So I thought I should share my general thoughts based on experience with both
continuous FEM & FV codes for these types of problems. And since this is
archived it'll probably come back to bite me, so keep in mind this is *my
opinion based on my personal research and application.*
1.) For high-order discretizations of elliptic equations, continuous FEM is the
go-to method.
2.) For high-order discretizations of hyperbolic equations, DG methods are
about the best you can do, particularly unstructured. Combined with an
explicit, multgrid-accelerated scheme, it is a powerful technique.
3.) The problems come in when you have both:
Continuous FEM: diffusive terms "easy," convective terms "hard &
researchy"
Discontinous FEM: convective terms easy, diffusive terms "hard & researchy"
4.) Shock capturing is a challenge for every scheme. In the F.V. world we rely
on good alignment of the mesh to shockwaves or else the results really suffer.
*Everything* does something O(h) at the shock, so you either live with it, or
control h directly via refinement, or indirectly. It is this latter category
that I would contend has been the recent trend in the DG community with
'subcell shock capturing'. In my mind taking a p=5 element and replacing it
with 256 p=1 elements for shock capturing purposes is just h refinement by
another name.
5.) For 2nd-order accurate discretizations it is hard to beat classic F.V.
High order F.V. with its broad stencils is a nightmare.
6.) For things like the compressible N-S equations, the underlying
characteristics-based behavior of the system is important, and the most
effective numerical methods deal with the characteristic form of the equations
at some level. Historically SUPG/GLS has only been concerned with the "flow
direction" and, again in my opinion, been at a disadvantage because they did
not treat this crucial aspect. That has been the motivation for the recent tau
stuff I alluded to earlier. For a derivation of that tau from a
characteristics argument see
https://github.com/libMesh/libmesh/wiki/documents/presentations/2013/fins_FEF2013.pdf
The attached paper shows some good results - basically with work you can make
either scheme give high order results, but there is no panacea.
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