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Chapter 3: Discrete operators based on reproducing kernels
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This gallery contains examples and tutorials for Chapter 3.

Each example includes code and outputs, such as plots or results, to help you understand
how the library works.

Below are the examples for this chapter:


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    <div class="sphx-glr-thumbcontainer" tooltip="This section introduces the concept of partition of unity in the context of kernel methods and how CodPy implements it via projection operators.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_3_2_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_3_2.py`

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      <div class="sphx-glr-thumbnail-title">3.3.2 Partition of unity</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial illustrates how to approximate gradients of a multivariate function using kernel-based operators provided by CodPy. It also introduces how the CodPy API implements these differential operators.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_1_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_4_1.py`

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      <div class="sphx-glr-thumbnail-title">3.4.1 Gradient Operator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial introduces the divergence operator in CodPy, based on the transpose of the gradient operator. The divergence is a key concept in vector calculus and is fundamental in the formulation of many differential equations.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_2_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_4_2.py`

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      <div class="sphx-glr-thumbnail-title">3.4.2 Divergence Operator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The inverse Laplace operator is a useful tool in many fluid mechanics, image processing, signal filtering etc. In the CodPy framework, this operator is implemented as the pseudo-inverse of the Laplacian matrix built from a kernel.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_3_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_4_3.py`

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      <div class="sphx-glr-thumbnail-title">3.4.3  Inverse Laplace operator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip=" Given a kernel gradient operator ∇ₖ, CodPy defines an integral-type inverse operator denoted by:">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_4_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_4_4.py`

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      <div class="sphx-glr-thumbnail-title">3.4.4 Integral operator - inverse gradient operator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip=" This section describes the Leray operator and its role in the Helmholtz–Hodge decomposition, which is foundational in the analysis of incompressible fluid flows, turbulence, and vector field analysis.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_5_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_4_5.py`

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      <div class="sphx-glr-thumbnail-title">3.4.5  Integral operator - inverse divergence operator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Leray-orthogonal operator plays a fundamental role in fluid dynamics, particularly in the mathematical formulation and numerical modeling of incompressible flows governed by the Euler or Navier–Stokes equations.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_6_thumb.png
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  :ref:`sphx_glr_auto_ch3_ch3_4_6.py`

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      <div class="sphx-glr-thumbnail-title">3.4.6 Leray-orthogonal operator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="This section describes the Leray operator and its role in the Helmholtz–Hodge decomposition, which is foundational in the analysis of incompressible fluid flows, turbulence, and vector field analysis.">

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  .. image:: /auto_ch3/images/thumb/sphx_glr_ch3_4_7_thumb.png
    :alt:

  :ref:`sphx_glr_auto_ch3_ch3_4_7.py`

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      <div class="sphx-glr-thumbnail-title">3.4.7 Leray operator and Helmholtz-Hodge decomposition</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /auto_ch3/ch3_3_2
   /auto_ch3/ch3_4_1
   /auto_ch3/ch3_4_2
   /auto_ch3/ch3_4_3
   /auto_ch3/ch3_4_4
   /auto_ch3/ch3_4_5
   /auto_ch3/ch3_4_6
   /auto_ch3/ch3_4_7



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

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