DUNE PDELab (git)
Integrate monomials over the reference element. More...
#include <dune/pdelab/finiteelement/l2orthonormal.hh>
Detailed Description
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
The documentation for this class was generated from the following file:
- dune/pdelab/finiteelement/l2orthonormal.hh