basix More...

Functions
xt::xtensor< double, 2 >	compute_jacobi_deriv (double a, std::size_t n, std::size_t nderiv, const xtl::span< const double > &x)
std::vector< double >	compute_gauss_jacobi_points (double a, int m)
std::pair< xt::xarray< double >, std::vector< double > >	compute_gauss_jacobi_rule (double a, int m)
	Gauss-Jacobi quadrature rule (points and weights) More...
std::pair< xt::xarray< double >, std::vector< double > >	make_quadrature_line (int m)
std::pair< xt::xarray< double >, std::vector< double > >	make_quadrature_triangle_collapsed (std::size_t m)
std::pair< xt::xarray< double >, std::vector< double > >	make_quadrature_tetrahedron_collapsed (std::size_t m)
std::pair< xt::xarray< double >, std::vector< double > >	make_quadrature (const std::string &rule, cell::type celltype, int m)
std::pair< xt::xarray< double >, std::vector< double > >	make_gll_line (int m)
std::pair< xt::xarray< double >, std::vector< double > >	compute_gll_rule (int m)
	GLL quadrature rule (points and weights)

Detailed Description

basix

Integration using Gauss-Jacobi quadrature on simplices. Other shapes can be obtained by using a product.

Todo:

• pyramid

Function Documentation

compute_gauss_jacobi_points()

std::vector< double > basix::quadrature::compute_gauss_jacobi_points	(	double	a,
		int	m
	)

Finds the m roots of \(P_{m}^{a,0}\) on [-1,1] by Newton's method.

Parameters

[in]	a	weight in Jacobi (b=0)
[in]	m	order

Returns

list of points in 1D

Computes the m roots of \(P_{m}^{a,0}\) on [-1,1] by Newton's method. The initial guesses are the Chebyshev points. Algorithm implemented from the pseudocode given by Karniadakis and Sherwin

compute_gauss_jacobi_rule()

std::pair< xt::xarray< double >, std::vector< double > > basix::quadrature::compute_gauss_jacobi_rule	(	double	a,
		int	m
	)

Gauss-Jacobi quadrature rule (points and weights)

Note

Computes on [-1, 1]

compute_jacobi_deriv()

xt::xtensor< double, 2 > basix::quadrature::compute_jacobi_deriv	(	double	a,
		std::size_t	n,
		std::size_t	nderiv,
		const xtl::span< const double > &	x
	)

Evaluate the nth Jacobi polynomial and derivatives with weight parameters (a, 0) at points x

Parameters

[in]	a	Jacobi weight a
[in]	n	Order of polynomial
[in]	nderiv	Number of derivatives (if zero, just compute polynomial itself)
[in]	x	Points at which to evaluate

Returns

Array of polynomial derivative values (rows) at points (columns)

make_gll_line()

std::pair< xt::xarray< double >, std::vector< double > > basix::quadrature::make_gll_line

(

int

m

)

Compute GLL line quadrature rule on [0, 1]

Parameters

m

order

Returns

list of 1D points, list of weights

make_quadrature()

std::pair< xt::xarray< double >, std::vector< double > > basix::quadrature::make_quadrature	(	const std::string &	rule,
		cell::type	celltype,
		int	m
	)

Utility for quadrature rule on reference cell

Parameters

[in]	rule	Name of quadrature rule (or use "default")
[in]	celltype
[in]	m	Maximum degree of polynomial that this quadrature rule will integrate exactly

Returns

List of points and list of weights. The number of points arrays has shape (num points, gdim)

make_quadrature_line()

std::pair< xt::xarray< double >, std::vector< double > > basix::quadrature::make_quadrature_line

(

int

m

)

Compute line quadrature rule on [0, 1]

Parameters

m

order

Returns

list of points, list of weights

make_quadrature_tetrahedron_collapsed()

std::pair< xt::xarray< double >, std::vector< double > > basix::quadrature::make_quadrature_tetrahedron_collapsed

(

std::size_t

m

)

Compute tetrahedron quadrature rule on [0, 1]x[0, 1]x[0, 1]

Parameters

[in]

m

order

Returns

List of points, list of weights. The number of points arrays has shape (num points, gdim)

make_quadrature_triangle_collapsed()

std::pair< xt::xarray< double >, std::vector< double > > basix::quadrature::make_quadrature_triangle_collapsed

(

std::size_t

m

)

Compute triangle quadrature rule on [0, 1]x[0, 1]

Parameters

[in]

m

order

Returns

list of points, list of weights