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I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics:

(Discrete Exterious Calculus is the newly developed subject mainly used in numerical computing, one reference is, for example, Hirani's PhD thesis: Discrete Exterior Calculus)

  1. Has any kind of convergence property been proved?I mean, under what conditions, the numerical solution of DEC scheme will converge to the the actual solution of PDE. I checked many literature and didn't see any material concerning the convergence property, because I am doing engineer problem in computer and if we can't gurantee it will converge then the precision will be a problem.

  2. What is the current status of using DEC to numerically solve equations/simulation in fluid mechnanics, elasticity and electromagnetism, respectively? Should anyone give me some relevant papers, I have found some but just don't know if I missed anything.

Thanks for any help!

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2 Answers 2

Hello!

I am working not exactly with DEC but with an arbitrary order method in between DEC and Finite Element Exterior Calculus (from Arnold, Falk and Winther).

You have a nice review on compatible and mimetic method by Perot in this paper:

http://www.annualreviews.org/doi/abs/10.1146/annurev-fluid-122109-160645

And some of his presentation on something in the lines of DEC:

http://www.sciencedirect.com/science/article/pii/S0021999101969736

And you have this paper from Ern, where he shows some convergence analysis for Elliptic problems.

A paper we have been working on in my group is:

http://arxiv.org/abs/1111.4304

Hope this helps.

For the sake of my curiosity, what is the problem you want to solve with DEC?

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@Artur Palha:Thanks!I am working on numerically solve equations in fluid mechanics, elasticity and electromagnetism using DEC, and develop software for this, however, it seems that the solution is still not completely found. What's your opinion?Thanks! –  HYYY Mar 22 '13 at 14:52

This is an answer to the first question. As far as I know, there is no proof of convergence that can be called close to being general. That said, convergence proofs can be obtained in special cases, by relating them to other discretization methods. For example, Yee's scheme is an instance of DEC, and its convergence is classical. Also, there is a "speculation" that a convergence proof can be given by interpreting DEC as a mass lumped Finite Element Exterior Calculus (FEEC), and using the recent FEEC variational crime framework of Holst and Stern, from http://arxiv.org/abs/1005.4455. I don't know if this program can be or has been implemented.

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@timur: Can you give a source for that speculation? –  shuhalo Mar 17 '13 at 22:09
    
@Martin: There is no printed source, but you may want to check the linked article. –  timur Mar 18 '13 at 11:49

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