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Are there guidelines to follow in selecting the right FE discretization. Specifically, for subsurface flow models like Darcy's equation (both single and multi-phase), Richard's equation, etc.

I know one of the most important issues in the fluid mechanics community is the issue of local mass conservation, which the standard classical Galerkin method does not satisfy. That said, there are so many discretizations available out there (e.g., DG, RT0, P2/(P1-P0), least squares FEM, etc). What factors should be taken into account when deciding on a FE formulation to use for my problem? (assuming that they are compatible with the governing equations)

I had these in mind:

1) User implementation - probably the most obvious one. How easy it is to implement in your specific application? Why spend weeks or months attempting to implement a certain formulation when one can achieve the same solution using another formulation that's easier to use?

2) Numerical accuracy - what is the mathematical rate of convergence for each method? This may be important for problems in which there is no analytical solution to compare to.

3) Problem size - that is, how many degrees of freedom each formulation ends up having? If one does not have access to super computers, this may lead to memory issues when running on a laptop or PC

4) Solvers/preconditioners - certain formulations (namely the classical mixed ones) result in saddle-point structures and render standard iterative solvers/preconditioners useless.

5) Parallel performance - How do they scale across multiple processing cores? I think this would be a factor to consider particularly during the assembly, because certain formulations like DG are completely local whereas nodal degrees of freedom may be shared by multiple processors, hence communication overhead. Or would this be more of a user/software implementation issue?

That's all the main ones I could think of for now. Are any of the above valid points to consider? And/or are there other important factors that I missed?

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    $\begingroup$ I think these are all great points. $\endgroup$ – Wolfgang Bangerth May 22 '15 at 6:40
  • $\begingroup$ @WolfgangBangerth That's good to know :) Though I guess my question now is are there any papers, studies, or online lectures that have already done this sort of study? The closest thing I could find is (researchgate.net/profile/Xiu_Ye/publication/…) but it doesn't seem to address the last two points I mentioned and is trying to advocate their method. Does anyone have references on 4) specifically for RT0 and/or other mixed first-order discretizations $\endgroup$ – Justin May 22 '15 at 12:26

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