Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType > Class Template Reference

Integrate monomials over the reference element. More...

#include <dune/pdelab/finiteelement/l2orthonormal.hh>

Detailed Description

template<typename FieldType, int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>
class Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >

Integrate monomials over the reference element.

Computes an L_2 orthonormal basis of P_k on the given reference element. The basis polynomials are stored in a monomial representation. With the matrix coeffs private to this class we have

\[ phi_i(x) = \sum_{j=0}{n_k-1} c[i][j] x^{\alpha_j} \qquad (1) \]

with n_k : the dimension of P_k alpha_j : the exponents of the j-th monomial

The class can be used to evaluate polynomials with any degree l smaller or equal to the compile-time parameter k.

Calculating derivatives. From (1) we have

\begin{align*} \partial_s \phi_i(x) &= \sum_{j=0}{n_k-1} c[i][j] \partial_s x^{(\alpha_{j1},...,\alpha_{jd})} \\ &= \sum_{j=0}{n_k-1} c[i][j] * \alpha_js * x^{\beta_j} \end{align*}

where beta_jr = alpha_jr-1 if r=s and alpha_jr else.

Template Parameters

FieldType	Type to represent coefficients after computation.
k	The polynomial degreee.
d	The space dimension.
GeometryType::BasicType	The reference element
ComputationFieldType	Type to do computations with. Might be high precission.
basisType	Type of the polynomial basis. eiter Pk or Qk

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

• dune/pdelab/finiteelement/l2orthonormal.hh