.. _diffusion-poisson_nonlinear_parametric:

diffusion/poisson_nonlinear_parametric.py
=========================================

**Description**


Nonlinear diffusion with a field-dependent coefficient and parametric sweep.

Solve the problem:

.. math::

   -\nabla \cdot \left( (1 + \alpha u^2)\nabla u \right)
   = \sin(\pi x)\sin(\pi y)
   \quad \text{in } \Omega,

with homogeneous Dirichlet boundary conditions:

.. math::

   u = 0 \quad \text{on } \partial \Omega.

The diffusion coefficient depends on the solution:

.. math::

   c(u) = 1 + \alpha u^2.

The problem is solved for several values of :math:`\alpha`, and the
solutions are compared to the linear case :math:`\alpha = 0`.

Usage Examples
--------------

- Run with the default parameters::

    sfepy-run sfepy/examples/diffusion/poisson_nonlinear_parametric.py
    sfepy-view output/poisson_nonlinear_parametric/square_unit_tri_alpha_*.vtk -2

- Use custom values of :math:`\alpha`, show :math:`\nabla u`::

    sfepy-run sfepy/examples/diffusion/poisson_nonlinear_parametric.py -d "alphas=[1e5,5e5,1e6]"
    sfepy-view output/poisson_nonlinear_parametric/square_unit_tri_alpha_*.vtk -2 -f u:gu:p0 u:p0 --no-scalar-bars


.. image:: /../doc/images/gallery/diffusion-poisson_nonlinear_parametric.png


:download:`source code </../sfepy/examples/diffusion/poisson_nonlinear_parametric.py>`

.. literalinclude:: /../sfepy/examples/diffusion/poisson_nonlinear_parametric.py