# Is Discrete Exterior Calculus currently a focusing point in numerial computing world or simulation in industry,

I am just wondering if Discrete Exterior Calculus, as a new numerical method , is good at numericall solving problems in elasticity, fluids or other physical/real area.

Speaking from a computational electromagnetics background, I think it is a very elegant way to discretize problems. I have used it with success in eigenmode and boundary value problems. It is probably less accurate than a strict finite element discretization if you go with diagonal Hodge stars (lumped mass approximation), but I think it still achieves the same asymptotic convergence rate if you compute your Hodge stars carefully (it's probably trickier in electromagnetics than in continuum mechanics). So it is probably just a small constant factor worse (in theory, in practice it may be negligible).

DEC greatly simplifies the formulations of the problems, and allows you to focus more on the physics of the problem. The construction of the Hodge stars forces you to think about the meaning of the constitutive relations and what is the physically meaningful way of performing spatial averages. It also seems to preserve many of the important symmetries of the continuous problems in the discrete setting, and it may be easier to prove these than in a finite element setting.

Finally, as someone who writes code, I appreciate not having to perform quadrature during matrix assembly. Instead, you can typically compute the Hodge stars using analytic spatial averages using assumed forms of spatial variation. In electromagnetics where we have piecewise constant material properties over space, these averages can be computed exactly, making the entire problem smooth with respect to small perturbations in the spatial geometry. This greatly aids any optimization you may want to wrap around your method.

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 an answer on the mathoverflow blog, 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 L2-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).

Discrete Exterior Calculus (DEC) has pros and cons:

Pros:

Ease of "use" For a student, it is quite easy to assemble a discretization and a solver for a simple PDE, e.g. Laplace/Poisson on curved surface (Laplace Beltrami). It made the method very popular in Computer Graphics, after the Caltech geometry lab. made some courses at the main graphics conference (SIGGRAPH), see reference in other answer. It is especially the case in Computer Graphics, where students know matrices well, but are less familiar with integrals. Using DEC, they can "play lego" and solve simple PDEs without suffering too much.

Making some computations simpler DEC is an emanation of EC (Exterior Calculus, invented by Elie Cartan from 1899 to 1945). Central in EC, there is the notion of forms ("things to be integrated") and chains ("integration domains"), and a duality between them. Several theorems (Stokes, Green-Gauss, Ostrogradsky, and the fundamental theorem of analysis) are a special case of this duality. Not only this is elegant, but also this makes computations simpler in some cases (like in electromagnetics), and avoids referring to the parameterization of the objects in many cases (e.g. when manipulating vector fields over surfaces, or in the curved space-time setting of relativity).

Exhibiting untrivial degrees of freedom By explaining the relation between forms ("things to be integrated") and chains ("integration domains"), EC can exhibit untrivial parameterization of objects, such as vector fields on surfaces, and explain the relations between the topology of the vector field and the underlying surface (homology: topology of curves traced on the surface, co-homology: topology of vector fields), see [1] for a deep study. We used it in [2,3] to study the topological degrees of freedom of discrete vector fields. An impressive example of the power of this type of reasoning is Gortler et.al's proof of Tutte's planar embedding theorem [4]. Previous proofs of this theorem (by Tutte, and later by Colin de Verdière) require a certain background on graph theory to be understood. The proof by Gortler et. al is much more accessible, it introduces some discrete differential forms defined on the edges of the graph, and then it obtains the result by an easy-to-understand counting argument.

Cons:

The simplified version of DEC promoted by the Caltech geometry lab. hides many details under the hood. While it is OK for deriving Laplace and Poisson equations on both the Euclidean and curved settings, some problems are quickly encountered when discretizing more complicated equations, because it does not incitate to ask questions concerning the convergence to the continuous setting and/or the properties that are preserved by the discretization (identities with div/grad/curl, referred to as the Hodge complex, studied by mathematicians such as Jenny Harisson and Robert Kotiuga). The way it is used in Computer Graphics (mainly for the Laplace equation) does not bring more in most cases than the classical P1 FEM Laplacian. I tend to prefer the classical P1 FEM Laplacian, because it both gives you the formula of the discretization and explains you why. Another aspect is the way the form of DEC promoted by Caltech mixes the Hodge star and the inner product. While it makes it easy to assemble a discrete Laplacian as a single matrix, it does not tell how to project a function onto the so-defined function space, and makes it difficult also to understand how orthogonality acts: you obtain a matrix equal to $B^{-1}A$ where $B$ is the condensed matrix of the inner product and $A$ the stiffness matrix, and you no longer see that the eigenvectors are orthogonal with respect to the inner product discretized in $B$, because you don't see $B$.

Conclusion/Summary: EC and DEC is a powerful theory for studying complicated problems (electromagnetics, vector fields on surfaces of arbitrary topology). The way it is used in Computer Graphics makes it simple for students who do not know integrals to do simple things. For simple things I tend to prefer the classical FEM formulation, where the complete deduction path is easier to follow from the theory down to the discretization together with the theoretical guarantees. For complicated things it can be very elegant and efficient (provided that all the reasoning path is preserved rather than just "playing lego" with some forms of the discrete Hodge star and the discrete exterior derivative).

[1] Douglas Arnold, Finite Element Exterior Calculus, 2006

[2] N-Symmetry direction field design, ACM Trans. Graph., 2008

[3] Geometry-aware direction field processing, ACM Trans. Graph., 2009

[4] An elementary proof of Tutte's Planar Embedding Theorem, Gortler, Gotsmann, Thurston, 2006, Computer Aided Geometric Design

I would say there appears to be some interest, but it's not exploding. It's a bit too finite volume-y for my tastes, but I'm a finite elements person.

I'd be very interested in the answer to this question as far a HPC and general scientific computing is concerned, but there have certainly been lots of nice results in computational geometry, for example in many of the publications and references here : http://www.geometry.caltech.edu/pubs.html