.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_ch7\ch7_invLaplace.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_ch7_ch7_invLaplace.py>`
        to download the full example code.

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

.. _sphx_glr_auto_ch7_ch7_invLaplace.py:


====================================
7.01 Inverse Laplace Operator
====================================
We reproduce here the figure 7.1 and 7.2 of the book.
Utilitary functions can be found next to this file. Here, we only define codpy-related functions.

.. GENERATED FROM PYTHON SOURCE LINES 10-12

Necessary Imports
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 12-28

.. code-block:: Python

    import os 
    import sys 

    import matplotlib.pyplot as plt 
    from codpy import core

    try:
        CURRENT_DIR = os.path.dirname(os.path.abspath(__file__))
    except NameError:
        CURRENT_DIR = os.getcwd()
    data_path = os.path.join(CURRENT_DIR, "data")
    PARENT_DIR = os.path.abspath(os.path.join(CURRENT_DIR, ".."))
    sys.path.insert(0, PARENT_DIR)

    from utils.ch7.ch7_utils import PDE1, PDE2








.. GENERATED FROM PYTHON SOURCE LINES 29-37

Problem statement
------------------------
We solve the following Poisson equation with Dirichlet boundary conditions: 
$$\Delta u  = f, \quad \text{ supp } u  \subset \Omega, \quad u_{\partial \Omega}=0$$
where $f$ is sufficient regular and $\Omega$ is a sufficient regular domain.
To do so we select a mesh, and a kernel, which approximates this equation by approximating the solution as a function $u \in \mathcal{H}_k$
This leads to the equation $(\Delta_k u)(X) = f(X)$, $\Delta_k$ being the approximation of the Laplace-Beltrami operator defined in the book.
A solution to this equation is computed as $u = (\Delta_k)^{-1} f$.

.. GENERATED FROM PYTHON SOURCE LINES 39-42

Regular mesh
------------------------
This figure displays a regular mesh for the domain $\Omega = [0,1]^2$, where $f$ is plotted in the left-hand side, and the solution $u$ in the right-hand side.

.. GENERATED FROM PYTHON SOURCE LINES 42-46

.. code-block:: Python


    PDE1() 
    plt.show()




.. image-sg:: /auto_ch7/images/sphx_glr_ch7_invLaplace_001.png
   :alt: f(x), solution
   :srcset: /auto_ch7/images/sphx_glr_ch7_invLaplace_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 47-50

Irregular mesh
------------------------
This figure computes a Poisson equation on an unstructured mesh generated by a bimodal Gaussian random variable, with $f$ plotted on the left and the solution $u$ on the right.

.. GENERATED FROM PYTHON SOURCE LINES 50-53

.. code-block:: Python


    PDE2() 
    plt.show()



.. image-sg:: /auto_ch7/images/sphx_glr_ch7_invLaplace_002.png
   :alt: f(x), solution
   :srcset: /auto_ch7/images/sphx_glr_ch7_invLaplace_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.759 seconds)


.. _sphx_glr_download_auto_ch7_ch7_invLaplace.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: ch7_invLaplace.ipynb <ch7_invLaplace.ipynb>`

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

      :download:`Download Python source code: ch7_invLaplace.py <ch7_invLaplace.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: ch7_invLaplace.zip <ch7_invLaplace.zip>`


.. only:: html

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

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