Reference documentation for deal.II version Git c064f80013 2020-07-04 08:24:28 -0400
Polynomials::Hierarchical Class Reference

#include <deal.II/base/polynomial.h>

Inheritance diagram for Polynomials::Hierarchical:
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## Public Member Functions

Hierarchical (const unsigned int p)

Public Member Functions inherited from Polynomials::Polynomial< double >
Polynomial (const std::vector< double > &coefficients)

Polynomial (const unsigned int n)

Polynomial (const std::vector< Point< 1 >> &lagrange_support_points, const unsigned int evaluation_point)

Polynomial ()

double value (const double x) const

void value (const double x, std::vector< double > &values) const

void value (const double x, const unsigned int n_derivatives, double *values) const

unsigned int degree () const

void scale (const double factor)

void shift (const number2 offset)

Polynomial< doublederivative () const

Polynomial< doubleprimitive () const

Polynomial< double > & operator*= (const double s)

Polynomial< double > & operator*= (const Polynomial< double > &p)

Polynomial< double > & operator+= (const Polynomial< double > &p)

Polynomial< double > & operator-= (const Polynomial< double > &p)

bool operator== (const Polynomial< double > &p) const

void print (std::ostream &out) const

void serialize (Archive &ar, const unsigned int version)

virtual std::size_t memory_consumption () const

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&) noexcept

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&) noexcept

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

## Static Public Member Functions

static std::vector< Polynomial< double > > generate_complete_basis (const unsigned int degree)

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Static Private Member Functions

static void compute_coefficients (const unsigned int p)

static const std::vector< double > & get_coefficients (const unsigned int p)

## Static Private Attributes

static std::vector< std::unique_ptr< const std::vector< double > > > recursive_coefficients

Protected Member Functions inherited from Polynomials::Polynomial< double >
void transform_into_standard_form ()

Static Protected Member Functions inherited from Polynomials::Polynomial< double >
static void scale (std::vector< double > &coefficients, const double factor)

static void shift (std::vector< double > &coefficients, const number2 shift)

static void multiply (std::vector< double > &coefficients, const double factor)

Protected Attributes inherited from Polynomials::Polynomial< double >
std::vector< doublecoefficients

bool in_lagrange_product_form

std::vector< doublelagrange_support_points

double lagrange_weight

## Detailed Description

Hierarchical polynomials of arbitrary degree on [0,1].

When Constructing a Hierarchical polynomial of degree p, the coefficients will be computed by a recursion formula. The coefficients are stored in a static data vector to be available when needed next time.

These hierarchical polynomials are based on those of Demkowicz, Oden, Rachowicz, and Hardy (CMAME 77 (1989) 79-112, Sec. 4). The first two polynomials are the standard linear shape functions given by $$\phi_{0}(x) = 1 - x$$ and $$\phi_{1}(x) = x$$. For $$l \geq 2$$ we use the definitions $$\phi_{l}(x) = (2x-1)^l - 1, l = 2,4,6,...$$ and $$\phi_{l}(x) = (2x-1)^l - (2x-1), l = 3,5,7,...$$. These satisfy the recursion relations $$\phi_{l}(x) = (2x-1)\phi_{l-1}, l=3,5,7,...$$ and $$\phi_{l}(x) = (2x-1)\phi_{l-1} + \phi_{2}, l=4,6,8,...$$.

The degrees of freedom are the values at the vertices and the derivatives at the midpoint. Currently, we do not scale the polynomials in any way, although better conditioning of the element stiffness matrix could possibly be achieved with scaling.

Calling the constructor with a given index p will generate the following: if p==0, then the resulting polynomial is the linear function associated with the left vertex, if p==1 the one associated with the right vertex. For higher values of p, you get the polynomial of degree p that is orthogonal to all previous ones. Note that for p==0 you therefore do not get a polynomial of degree zero, but one of degree one. This is to allow generating a complete basis for polynomial spaces, by just iterating over the indices given to the constructor.

On the other hand, the function generate_complete_basis() creates a complete basis of given degree. In order to be consistent with the concept of a polynomial degree, if the given argument is zero, it does not return the linear polynomial described above, but rather a constant polynomial.

Definition at line 495 of file polynomial.h.

## ◆ Hierarchical()

 Polynomials::Hierarchical::Hierarchical ( const unsigned int p )

Constructor for polynomial of degree p. There is an exception for p==0, see the general documentation.

Definition at line 1004 of file polynomial.cc.

## ◆ generate_complete_basis()

 std::vector< Polynomial< double > > Polynomials::Hierarchical::generate_complete_basis ( const unsigned int degree )
static

Return a vector of Hierarchical polynomial objects of degrees zero through degree, which then spans the full space of polynomials up to the given degree. Note that there is an exception if the given degree equals zero, see the general documentation of this class.

This function may be used to initialize the TensorProductPolynomials, AnisotropicPolynomials, and PolynomialSpace classes.

Definition at line 1161 of file polynomial.cc.

## ◆ compute_coefficients()

 void Polynomials::Hierarchical::compute_coefficients ( const unsigned int p )
staticprivate

Compute coefficients recursively.

Definition at line 1011 of file polynomial.cc.

## ◆ get_coefficients()

 const std::vector< double > & Polynomials::Hierarchical::get_coefficients ( const unsigned int p )
staticprivate

Get coefficients for constructor. This way, it can use the non- standard constructor of Polynomial.

Definition at line 1145 of file polynomial.cc.

## ◆ recursive_coefficients

 std::vector< std::unique_ptr< const std::vector< double > > > Polynomials::Hierarchical::recursive_coefficients
staticprivate

Vector with already computed coefficients. For each degree of the polynomial, we keep one pointer to the list of coefficients; we do so rather than keeping a vector of vectors in order to simplify programming multithread-safe. In order to avoid memory leak, we use a unique_ptr in order to correctly free the memory of the vectors when the global destructor is called.

Definition at line 540 of file polynomial.h.

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