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June 26, 2003, 14:25 
Is FEM Local Conservative

#1 
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Dear Friends I want to raise this issue again. It has been discussed many times by many big names in CFD. If we are working with classical stabilized FEM method (say GLS), and we are not doing any "numerical crime" . Can we get local conservative solution for Incompressible flow? I know some "special" elements will probably give locally conserved solution but I am talking about equal order elements i.e linear P Linear V, or quadratic P quadratic V elements. If some one is intersted in this discussion and he feels I have not explined my problem properly, please dont hesitate to ask? I am interested in comments on the method itself and not on the results produced by some people in the past Thanks. Iwould be eagerly waiting for ur comments


June 26, 2003, 16:05 
Re: Is FEM Local Conservative

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It's easy: classical FEM cannot be made locally conservative no matter how hard you try. The root of the problem is that the solution in the FEM is required to be continuous across the face and that's the end of story.
An extension to "conservative FEM" (read "generalised FVM") exists: it is called the discontinuous Galerkin formulation. It has been made specifically to enforce local conservation in the FEM and mathematically it is interesting. However, it is still young (with all the teeting problems that implies!) and costs an absolute fortune in terms of storage and execution time. 

June 26, 2003, 16:36 
Re: Is FEM Local Conservative

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Thank you for the prompt and accurate reply. I absoloutely agree with you. I know the future method is the Discontinuous Galerkin (DG) or Discontinuous Spectral Element Method. I just want to exhaust all possible options for classical FEM. Following are some of my observations/questions; 1) Do you think that resiual free bubble can solve advection dominant flows (ref. Brezzi and Leo Franca) 2) Constant pressure, linear velocity element works in classical setting. I think that it is mainly due to the fact that the variational form for continuity equation does not have the pressure weighting function. So in a sense the continuity is enforced "STRONGLY" whereas in all the other cases it is imposed "WEAKLY". Also note that constant P linear V FEM is similar to FVM multigrid formulation which are proven to work. What are ur comments? 3) Have you seen any evidence that arbitrary equal order interpolation functions work in DG formulations 4) Could you give me the reference of the paper in which I can see the formulation and the benchmark results for incompressible flow,using DG formulation. I am interested in equalorder mixed formulations only. Thank you again R.A.Khurram


June 27, 2003, 00:53 
Re: Is FEM Local Conservative

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You may visit:
http://www.math.umn.edu/~cockburn/ http://www.ticam.utexas.edu/ http://www.dam.brown.edu/scicomp/pub...blications.htm You'll find some interesting paper about DG. regards Pran 

June 27, 2003, 08:12 
Re: Is FEM Local Conservative

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Hmm, too many questions I cannot answer properly... let's see:
ad 1) No idea. I am not familiar with this paper. However, what I do know is that preserving boundedness is critical for a numerical methods. E.g. if you've got a variable that represents concentration it is bounded between 0 and 1  the numerical method should be able to carry this through without cutoffs. You will never be able to preserve boundedness unless you've got conservative fluxes and classical FEM cannot give you that. ad 2) I think I agree but I'm not sure. In the FVM we've got something caleed the "pressure staggering problem" (unrelated to conservation) which is solved by introducing some damping terms equivalent to using different orders of interpolation for p and u. As fas as I know, people have managed to get equalorder pu FEM solvers to work as well (the trick is called Arakawa correction or something like that). Correct me if I'm wrong, but "constant per element" p gives you interelement jumps, which is what you need to create a conservative scheme. ad 3) and 4) Yup, but with scores of other problems  according to the authors, sometimes the solvers just "blow up", there are problems with anisotropic elements, there was no indication how expensive this is, I've no idea if the matrices are good enough for iterative solvers (which is crucial!), nobody tried it for a significantly higher order (say, 6th order elements and everybody claims it is possible!) and generally a "large 3D mesh would have maybe a couple of hundred elements. I've been sitting at the 2nd MIT CFD conference last week and there has been some papers describing exactly what you are talking about so have a look at the conference proceeedings and the web site. There should also be a list of presentations session by session so you can contact the people in the know directly rather than getting it secondhand. Aha: http://www.secondmitconference.org 

July 1, 2003, 00:12 
Re: Is FEM Local Conservative

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Some of the responses to the original question are sheer nonsense. The finite element method applied to incompressible flow is neither conservative nor nonconservative. Most IMPLEMENTATIONS of the method are not strongly massconserving, and this includes all but two implementations with Lagrange basis elements. The FEM implementation will be NECESSARILY conservative if the basis functions/elements are divergencefree within the element and the normal component of flow is continuous across element interfaces.
In two dimensions there are at least three Hermite elements which satisfy these properties. The velocity elements are the curl of a sufficientlycontinuous stream function element. One, which is cubiccomplete in the stream function (quadraticcomplete in the velocity) has stream function and velocities as its three degrees of freedom. Two more, which are quartic and quinticcomplete in the stream function, additionally have second derivatives of the stream function as degrees of freedom. There is at least one basis element in three dimensions (linearlycomplete in the velocities) with the necessary properties. These Hermite bases need no constraint. Pressure is totally decoupled from the equation for fluid motion so its representation and boundary conditions are not an issue. 

July 1, 2003, 09:09 
Re: Is FEM Local Conservative

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Hehe, a mathematician! Nice to hear there's a difference between the FEM (as in a book in a drawer) and an implementation (as in software that actually does something).
To get to the point, there is such a thing as conservativeness even for compressible flows and the vorticity formulation does not do those! What you have done here is to take the curl of the equation before discretising it in order to enforce the divergencefree condition rather than described a discretisation practice which is conservative  not what we are talking about. By the way, are there any (commercial or opensource) CFD packages I can try with the quartic and quinticcomplete Hermite elements because I'd like to see how efficient (read: realistic to use in real life). Also, I live in a 3D world so a 3D extension would be nice For the quotation on sheer nonsense, please look at the beginning of the previous message  I think it applies here! Would you like to try again? 

July 1, 2003, 09:37 
Re: Is FEM Local Conservative

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Are there any commercial codes you're aware of that you can use for a simple 1D conv/div nozzle with a shock? What exactly is your point? Are you implying you're alone in the 3D world?


July 2, 2003, 01:12 
Re: Is FEM Local Conservative

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The mention of stream functions or vector potentials does not imply anything about stream function vorticity formulations. The mathematicians (which I am not) assert that the existance of a divergencefree vector field implies the existance of a stream function or vector potential. P. Gresho in his book remarks that when people start trying to produce divergencefree formulations, the result starts looking like a stream functions are involved.
I am talking about primative variable formulations. I am talking about two and threedimensions. When you deride quartic and quintic bases, I wonder if you have heard of prefinement. Impractical computations (including benchmarks) are valued to check the effectiveness of more practical methods. People have been thrashing around for several decades trying to make the incompressible simplification simple. These Hermite elements do that. You trade some added complexity for complete freedom from pressure poison equations and LBB constraints. No continuity constraints to be applied within time integration steps. No need for fractional multistep methods. Incompressible flow is no longer mixed. Except for the usual complications of turbulence, it is truly simple. The method has been described at conferences for several years and two papers on the subject submitted to Math Comp can be found at http://j.t.holdeman.home.att.net/research/ , with more on the way. 

July 2, 2003, 02:47 
Re: Is FEM... URL correction

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The correct URL for the papers mentioned is,
http://j.t.holdeman.home.att.net/research.htm Sorry, I didn't check it before posting. 

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