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Old   December 23, 2013, 12:28
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Filippo Maria Denaro
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Originally Posted by praveen View Post
Going back to the original question about evaluating solution at some arbitrary point in the domain.

Galerkin, petrov-galerkin and DGFEM all give us a function as a solution, so in this case it is very clear.

You can reinterpret some Godunov FVMs as a special type of petrov-galerkin DG method for hyperbolic problems. This is done in some papers of Barth. This involves a reconstruction process and one can say to the original question that you use the reconstruction to evaluate the solution at some desired point.

It may not be possible to do this in general, e.g., say for Navier-Stokes. One usually constructs schemes for inviscid and viscous terms using different ideas, and I dont think it is possible to give a proper re-intepretation as an FEM. Remember that the purpose of this re-interpretation is to give us a function that can be evaluated at a desired point.

In many other FVM, the FV method itself tells you how to reconstruct the face values but may not tell you what is the reconstructed function inside the cells. In this case you have to use some reconstruction process of sufficient accuracy which may have nothing to do with the FVM.

well, we can have different opinions, no problem... what I think is that if you have two finite lenght vectors containing the nodal values the first for FEM solution, the second for FVM solution, you have guidelines for both of them to compute values in every points you want. In FEM you use your functional basis, in FVM use the reconstruction congruent to the adopted scheme. Of course, in practice you can use any type of interpolation (even spectral) for both FEM and FVM vectors, but that is not congruent and does not add nothing of more accurate ...
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Old   December 23, 2013, 12:29
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I don't want to be annoying on this but i only partly agree, that is:

- you're right, FV practice is not based on this comparison with FEM and viscous terms are usually treated in a very different manner, so that a unique function underlying the functional element for the cell cannot be identified. Still, this is not a requirement of the method and i can imagine having a centered stencil (interpolation element) having suffiecient points to give me a consistent treatment for all the terms. For example, given a certain stencil i could use a Radial Basis Function to interpolate, as some of them are infinitely differentiable.

- i don't know exactly how it works for FEM/Spectral element, but i remember that some methods only have C0 continuity at the interface between elements (if they are not continuous at all as in DG). I wonder thus how diffusive terms are treated in this case? I remember some additional terms appearing (as those due to the Neumann boundary conditions), which are not usually considered and go to 0 only for an infinite number of degrees of freedom. Isn't this a sort of approximation similar to the one used in the treatment of diffusive terms in FV?

- how you evaluate fluxes on the faces in FV is exactly the same as in DG. But these methods are based on local reconstruction within the cells, which exist in FV too!

To me, it seems arbitrary to say that it has nothing to do with the FV, as it is at the core of the different discretization methods in FV: you have the Roe solver which is mostly common to every method and then the reconstruction within the cells, which defines the specific scheme. This is how you achieve a higher accuracy in FV.

The main problem remaining in FV before high accuracy can be really achieved is the fact that the solution still remains in the form of a FV averaged function, thus only a 2nd order approximation of the function. Still, it remains a function. In this case, deconvolution methods have to be adopted to really go beyond second order.

Let me put it in another way. How is it possible for a FV to go beyond 1st order accuracy if it would be stacked to a 0-order V_h. There is no way for a computed solution to be nth order accurate without being in some functional space that has similar characteristics to those used in FEM of the same order. Thus, if i want the function value in a point, once i have the FV solution, i just use the underlying function in the exact same way it is done in FEM. The fact that it can easily become tedious and is certainly not common among practicioners does not invalidate the reasoning.

Also, the fact that viscous terms may have be treated inconsistently is not different form the different treatment in time of convective and diffusive terms. Still, this inconsistency would be immaterial in the same way the non exactness of Gauss integration formula affects some FEM methods using it (e.g., SEM). That is, up the accuracy order, the function and the computed solution are exactly the same, the inconsistency is on higher order terms.

In conclusion, you are right, the common approach to 2nd order FV method does not identify a unique function to interpolate, while FEM does. However, it seems to me that it is not a fault of the method but just due to a common practice, which can be easily avoided. Still, this practice is especially effective for 2nd order codes.
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Old   December 24, 2013, 12:21
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Thank you for this conversation. It is the Galerkin method with which I am most familiar, so it seemed strange to me that something like FVM would not have a similar result, and not be able to generate intermediate solutions. But this answers my question, and it seems that this underlying difference has led to a culture that does not expect to be able to obtain intermediate data, so postprocessing has to be explicit to generate these data, and some workers may be unaware of this ability of their postprocessing tools or unskilled in obtaining these data.
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fvm, interpolation, numerical

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