DUNE PDELab (git)

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
FieldTypeType to represent coefficients after computation.
kThe polynomial degreee.
dThe space dimension.
GeometryType::BasicTypeThe reference element
ComputationFieldTypeType to do computations with. Might be high precission.
basisTypeType of the polynomial basis. eiter Pk or Qk

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