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peter grafenberger October 25, 1999 05:29

FV discretisation on tet meshes

i'm looking for some theoretical and analytical stuff on finite volume formulation/discretisation on tet meshes. for example how to formulate ud or central schemes on tets, how to estimate the truncation errors and so on. so can you tell me some papers dealing with these things.


ulrich bieder October 25, 1999 11:43

Re: FV discretisation on tet meshes
hi peter or (susanne?),

you are looking for quite exotic stuff. can you please specify you discretisation method. we are working on non-conforming elements for the velocity and constant or P1-iso_P1-bubble discretisation for the pressure, respectively. this discretisation is very well adapted to the needs of LES. for this discretisation we have internal reports or pdf-files available.

please contact me if you are interested.

a good reference on finite volumes is always r.a.nicolaides. you can find information in the web

salut ulrich

clifford bradford October 25, 1999 13:18

Re: FV discretisation on tet meshes
this isn't very high tech, i'd say you can find this in any relatively advanced CFD book. try Hirsch's book which most people seem to like. or you can try Tannehill Anderson and Pletcher's CFD book.

Andreas Haselbacher October 26, 1999 04:39

Re: FV discretisation on tet meshes
Hello Peter,

simple upwind and central schemes are easily formulated on tetrahedral grids with the finite-volume method. The central scheme on the median-dual control volume on a tetrahedral grid is equivalent (shown by many people) to the Galerkin finite-element discretisation with linear basis functions.

To get simple schemes implemented on tetrahedral grids is actually quite simple. If you want more accurate ones, it becomes quite tricky. In general, as long as the tetrahedra are not too highly stretched, the discretisation will work very well. It's when the tetrahedra are highly stretched, to capture boundary layers, for example, that numerical difficulties appear.

There is quite a large body of literature on discretisation on tetrahedral grids. Have you tried searching for relevant keywords on the NASA Technical Report Server (

You might find the following reports/papers of interest:

M.B. Giles. `Stability analysis of a Galerkin/Runge-Kutta Navier-Stokes discretisation on unstructured tetrahedral grids'. Journal of Computational Physics, 132:201-214, 1997 (Also available in ps format from Prof. Giles' homepage:

T.J. Barth. Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations, High Resolution Upwind and TVD Methods for the Compressible Flow Equations, Von Karman Inst. for Fluid Dynamics, 1994.

T.J. Barth. Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations. AGARD, Special Course on Unstructured Grid Methods for Advection Dominated Flows, R-787, May 1992

Martin Berzins. Mesh Quality: A Function of Geometry, Error Estimates or Both? 7th International Meshing Roundtable, October 1998, Available from: (When you're at the site, check also the other years of the conference for relevant articles by Berzins and other people.)

If you have any more questions, just email me.



peter grafenberger October 27, 1999 07:33

Re: FV discretisation on tet meshes
hello ulrich,

at first thanks for your help. i'm just looking for how to make upwind differencing on tets (to compare this with the hex formulation - so the question is how to deal with the missing faces)


peter grafenberger October 27, 1999 07:39

Re: FV discretisation on tet meshes

i'm sure that it isn't very high tech - it's implemented in every cfd code i know, but nevertheless i can't find it. hirsch's books is alway a good place to start, if you don't have too much questions. but in hirsch's books there are only hexahedral meshes (and schemes are always denoted for the cartesian ones) - but maybe my versions of his books are too old - i'll check this. the other book you mentioned i did't know, so mybe you can tell me the title.

thanks a lot


peter grafenberger October 27, 1999 07:44

Re: FV discretisation on tet meshes
hello andreas,

thanks a lot for your answer. i'll check all these papers as well as i will take a look on the nasa database and after this i may come back to you. concerning the first remark you made - do you have such a paper where the equivalence of fem and fv is shown in that case.

thanks again



Andreas Haselbacher October 29, 1999 07:32

Re: FV discretisation on tet meshes
Hello Peter,

for the equivalence of the diffusion terms, you can have a look at the references by Barth from 1992 and 1994.

For the equivalence of the convection terms, you can show that yourself very quickly. You should see that the linear Galerkin approach gives the same as the centred finite-volume method on the median dual. If the discretisation is non-centred (i.e., an upwind scheme), I cannot remember having seen a proof of equivalence, although there might have been some work by French research groups, especially at INRIA. I guess that there is an equivalence as well in certain cases.

Again, if you have any more questions, just email me.



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