# Finite element exterior calculus, homological techniques, and applications

@article{Arnold2006FiniteEE, title={Finite element exterior calculus, homological techniques, and applications}, author={Douglas N. Arnold and Richard S. Falk and Ragnar Winther}, journal={Acta Numerica}, year={2006}, volume={15}, pages={1 - 155} }

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus…

## 837 Citations

### Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green–Naghdi Equations

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- 2018

The study of finite element methods for the numerical solution of differential equations is one of the gems of modern mathematics, boasting rigorous analytical foundations as well as unambiguously…

### An hp-hierarchical framework for the finite element exterior calculus

- MathematicsArXiv
- 2020

The aim of this essay is the exposition of a simple, efficiently-implementable framework for general hp-adaptivity applicable to the FEEC on higher-dimensional manifolds and the power of the method as an engineering tool is demonstrated.

### Finite element exterior calculus: From hodge theory to numerical stability

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- 2010

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we…

### Finite element differential forms on cubical meshes

- MathematicsMath. Comput.
- 2014

A family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions which contains elements of all polynomial degrees and all form degrees can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus.

### Finite element differential forms

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- 2007

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to…

### Finite Element Exterior Calculus for Evolution Problems

- Mathematics
- 2012

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood…

### The chain collocation method: A spectrally accurate calculus of forms

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This paper describes work in progress, towards the formulation, implementation and testing of compatible discretization of differential equations, using a combination of Finite Element Exterior…

### Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2020

A general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods is developed, and its usefulness is demonstrated by establishing convergence results for DEC beyond the Poisson problem in two dimensions.

### Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus

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- 2008

A numerical method is derived for Darcy flow, and also for Poisson’s equation in mixed (first order) form, based on discrete exterior calculus (DEC), which develops a discretization for a spatially dependent Hodge star that varies with the permeability of the medium.

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