Questions on Discrete Exterior Calculus in numerical computing

Question

I have several questions about Discrete Exterior Calculus (DEC) in numerical methods for solving partial differential equations in physics:

(Discrete Exterious Calculus is a 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 will the numerical solution of a DEC scheme converge to the actual solution of the PDE? I checked many papers in the literature and didn't see any material concerning the convergence property, because I am studying an engineering problem on a computer and if we can't guarantee it will converge then the precision will be a problem.

2. What is the current status of using DEC to numerically solve equations or create simulations 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.

Answer 1

This response is a few years late, but I feel those questions are still relevant today. In the recent years new applications of DEC appeared in fields such as computer graphics, geometry processing, Navier-Stokes equations and Darcy flow. In the introduction of the paper suggested below, you will find a quick overview of the fields (including linear elasticity, electrodynamics and variational integrators) in which DEC has been used (some of the authors cited have been quite active in the DEC literature).

As timur said in the previous answer, convergence can be obtained in special cases by relating DEC with other methods known to converge. However, serious attempts at developing a general framework to tackle convergence problems were undertaken. Recently, we proved convergence of the DEC solutions for the Poisson problem (on functions, i.e. $0$-forms) in arbitrary dimension in the discrete $L^2$-norm. Many problems and questions related to the asymptotic behaviours of the discrete solutions in other norms remain open, but the following is a welcomed step toward a better understanding of the theory : https://arxiv.org/abs/1611.03955 (Convergence of Discrete Exterior Calculus Approximations for Poisson Problems, Erick Schulz and Gantumur Tsogtgerel, 2016).

EDIT (2018): An updated version of the paper I posted two years ago was uploaded on the arxiv to make some parts easier to read. Also, a bulk of numerical evidences of convergence over non-delaunay meshes was recently published by Mamdouh S. Mohamed, Anil N. Hirani and Ravi Samtaney: https://arxiv.org/abs/1802.04506 (Numerical Convergence of Discrete Exterior Calculus on Arbitrary Surface Meshes).

I hope this helps!

Answer 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?

Answer 3

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.