Defines the set of hierarchical monomials in the 1D reference element (the interval \([-1.1]\)) of the form \(x^\alpha\), for \(\alpha=0,\dots,p+1\). More...

#include <ptems/spaces/monomials.hpp>

Inheritance diagram for ptems::Monomials< 1 >:

Collaboration diagram for ptems::Monomials< 1 >:

Static Public Member Functions
template<typename T >
static T *	Polynomials (const ReferenceElement< 1 > &, const Vector< 1 > &pt, T *result, std::size_t polydeg)
	Computes the value of the first \(p+1\) Legendre polynomials on the reference interval \([-1,1]\). More...

template<typename T >
static FuncAndGradData< 1, T > *	Gradient (const ReferenceElement< 1 > &, const Vector< 1 > &pt, FuncAndGradData< 1, T > *result, std::size_t polydeg)
	Computes the value of the first \(p+1\) Legendre polynomials, and their first order derivatives, on the reference interval \(4[-1.1]\). More...

template<typename T >
static FuncGradAndHessianData< 1, T > *	Hessian (const ReferenceElement< 1 > &, const Vector< 1 > &pt, FuncGradAndHessianData< 1, T > *result, std::size_t polydeg)
	Computes the value of the first \(p+1\) Legendre polynomials, and their first and second order derivatives, on the reference interval \(4[-1.1]\). More...

template<typename T = double>
constexpr static T	Integrate (const ReferenceElement< 1 > &, [[maybe_unused]] std::size_t polydeg, std::size_t i)
	Returns the value of integrating over the reference element the specified monomial. More...

template<typename T = double>
constexpr static T	Integrate (const ReferenceElement< 1 > &refElement, std::size_t polydeg, std::size_t i, std::size_t j)
	Returns the value of integrating over the reference element the multiplication of the specified monomials. More...

template<typename T = double>
static Vector< 1 >	EstimateAnalyticity ([[maybe_unused]] const ReferenceElement< 1 > &refElement, [[maybe_unused]] std::size_t polydeg, [[maybe_unused]] T *coefficients)
	Estimates the analyticity of a function with the current basis and specified coefficients. More...

static constexpr std::size_t	NumberDoFs ([[maybe_unused]] const ReferenceElement< 1 > &refElement, std::size_t polydeg)
	Gets the number of degrees of freedom required for a polynomial of the specified degree on the specified reference element. More...

Detailed Description

Defines the set of hierarchical monomials in the 1D reference element (the interval \([-1.1]\)) of the form \(x^\alpha\), for \(\alpha=0,\dots,p+1\).

Member Function Documentation

◆ EstimateAnalyticity()

template<typename T = double>

static Vector<1> ptems::Monomials< 1 >::EstimateAnalyticity	(	[[maybe_unused] ] const ReferenceElement< 1 > &	refElement,
		[[maybe_unused] ] std::size_t	polydeg,
		[[maybe_unused] ] T *	coefficients
	)

inlinestatic

Estimates the analyticity of a function with the current basis and specified coefficients.

Template Parameters

T	The type of the coefficients (default=double)

Parameters

refElement	The reference element
polydeg	The polynomial degree of the function
coefficients	Array of coefficients for each basis. Must be length of NumberDoFs

Returns: Vector of analyticity for each dimension in range [0,1], or -1 if analyticity computation invalid

◆ Gradient()

template<typename T >

static FuncAndGradData<1,T>* ptems::Monomials< 1 >::Gradient	(	const ReferenceElement< 1 > &	,
		const Vector< 1 > &	pt,
		FuncAndGradData< 1, T > *	result,
		std::size_t	polydeg
	)

inlinestatic

Computes the value of the first \(p+1\) Legendre polynomials, and their first order derivatives, on the reference interval \(4[-1.1]\).

Template Parameters

T	The type of the result

Parameters

pt	The point to compute the values of the Legendre polynomial at
result	Array of at least `polydeg+1` entries to store the result into
polydeg	The polynomial degree of the Legendre basis functions

Returns: Pointer to the entry AFTER the last basis filled (=result+polydeg+1)

◆ Hessian()

template<typename T >

static FuncGradAndHessianData<1,T>* ptems::Monomials< 1 >::Hessian	(	const ReferenceElement< 1 > &	,
		const Vector< 1 > &	pt,
		FuncGradAndHessianData< 1, T > *	result,
		std::size_t	polydeg
	)

inlinestatic

Computes the value of the first \(p+1\) Legendre polynomials, and their first and second order derivatives, on the reference interval \(4[-1.1]\).

Template Parameters

T	The type of the result

Parameters

pt	The point to compute the values of the Legendre polynomial at
result	Array of at least `polydeg+1` entries to store the result into
polydeg	The polynomial degree of the Legendre basis functions

Returns: Pointer to the entry AFTER the last basis filled (=result+polydeg+1)

◆ Integrate() [1/2]

template<typename T = double>

constexpr static T ptems::Monomials< 1 >::Integrate	(	const ReferenceElement< 1 > &	,
		[[maybe_unused] ] std::size_t	polydeg,
		std::size_t	i
	)

inlinestaticconstexpr

Returns the value of integrating over the reference element the specified monomial.

Note: This is usually an analytical value

Template Parameters

T	The type of the result (default=double)

Parameters

polydeg	The maximum polynomial degree of the basis
i	The index of the monomial \(M_i\)

Returns: T The integration over the reference interval \(\int_{-1}^1 M_i M_j \dx\)

◆ Integrate() [2/2]

template<typename T = double>

constexpr static T ptems::Monomials< 1 >::Integrate	(	const ReferenceElement< 1 > &	refElement,
		std::size_t	polydeg,
		std::size_t	i,
		std::size_t	j
	)

inlinestaticconstexpr

Returns the value of integrating over the reference element the multiplication of the specified monomials.

Note: This is usually an analytical value

Template Parameters

T	The type of the result (default=double)

Parameters

refElement	The reference element
polydeg	The maximum polynomial degree of the monomials
i	The index of the first monomial \(M_i\)
j	The index of the second monomial \(M_j\)

Returns: T The integration over the reference interval \(\int_{-1}^1 M_i M_j \dx\)

◆ NumberDoFs()

static constexpr std::size_t ptems::Polynomials< 1 >::NumberDoFs	(	[[maybe_unused] ] const ReferenceElement< 1 > &	refElement,
		std::size_t	polydeg
	)

inlinestaticconstexprinherited

Gets the number of degrees of freedom required for a polynomial of the specified degree on the specified reference element.

Parameters

refElement	The reference interval (parameter is ignored)
polydeg	The degree of the polynomial

Returns: The number of degrees of freedom

◆ Polynomials()

template<typename T >

static T* ptems::Monomials< 1 >::Polynomials	(	const ReferenceElement< 1 > &	,
		const Vector< 1 > &	pt,
		T *	result,
		std::size_t	polydeg
	)

inlinestatic

Computes the value of the first \(p+1\) Legendre polynomials on the reference interval \([-1,1]\).

Template Parameters

T	The type of the result

Parameters

pt	The point to compute the values of the Legendre polynomial at
result	Array of at least `polydeg+1` entries to store the result into
polydeg	The polynomial degree of the Legendre basis functions

Returns: Pointer to the entry AFTER the last basis filled (=result+polydeg+1)

The documentation for this struct was generated from the following file:

ptems/spaces/monomials.hpp

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ EstimateAnalyticity()

◆ Gradient()

◆ Hessian()

◆ Integrate() [1/2]

◆ Integrate() [2/2]

◆ NumberDoFs()

◆ Polynomials()