.. _convection-difusion:

Stabilized convection-difusion
==============================

Convection diffusion equation
-----------------------------

Find approximate solution to following linear PDE

.. math::
   u_t + \mathbf{b}\cdot\nabla{u} - \operatorname{div}(K \nabla u) &=
   f
        \quad\text{ in }\Omega\times[0, T], \\
   u &= u_\mathrm{D}
        \quad\text{ in }\Gamma\times[0, T], \\
   u &= u_0
        \quad\text{ on }\Omega\times{0} \\

using :math:`\theta`-scheme discretization in time and arbitrary FE
discretization in space with data

* :math:`\Omega = [0, 1]^2`
* :math:`T = 10`
* :math:`K = \frac{1}{\mathrm{Pe}}`
* :math:`\mathbf{b} = \left( -(y-\tfrac{1}{2}), x-\tfrac{1}{2}\right)`
* :math:`f = \vec{0}`
* :math:`u_0(\mathbf{x}) = \left( 1 - 25
  \operatorname{dist}\left(\mathbf{x}, \left[\frac{1}{4},
  \frac{1}{4}\right]\right)
  \right)
  \chi_{ B_{1/5}\left(\left[\frac{1}{4}, \frac{1}{4}\right]\right) }`
* :math:`u_\mathrm{D} = 0`

where :math:`\chi_X` is a characteristic function of set :math:`X`,
:math:`B_R(\mathbf{z})` is a ball of radius :math:`R` and center
:math:`\mathbf{z}` and :math:`\operatorname{dist}(\mathbf{p},\mathbf{q})`
is Euclidian distance between points :math:`\mathbf{p}`,
:math:`\mathbf{q}`.


.. only:: solution

Reference solution
------------------


* simple straight-forward Galerkin FEM
  
  .. literalinclude:: test_no.py
                      
* with SUPG stabilization
  
  .. literalinclude:: test_supg.py
                      
* with IP (interior penalty) approximation
  
  .. literalinclude:: test_ip.py
                      
* solved in GLS (Galerkin least squares) formulation
  
  .. literalinclude:: test_gls.py