New examples for Mesh

docs/examples/Mesh_Operators_CahnHilliard.rst

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+.. _examples_Mesh_Operators_CahnHilliard:
+
+.. --------------------------------- ..
+..                                   ..
+..    THIS FILE IS AUTO GENEREATED   ..
+..                                   ..
+..    SimPEG/Examples/__init__.py    ..
+..                                   ..
+.. --------------------------------- ..
+
+
+Mesh: Operators: Cahn Hilliard
+==============================
+
+This example is based on the example in the FiPy_ library.
+Please see their documentation for more information about the Cahn-Hilliard equation.
+
+The "Cahn-Hilliard" equation separates a field \\( \\phi \\) into 0 and 1 with smooth transitions.
+
+.. math::
+
+    \frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \left( \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi \right)
+
+Where \\( f \\) is the energy function \\( f = ( a^2 / 2 )\\phi^2(1 - \\phi)^2 \\)
+which drives \\( \\phi \\) towards either 0 or 1, this competes with the term
+\\(\\epsilon^2 \\nabla^2 \\phi \\) which is a diffusion term that creates smooth changes in \\( \\phi \\).
+The equation can be factored:
+
+.. math::
+
+    \frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \psi \\
+    \psi = \frac{\partial^2 f}{\partial \phi^2} (\phi - \phi^{\text{old}}) + \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi
+
+Here we will need the derivatives of \\( f \\):
+
+.. math::
+
+    \frac{\partial f}{\partial \phi} = (a^2/2)2\phi(1-\phi)(1-2\phi)
+    \frac{\partial^2 f}{\partial \phi^2} = (a^2/2)2[1-6\phi(1-\phi)]
+
+The implementation below uses backwards Euler in time with an exponentially increasing time step.
+The initial \\( \\phi \\) is a normally distributed field with a standard deviation of 0.1 and mean of 0.5.
+The grid is 60x60 and takes a few seconds to solve ~130 times. The results are seen below, and you can see the
+field separating as the time increases.
+
+.. _FiPy: http://www.ctcms.nist.gov/fipy/examples/cahnHilliard/generated/examples.cahnHilliard.mesh2DCoupled.html
+
+
+
+.. plot::
+
+    from SimPEG import Examples
+    Examples.Mesh_Operators_CahnHilliard.run()
+
+.. literalinclude:: ../../SimPEG/Examples/Mesh_Operators_CahnHilliard.py
+    :language: python
+    :linenos:

docs/examples/Mesh_QuadTree_FaceDiv.rst

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+.. _examples_Mesh_QuadTree_FaceDiv:
+
+.. --------------------------------- ..
+..                                   ..
+..    THIS FILE IS AUTO GENEREATED   ..
+..                                   ..
+..    SimPEG/Examples/__init__.py    ..
+..                                   ..
+.. --------------------------------- ..
+
+
+Mesh: QuadTree: FaceDiv
+=======================
+
+
+
+
+
+.. plot::
+
+    from SimPEG import Examples
+    Examples.Mesh_QuadTree_FaceDiv.run()
+
+.. literalinclude:: ../../SimPEG/Examples/Mesh_QuadTree_FaceDiv.py
+    :language: python
+    :linenos: