Editing Appendix/Ramblings/ForCohlHoward (section)

===Perhaps We Should Consider the FEM===

Although I have had virtually no experience developing or using numerical algorithms that fall into the category of the Finite-Element Method (FEM), it is my understanding that such algorithms have the following features:
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They can be used to follow the evolution of ''incompressible'' &#8212; and, for example, uniform-density &#8212; fluid systems.
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They generally (always?) incorporate an ''implicit'' time-integration scheme in which case time steps are not constrained by &#8212; and can be much larger than &#8212; the often severely limiting sound-crossing times encountered in ''explicit'' FVM schemes.  
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They are particularly good at identifying ''surfaces'' and modeling the time-evolution of surface distortions.
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If you (Howard) and Sorokanich decide to build a hydro-code based on the Finite-Element Method, <font color="red">a potentially helpful reference</font> is:  
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{{ Meier99figure }}
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I interacted with David Meier at a couple of different meetings back at the time he was developing this FEM algorithm. I'm not sure what astrophysics problems he tackled with this code or, ultimately, what he did with it; for example, I have never seen "the second paper in this series" to which he refers in the abstract.

In the context of our examination of the stability of Riemann ellipsoids, an FEM might permit us to reduce the dimensionality of the problem.  For example, ignore details associated with the interior of the 3D configuration and focus on modeling the distortion of its 2D surface.  

We might even be able to model (for the first time!) the topological transformation that is experienced by the surface when a rapidly spinning, incompressible ellipsoid ''fissions'' into a pair of disconnected tear-drop shaped objects in orbit about one another.  See, for example, the dissertation research of [https://www.semanticscholar.org/paper/A-hybrid-variational-level-set-approach-to-handle-Walker/7ee624ab9ffe45241cb0e5a0ce0898a6da201e7b Shawn W. Walker (2007)] (LSU Mathematics &amp; CCT), or more recent papers that Walker has coauthored with his dissertation advisor, Ricardo H. Nochetto (U. Maryland, Mathematics).