Reference documentation for deal.II version 9.2.0

#include <deal.II/base/tensor_product_polynomials.h>
Public Member Functions  
AnisotropicPolynomials (const std::vector< std::vector< Polynomials::Polynomial< double >>> &base_polynomials)  
void  evaluate (const Point< dim > &unit_point, std::vector< double > &values, std::vector< Tensor< 1, dim >> &grads, std::vector< Tensor< 2, dim >> &grad_grads, std::vector< Tensor< 3, dim >> &third_derivatives, std::vector< Tensor< 4, dim >> &fourth_derivatives) const override 
double  compute_value (const unsigned int i, const Point< dim > &p) const 
template<int order>  
Tensor< order, dim >  compute_derivative (const unsigned int i, const Point< dim > &p) const 
Tensor< 1, dim >  compute_grad (const unsigned int i, const Point< dim > &p) const 
Tensor< 2, dim >  compute_grad_grad (const unsigned int i, const Point< dim > &p) const 
std::string  name () const override 
virtual std::unique_ptr< ScalarPolynomialsBase< dim > >  clone () const override 
Public Member Functions inherited from ScalarPolynomialsBase< dim >  
ScalarPolynomialsBase (const unsigned int deg, const unsigned int n_polynomials)  
ScalarPolynomialsBase (ScalarPolynomialsBase< dim > &&)=default  
ScalarPolynomialsBase (const ScalarPolynomialsBase< dim > &)=default  
virtual  ~ScalarPolynomialsBase ()=default 
unsigned int  n () const 
unsigned int  degree () const 
virtual std::size_t  memory_consumption () const 
Private Member Functions  
void  compute_index (const unsigned int i, std::array< unsigned int, dim > &indices) const 
Static Private Member Functions  
static unsigned int  get_n_tensor_pols (const std::vector< std::vector< Polynomials::Polynomial< double >>> &pols) 
Private Attributes  
const std::vector< std::vector< Polynomials::Polynomial< double > > >  polynomials 
Anisotropic tensor product of given polynomials.
Given onedimensional polynomials \(P^x_1(x), P^x_2(x), \ldots\) in \(x\)direction, \(P^y_1(y), P^y_2(y), \ldots\) in \(y\)direction, and so on, this class generates polynomials of the form \(Q_{ijk}(x,y,z) = P^x_i(x)P^y_j(y)P^z_k(z)\). (With obvious generalization if dim
is in fact only 2. If dim
is in fact only 1, then the result is simply the same set of onedimensional polynomials passed to the constructor.)
If the elements of each set of base polynomials are mutually orthogonal on the interval \([1,1]\) or \([0,1]\), then the tensor product polynomials are orthogonal on \([1,1]^d\) or \([0,1]^d\), respectively.
The resulting dimdimensional
tensor product polynomials are ordered as follows: We iterate over the \(x\) coordinates running fastest, then the \(y\) coordinate, etc. For example, for dim==2
, the first few polynomials are thus \(P^x_1(x)P^y_1(y)\), \(P^x_2(x)P^y_1(y)\), \(P^x_3(x)P^y_1(y)\), ..., \(P^x_1(x)P^y_2(y)\), \(P^x_2(x)P^y_2(y)\), \(P^x_3(x)P^y_2(y)\), etc.
Definition at line 287 of file tensor_product_polynomials.h.
AnisotropicPolynomials< dim >::AnisotropicPolynomials  (  const std::vector< std::vector< Polynomials::Polynomial< double >>> &  base_polynomials  ) 
Constructor. base_polynomials
is a table of onedimensional polynomials. The number of rows in this table (the first index when indexing into base_polynomials
) needs to be equal to the space dimension, with the elements of each row (i.e., the second index) giving the polynomials that shall be used in this particular coordinate direction.
Since we want to build anisotropic polynomials, the dim
sets of polynomials passed in as arguments may of course be different, and may also vary in number.
The number of tensor product polynomials is Nx*Ny*Nz
, or with terms dropped if the number of space dimensions is less than 3.
Definition at line 454 of file tensor_product_polynomials.cc.

overridevirtual 
Compute the value and the first and second derivatives of each tensor product polynomial at unit_point
.
The size of the vectors must either be equal 0
or equal this>n()
. In the first case, the function will not compute these values.
If you need values or derivatives of all tensor product polynomials then use this function, rather than using any of the compute_value
, compute_grad
or compute_grad_grad
functions, see below, in a loop over all tensor product polynomials.
Implements ScalarPolynomialsBase< dim >.
Definition at line 580 of file tensor_product_polynomials.cc.
double AnisotropicPolynomials< dim >::compute_value  (  const unsigned int  i, 
const Point< dim > &  p  
)  const 
Compute the value of the i
th tensor product polynomial at unit_point
. Here i
is given in tensor product numbering.
Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each point value of the underlying (onedimensional) polynomials is (unnecessarily) computed several times. Instead use the compute
function, see above, with values.size()==this>n()
to get the point values of all tensor polynomials all at once and in a much more efficient way.
Definition at line 500 of file tensor_product_polynomials.cc.
Tensor<order, dim> AnisotropicPolynomials< dim >::compute_derivative  (  const unsigned int  i, 
const Point< dim > &  p  
)  const 
Compute the order
th derivative of the i
th tensor product polynomial at unit_point
. Here i
is given in tensor product numbering.
Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each derivative value of the underlying (onedimensional) polynomials is (unnecessarily) computed several times. Instead use the evaluate() function, see above, with the size of the appropriate parameter set to n() to get the point value of all tensor polynomials all at once and in a much more efficient way.
order  The derivative order. 
Tensor< 1, dim > AnisotropicPolynomials< dim >::compute_grad  (  const unsigned int  i, 
const Point< dim > &  p  
)  const 
Compute the grad of the i
th tensor product polynomial at unit_point
. Here i
is given in tensor product numbering.
Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each derivative value of the underlying (onedimensional) polynomials is (unnecessarily) computed several times. Instead use the compute
function, see above, with grads.size()==this>n()
to get the point value of all tensor polynomials all at once and in a much more efficient way.
Definition at line 516 of file tensor_product_polynomials.cc.
Tensor< 2, dim > AnisotropicPolynomials< dim >::compute_grad_grad  (  const unsigned int  i, 
const Point< dim > &  p  
)  const 
Compute the second derivative (grad_grad) of the i
th tensor product polynomial at unit_point
. Here i
is given in tensor product numbering.
Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each derivative value of the underlying (onedimensional) polynomials is (unnecessarily) computed several times. Instead use the compute
function, see above, with grad_grads.size()==this>n()
to get the point value of all tensor polynomials all at once and in a much more efficient way.
Definition at line 544 of file tensor_product_polynomials.cc.

overridevirtual 
Return the name of the space, which is AnisotropicPolynomials
.
Implements ScalarPolynomialsBase< dim >.

overridevirtual 
A sort of virtual copy constructor, this function returns a copy of the polynomial space object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
Some places in the library, for example the constructors of FE_Poly, need to make copies of polynomial spaces without knowing their exact type. They do so through this function.
Implements ScalarPolynomialsBase< dim >.
Definition at line 739 of file tensor_product_polynomials.cc.

private 
Each tensor product polynomial \(p_i\) is a product of onedimensional polynomials in each space direction. Compute the indices of these one dimensional polynomials for each space direction, given the index i
.
Definition at line 470 of file tensor_product_polynomials.cc.

staticprivate 
Given the input to the constructor, compute n_pols
.
Definition at line 727 of file tensor_product_polynomials.cc.

private 
Copy of the vector pols
of polynomials given to the constructor.
Definition at line 409 of file tensor_product_polynomials.h.