* automatic shipping of information between different processors.
*
* To that end, the constructor of the class is provided with three main objects:
- * scalar FiniteElement @projection_fe, @p mass_quadrature and @p data_quadrature
+ * scalar FiniteElement @p projection_fe, @p mass_quadrature and @p data_quadrature
* Quadrature rules.
* First, the data located at @p data_quadrature of each cell is L2-projected
- * to the continuous space defined by a single FiniteElement @projection_fe .
+ * to the continuous space defined by a single FiniteElement @p projection_fe .
* This is achieved using FETools::compute_projection_from_quadrature_points_matrix().
* In doing so the mass matrix of this element is required, which will be calculated
* with the @p mass_quadrature rule . Should the cell now belong to another processor,
* A class to calculate expansion of a scalar FE field into Fourier series
* on a reference element. The exponential form of the Fourier series is
* based on completeness and Hermitian orthogonality of the set of exponential
- * functions \f$ \phi_{\bf k}({\bf x}) = \exp(2 \pi i\, {\bf k} \cdot {\bf x})\f$.
+ * functions $ \phi_{\bf k}({\bf x}) = \exp(2 \pi i\, {\bf k} \cdot {\bf x})$.
* For example in 1D the L2-orthogonality condition reads
* @f[
* \int_0^1 \phi_k(x) \phi_l^\ast(x) dx=\delta_{kl}.
* @f]
- * Note that \f$ \phi_{\bf k} = \phi_{-\bf k}^\ast \f$.
+ * Note that $ \phi_{\bf k} = \phi_{-\bf k}^\ast $.
*
* The arbitrary scalar FE field on the reference element can be expanded in
* the complete orthogonal exponential basis as
* \int_{[0,1]^d} u({\bf x}) \phi_{\bf k}^\ast ({\bf x}) d{\bf x}\,.
* @f]
* It is this complex-valued expansion coefficients, that are calculated by
- * this class. Note that \f$ u({\bf x}) = \sum_i u_i N_i({\bf x})\f$,
- * where \f$ N_i({\bf x}) \f$ are real-valued FiniteElement shape functions.
- * Consequently \f$ c_{\bf k} \equiv c_{-\bf k}^\ast \f$ and
- * we only need to compute \f$ c_{\bf k} \f$ for positive indices
- * \f$ \bf k \f$ .
+ * this class. Note that $ u({\bf x}) = \sum_i u_i N_i({\bf x})$,
+ * where $ N_i({\bf x}) $ are real-valued FiniteElement shape functions.
+ * Consequently $ c_{\bf k} \equiv c_{-\bf k}^\ast $ and
+ * we only need to compute $ c_{\bf k} $ for positive indices
+ * $ \bf k $ .
*
* @author Denis Davydov, 2016.
*/
void ensure_existence(const unsigned int fe_index);
/**
- * Angular frequencies \f$ 2 \pi {\bf k} \f$ .
+ * Angular frequencies $ 2 \pi {\bf k} $ .
*/
Table<dim, Tensor<1,dim> > k_vectors;
* @f[
* P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[x^2-1]^n.
* @f]
- * These polynomials are orthogonal with respect to the \f$ L^2 \f$ inner
- * product on the interval \f$ [-1;1] \f$
+ * These polynomials are orthogonal with respect to the $ L^2 $ inner
+ * product on the interval $ [-1;1] $
* @f[
* \int_{-1}^1 P_m(x) P_n(x) = \frac{2}{2n + 1} \delta_{mn}
* @f]
* and are complete.
- * A family of \f$ L^2 \f$-orthogonal polynomials on \f$ [0;1] \f$ can be
+ * A family of $ L^2 $-orthogonal polynomials on $ [0;1] $ can be
* constructed via
* @f[
* \widetilde P_m = \sqrt{2} P_m(2x-1).
* @f]
*
*
- * An arbitrary scalar FE field on the reference element \f$ [0;1] \f$ can be
+ * An arbitrary scalar FE field on the reference element $ [0;1] $ can be
* expanded in the complete orthogonal basis as
* @f[
* u(x)
* c_m = \frac{2m+1}{2}
* \int_0^1 u(x) \widetilde P_m(x) dx .
* @f]
- * This class calculates coefficients \f$ c_{\bf k} \f$ using
- * \f$ dim \f$-dimensional Legendre polynomials constructed from
- * \f$ \widetilde P_m(x) \f$ using tensor product rule.
+ * This class calculates coefficients $ c_{\bf k} $ using
+ * $ dim $-dimensional Legendre polynomials constructed from
+ * $ \widetilde P_m(x) $ using tensor product rule.
*
* @author Denis Davydov, 2016.
*/
projects / dealii.git / commitdiff