The Abstract Hodge-Dirac Operator and Its Stable Discretization

@article{Leopardi2014TheAH,
  title={The Abstract Hodge-Dirac Operator and Its Stable Discretization},
  author={Paul C. Leopardi and Ari Stern},
  journal={SIAM J. Numer. Anal.},
  year={2014},
  volume={54},
  pages={3258-3279}
}
This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge--Dirac operator, which is a square root of the abstract Hodge--Laplace operator considered by Arnold, Falk, and Winther [Bull. Amer. Math. Soc., 47 (2010), pp. 281--354]. Dirac-type operators are central to the field of Clifford analysis, where recently there has been considerable interest in their discretization. We prove a priori stability and convergence estimates, and show that… 
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34 References

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