.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/ex10_poisson-equation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_ex10_poisson-equation.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_ex10_poisson-equation.py:


Poisson Equation
----------------
The `Poisson equation <https://en.wikipedia.org/wiki/Poisson%27s_equation>`_

.. math::
    
   \text{div}(\boldsymbol{\nabla} v) + f = 0 \quad \text{in} \quad \Omega

with fixed boundaries on the bottom, top, left and right end-edges

.. math::
   
   v = 0 \quad \text{on} \quad \Gamma_v

and a unit load

.. math::
   
   f = 1 \quad \text{in} \quad \Omega

is solved on a unit rectangle with triangles.

.. GENERATED FROM PYTHON SOURCE LINES 27-37

The Poisson equation is transformed into integral form representation by the
`divergence (Gauss's) theorem <https://en.wikipedia.org/wiki/Divergence_theorem>`_.

.. math::

   \int_\Omega \boldsymbol{\nabla} v \cdot \boldsymbol{\nabla} u \ d\Omega
       = \int_\Omega  f \cdot v \ d\Omega

For the :func:`~felupe.newtonrhapson` to converge, the *linear form* of the Poisson
equation is also required.

.. GENERATED FROM PYTHON SOURCE LINES 37-71

.. code-block:: Python


    import felupe as fem
    from felupe.math import ddot, grad

    mesh = fem.Rectangle(n=2**5).triangulate()
    region = fem.RegionTriangle(mesh)
    scalar = fem.Field(region)
    field = fem.FieldContainer([scalar])


    @fem.Form(v=field, u=field)
    def a():
        "Container for a bilinear form."
        return [lambda v, u, **kwargs: ddot(grad(v), grad(u))]


    @fem.Form(v=field)
    def L():
        "Container for a linear form."
        return [lambda v, **kwargs: ddot(grad(v), grad(scalar)) - kwargs["scale"] * v]


    poisson = fem.FormItem(bilinearform=a, linearform=L, kwargs={"scale": 1.0})

    boundaries = {
        "bottom-or-left": fem.Boundary(field[0], fx=0, fy=0, mode="or"),
        "top-or-right": fem.Boundary(field[0], fx=1, fy=1, mode="or"),
    }

    step = fem.Step([poisson], boundaries=boundaries)
    job = fem.Job([step]).evaluate()

    view = mesh.view(point_data={"Field": scalar.values})
    view.plot("Field", show_undeformed=False).show()







.. tab-set::



   .. tab-item:: Static Scene



            
     .. image-sg:: /examples/images/sphx_glr_ex10_poisson-equation_001.png
        :alt: ex10 poisson equation
        :srcset: /examples/images/sphx_glr_ex10_poisson-equation_001.png
        :class: sphx-glr-single-img
     


   .. tab-item:: Interactive Scene



       .. offlineviewer:: /home/docs/checkouts/readthedocs.org/user_builds/felupe/checkouts/latest/docs/examples/images/sphx_glr_ex10_poisson-equation_001.vtksz







.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.515 seconds)


.. _sphx_glr_download_examples_ex10_poisson-equation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: ex10_poisson-equation.ipynb <ex10_poisson-equation.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: ex10_poisson-equation.py <ex10_poisson-equation.py>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_