Manicore::PEC< dimension > Class Template Reference

Implement the discrete spaces of DDR-PEC. More...

#include <pec.hpp>

Public Member Functions
	PEC (Mesh< dimension > const &mesh, int r, bool use_threads=true, std::array< int, dimension > const *dqr_p=nullptr, std::ostream &output=std::cout)
double	getTopScaling (size_t i_cell) const
	Return the characteristic scale \(h_T\) of a top dimensional cell \(T\). More...
Eigen::MatrixXd	get_mass (size_t k, size_t d, size_t i_cell) const
	Return the mass matrix for the k-forms on the i-th d-cell in the basis PL(r,k,d) More...
Eigen::MatrixXd	get_trace (size_t k, size_t d, size_t i_cell, size_t j_bd) const
	Return the trace for the k-forms on the i-th d-cell onto its j-th (d-1)-neighbour. More...
Eigen::MatrixXd	get_starTrace (size_t k, size_t d, size_t i_cell, size_t j_bd) const
	Return the Hodge dual of the trace for the Hodge dual of \((d-k)\)-forms on the i-th d-cell onto its j-th (d-1)-neighbour. More...
const Eigen::MatrixXd &	get_diff (size_t l, size_t d) const
	Return the image of the differential operator. More...
const Eigen::MatrixXd &	get_Koszul (size_t l, size_t d) const
	Return the image of the Koszul operator. More...
const Eigen::MatrixXd &	get_diff_as_degr (size_t l, size_t d) const
	Return the image of the differential operator. More...
const Eigen::MatrixXd &	get_trimmed (size_t l, size_t d) const
	Return the image of the trimmed polynomial basis. More...
const Eigen::MatrixXd &	get_reduced_Koszul_m1 (size_t l, size_t d) const
	Return the image of the Koszul operator. More...

Detailed Description

template<size_t dimension>
class Manicore::PEC< dimension >

Implement the discrete spaces of DDR-PEC.

This class compute and store the matrices of the differential operator, the Koszul operator, the trimmed polynomial basis, and the masses and traces of every cell.

The matrices for the differential, Koszul and trimmed basis are global and shared for all cells. The mass matrices and traces are computed and stored for each cell.

The basis used for the operator are given in exterior_algebra.hpp

Template Parameters

dimension

Dimension of the manifold

Constructor & Destructor Documentation

PEC()

template<size_t dimension>

PEC::PEC	(	Mesh< dimension > const &	mesh,
		int	r,
		bool	use_threads = true,
		std::array< int, dimension > const *	dqr_p = nullptr,
		std::ostream &	output = std::cout
	)

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Parameters

mesh	Mesh to use
r	Polynomial degree
use_threads	Enable pthreads parallelism
dqr_p	Degree of quadrature used to generate the mass matrices. It cannot be exact for generic metric and default to a pretty high degree. Use a lower degree if the metric and cells are almost flat.
output	Output stream for status messages.

Member Function Documentation

get_diff()

template<size_t dimension>

const Eigen::MatrixXd& Manicore::PEC< dimension >::get_diff ( size_t  l, size_t  d  ) const

inline

Return the image of the differential operator.

Returns

\( \mathcal{P}_r\Lambda^l(\mathbb{R}^d) \rightarrow \mathcal{P}_{r-1}\Lambda^{l+1}(\mathbb{R}^{d}) \)

Parameters

l	Form degree
d	Dimension

get_diff_as_degr()

template<size_t dimension>

const Eigen::MatrixXd& Manicore::PEC< dimension >::get_diff_as_degr ( size_t  l, size_t  d  ) const

inline

Return the image of the differential operator.

Returns

\( \mathcal{P}_r\Lambda^l(\mathbb{R}^d) \rightarrow \mathcal{P}_{r}\Lambda^{l+1}(\mathbb{R}^{d}) \)

Parameters

l	Form degree
d	Dimension

get_Koszul()

template<size_t dimension>

const Eigen::MatrixXd& Manicore::PEC< dimension >::get_Koszul ( size_t  l, size_t  d  ) const

inline

Return the image of the Koszul operator.

Returns

\( \mathcal{P}_r\Lambda^l(\mathbb{R}^d) \rightarrow \mathcal{P}_{r+1}\Lambda^{l-1}(\mathbb{R}^{d}) \)

Parameters

l	Form degree
d	Dimension

get_mass()

template<size_t dimension>

Eigen::MatrixXd PEC::get_mass	(	size_t	k,
		size_t	d,
		size_t	i_cell
	)		const

Return the mass matrix for the k-forms on the i-th d-cell in the basis PL(r,k,d)

Returns

Mass matrix in the basis \( \mathcal{P}_r\Lambda^k(\mathbb{R}^d)\)

Parameters

k	Form degree
d	Cell dimension
i_cell	Cell index

get_reduced_Koszul_m1()

template<size_t dimension>

const Eigen::MatrixXd& Manicore::PEC< dimension >::get_reduced_Koszul_m1 ( size_t  l, size_t  d  ) const

inline

Return the image of the Koszul operator.

Returns

\( \mathcal{P}_{r-1}\Lambda^l(\mathbb{R}^d) \rightarrow \mathcal{P}_{r}\Lambda^{l-1}(\mathbb{R}^{d}) \)

Parameters

l	Form degree
d	Dimension

get_starTrace()

template<size_t dimension>

Eigen::MatrixXd PEC::get_starTrace	(	size_t	k,
		size_t	d,
		size_t	i_cell,
		size_t	j_bd
	)		const

Return the Hodge dual of the trace for the Hodge dual of \((d-k)\)-forms on the i-th d-cell onto its j-th (d-1)-neighbour.

Returns

The matrix of the operator \(\star \text{tr} \star^{-1}\) in the basis \( \mathcal{P}_r\Lambda^{d-k}(\mathbb{R}^d) \rightarrow \mathcal{P}_r\Lambda^{d-1-k}(\mathbb{R}^{d-1}) \)

Parameters

k	Form degree
d	Cell dimension
i_cell	Cell index
j_bd	Relative index of the boundary element (e.g. between 0 and 2 for a triangle)

get_trace()

template<size_t dimension>

Eigen::MatrixXd PEC::get_trace	(	size_t	k,
		size_t	d,
		size_t	i_cell,
		size_t	j_bd
	)		const

Return the trace for the k-forms on the i-th d-cell onto its j-th (d-1)-neighbour.

Returns

Trace matrix in the basis \( \mathcal{P}_r\Lambda^k(\mathbb{R}^d) \rightarrow \mathcal{P}_r\Lambda^k(\mathbb{R}^{d-1}) \)

Parameters

k	Form degree
d	Cell dimension
i_cell	Cell index
j_bd	Relative index of the boundary element (e.g. between 0 and 2 for a triangle)

get_trimmed()

template<size_t dimension>

const Eigen::MatrixXd& Manicore::PEC< dimension >::get_trimmed ( size_t  l, size_t  d  ) const

inline

Return the image of the trimmed polynomial basis.

Returns

\( \mathcal{P}_r^{-}\Lambda^l(\mathbb{R}^d) \rightarrow \mathcal{P}_{r}\Lambda^{l}(\mathbb{R}^{d}) \)

Parameters

l	Form degree
d	Dimension

getTopScaling()

template<size_t dimension>

double Manicore::PEC< dimension >::getTopScaling ( size_t  i_cell ) const

inline

Return the characteristic scale \(h_T\) of a top dimensional cell \(T\).

Computed as the \(n\)th root of the volume of the cell

Parameters

i_cell

Cell index

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The documentation for this class was generated from the following files:

• src/DDR/pec.hpp
• src/DDR/pec.cpp