* try to implement a discontinuous Galerkin-based version of the equations
I'm trying to solve as described in
https://www.sciencedirect.com/science/article/pii/S0377042718302875
<https://www.sciencedirect.com/science/article/pii/S0377042718302875> in
sections 1 (equations) and 2.1 (dg discretization of the equations). Only the fine
scale equations, not the model reduction equations in section 2.2.
The goal here would not be to recover piecewise constant finite volume
formulations but actually to take advantage of the possibility to use
higher order elements. Therefore, I could try to do this in d>=2 space
dimensions using elements of order >= 1 to avoid the limitations discussed
below. This way, I can take full advantage of the deal.ii capabilities for
matrix assembly, mesh refinement etc.
If anyone has the time to take a look at the equations (17)--(20) in the
paper and sees something that doesn't seem possible with deal.ii, I would
appreciate it. However, the formulation looks pretty close to the one in
step-74.
If you start from (7), it's really just a heat equation coupled to a Laplace
equation. Looking at Fig. 5, the solution is also smooth. I don't think you
need anything like FV or DG discretizations -- they are good for problems with
non-smooth solutions. If you can separate the two parts of the domain into
individual cells, DG is fine but you might also get away with a continuous
element approach if you put the different solution variables that are
discontinuous along the interface into different vector conditions (-> step-46).
* if I want a version that really focuses and sticks to the finite
volume, piecewise constant discretization, I would try to see, how a
framework like OpenFoam could be used for this purpose. Because if I brute
force a finite volume discretization in deal.ii, I think I might lose some
of the nice features that it offers with FEM approaches like discontinuous
Galerkin FEM methods.
However, if you think I'm wrong and a finite volume implementation in
deal.ii could be well integrated with most of its tools, I'm very eager to
get feedback or hints on how I should approach this.
It's a framing problem. Many people think that FVM is simply a special case of
FEM where you choose discontinuous shape functions and define nonlinear fluxes
as appropriate. I don't think that FVM is conceptually different than FEM.
Whether you call it FVM or DG-FEM, it's two sides of the same coin.
Best
W>
--
------------------------------------------------------------------------
Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google Groups "deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to dealii+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/dealii/75693aa9-b0d4-49de-a030-3c67e90c5c9a%40colostate.edu.
