Vectorized computation of basis functions gradients
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By construction, the gradients of basis functions are constants on each element :math:`T_k.` 
So, we denote, :math:`\forall \il\in\ENS{1}{3},` by :math:`\vecb{G}^\il` the :math:`2 \times \nme` array defined,
:math:`\forall k\in\ENS{1}{\nme},` by

.. math::

   \vecb{G}^\il(:,k)= \GRAD\BasisFunc_{\me(\il,k)}(\q),\ \forall \q\in T_k.

On a triangle :math:`T_k,`
we denote by :math:`\vecb{D}^{12}=\q^{\me(1,k)}-\q^{\me(2,k)},`
:math:`\vecb{D}^{13}=\q^{\me(1,k)}-\q^{\me(3,k)}` and
:math:`\vecb{D}^{23}=\q^{\me(2,k)}-\q^{\me(3,k)}.` Then, we have

.. math::
    \GRAD\BasisFunc_{1}^k(\q)=\frac{1}{2|T_k|}
    \begin{pmatrix}
    \vecb{D}^{23}_y\\-\vecb{D}^{23}_x
    \end{pmatrix},\ \GRAD\BasisFunc_{2}^k(\q)=\frac{1}{2|T_k|}
    \begin{pmatrix}
    -\vecb{D}^{13}_y\\\vecb{D}^{13}_x
    \end{pmatrix},\ \GRAD\BasisFunc_{3}^k(\q)=\frac{1}{2|T_k|}
    \begin{pmatrix}
    \vecb{D}^{12}_y\\-\vecb{D}^{12}_x
    \end{pmatrix}.
    
With these formulas, we obtain the vectorized algorithm given in Algorithm :num:`gradientvec2dalgo`.

.. admonition:: Algorithm :num:`gradientvec2dalgo`

    .. _GradientVec2Dalgo:  
          
    .. figure::  images/GradientVec2D_algo.png
          :width: 600px
          :scale: 100%
          :align:   center
        
          Vectorized algorithm for computation of basis functions gradients in 2d