.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/elliptic/plot_01_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_api_gallery_pde_elliptic_plot_01_poisson_equation.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_pde_elliptic_plot_01_poisson_equation.py:


Poisson's equation: a potential with sources
============================================

Laplace's equation :math:`\nabla^2 u = 0` governs a potential in empty
space; Poisson added the source term, :math:`\nabla^2 u = f`, for the
potential inside matter. This script solves Poisson's equation on the
unit square for a pair of opposite point-like charges, then verifies the
five-point finite-difference solver's second-order accuracy against a
problem with a known solution.

.. GENERATED FROM PYTHON SOURCE LINES 14-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import solve_poisson_2d
    from mathematicskit.pde.visualizers import plot_field_2d


    def charges(X, Y):
        blob = lambda x0, y0: np.exp(-((X - x0) ** 2 + (Y - y0) ** 2) / 0.002)
        return -200.0 * (blob(0.35, 0.5) - blob(0.65, 0.5))









.. GENERATED FROM PYTHON SOURCE LINES 27-29

The potential of a dipole in a grounded box
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-37

.. code-block:: Python


    sol = solve_poisson_2d(charges, n=(81, 81))
    print(f"sparse direct solve: {(len(sol.x) - 2) ** 2} unknowns, residual {sol.residual_norm:.1e}")
    fig1, ax1 = plt.subplots(figsize=(5.5, 4.5))
    plot_field_2d(sol, ax=ax1, levels=30, cmap="RdBu_r")
    ax1.set_title(r"$\nabla^2 u = f$: two opposite charges, $u = 0$ on the box")
    fig1.tight_layout()




.. image-sg:: /api/gallery/pde/elliptic/images/sphx_glr_plot_01_poisson_equation_001.png
   :alt: $\nabla^2 u = f$: two opposite charges, $u = 0$ on the box
   :srcset: /api/gallery/pde/elliptic/images/sphx_glr_plot_01_poisson_equation_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    sparse direct solve: 6241 unknowns, residual 3.0e-11




.. GENERATED FROM PYTHON SOURCE LINES 38-42

Second-order convergence
------------------------
For :math:`f = -2\pi^2 \sin(\pi x)\sin(\pi y)` the exact solution is
:math:`\sin(\pi x)\sin(\pi y)`; halving the grid spacing quarters the error.

.. GENERATED FROM PYTHON SOURCE LINES 42-60

.. code-block:: Python


    ns = np.array([9, 17, 33, 65])
    errors = []
    for n in ns:
        s = solve_poisson_2d(lambda X, Y: -2 * np.pi**2 * np.sin(np.pi * X) * np.sin(np.pi * Y), n=(n, n))
        X, Y = np.meshgrid(s.x, s.y, indexing="ij")
        errors.append(np.max(np.abs(s.u - np.sin(np.pi * X) * np.sin(np.pi * Y))))
    h = 1.0 / (ns - 1)
    print(f"observed order: {np.polyfit(np.log(h), np.log(errors), 1)[0]:.2f}")
    fig2, ax2 = plt.subplots(figsize=(6, 4))
    ax2.loglog(h, errors, "o-", label="five-point Poisson solver")
    ax2.loglog(h, errors[0] * (h / h[0]) ** 2, "k--", label=r"$O(h^2)$")
    ax2.set_xlabel("$h$")
    ax2.set_ylabel("max error")
    ax2.legend()
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/elliptic/images/sphx_glr_plot_01_poisson_equation_002.png
   :alt: plot 01 poisson equation
   :srcset: /api/gallery/pde/elliptic/images/sphx_glr_plot_01_poisson_equation_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    observed order: 2.00





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

   **Total running time of the script:** (0 minutes 0.309 seconds)


.. _sphx_glr_download_api_gallery_pde_elliptic_plot_01_poisson_equation.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/elliptic/plot_01_poisson_equation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_poisson_equation.ipynb <plot_01_poisson_equation.ipynb>`

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

      :download:`Download Python source code: plot_01_poisson_equation.py <plot_01_poisson_equation.py>`

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

      :download:`Download zipped: plot_01_poisson_equation.zip <plot_01_poisson_equation.zip>`


.. only:: html

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

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