From 8c74829fce6e70844df5978fc13b1030f86c436d Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Fri, 17 Apr 2015 14:45:04 -0500 Subject: [PATCH] Document the polynomials one gets for PolynomialBDM for k=1. This is joint work with Zhen Tao and Arezou Ghesmati. --- include/deal.II/base/polynomials_bdm.h | 28 +++++++++++++++++++++----- 1 file changed, 23 insertions(+), 5 deletions(-) diff --git a/include/deal.II/base/polynomials_bdm.h b/include/deal.II/base/polynomials_bdm.h index c181983ae2..f657758e85 100644 --- a/include/deal.II/base/polynomials_bdm.h +++ b/include/deal.II/base/polynomials_bdm.h @@ -40,14 +40,32 @@ DEAL_II_NAMESPACE_OPEN * these shape functions are not sufficient, additional functions are added. * These are the following vector valued polynomials: * - *
In 2D:
The 2D-curl of the functions xk+1y - * and xyk+1.
In 3D:
For any i=0,...,k the + *
+ *
In 2D: + *
The 2D-curl of the functions xk+1y + * and xyk+1. + * Note the curl of a scalar function is given by $\text{curl}(f(x,y)) = + * \begin{pmatrix} f_{y}(x,y) \\ -f_{x}(x,y) \end{pmatrix}$. + * + * More specifically, for $k=1$, this space has shape functions + * @f{align*} + * \phi_0 = \begin{array}{cc} 1 \\ 0 \end{array}, + * \phi_1 = \begin{array}{cc} -\sqrt{3}+2\sqrt{3}x \\ 0 \end{array}, + * \phi_2 = \begin{array}{cc} -\sqrt{3}+2\sqrt{3}y \\ 0 \end{array}, + * \phi_3 = \begin{array}{cc} 0 \\ 1 \end{array}, + * \phi_4 = \begin{array}{cc} 0 \\ -\sqrt{3}+2\sqrt{3}x \end{array}, + * \phi_5 = \begin{array}{cc} 0 \\ -\sqrt{3}+2\sqrt{3}y \end{array}, + * \phi_6 = \begin{array}{cc} x^2 \\ -2xy \end{array}, + * \phi_7 = \begin{array}{cc} 2xy \\ -y^2 \end{array}, + * @f} + * + *
In 3D: + *
For any i=0,...,k the * curls of (0,0,xyi+1zk-i), * (xk-iyzi+1,0,0) and - * (0,xi+1yk-iz,0)
+ * (0,xi+1yk-iz,0) + *
* - * Note the curl of a scalar function is given by $\text{curl}(f(x,y)) = - * \begin{pmatrix} f_{y}(x,y) \\ -f_{x}(x,y) \end{pmatrix}$. * @todo Second derivatives in 3D are missing. * * @ingroup Polynomials -- 2.39.5