Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods using chain complexes. Its main application has been a comprehensive theory for finite element methods in computational electromagnetism, computational solid and fluid mechanics. FEEC was developed in the early 2000s by Douglas N. Arnold, Richard S. Falk and Ragnar Winther, among others. Finite element exterior calculus is sometimes called as an example of a compatible discretization technique, and bears similarities with

discrete exterior calculus In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes. DEC methods have proved to be very powerful in improving and analyzing finite element me ...

, although they are distinct theories. One starts with the recognition that the used

differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

are often part of complexes: successive application results in zero. Then, the phrasing of the differential operators of relevant differential equations and relevant boundary conditions as a

Hodge Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

. The Hodge Laplacian terms are split using the Hodge decomposition. A related variational

saddle-point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...

formulation for mixed quantities is then generated.

Discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...

to a mesh-related subcomplex is done requiring a collection of

projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...

operators which

commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

with the differential operators. One can then prove uniqueness and optimal convergence as function of mesh density. FEEC is of immediate relevancy for

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the cl ...

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

Stokes flow Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of fluid flow where advective iner ...

. For the important

de Rham complex In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...

, pertaining to the grad, curl and div operators, suitable family of elements have been generated not only for tetrahedrons, but also for other shaped elements such as bricks. Moreover, also conforming with them, prism and pyramid shaped elements have been generated. For the latter, uniquely, the shape functions are not polynomial. The quantities are 0-forms (scalars), 1-forms (gradients), 2-forms (fluxes), and 3-forms (densities). Diffusion, electromagnetism, and elasticity, Stokes flow, general relatively, and actually all known complexes, can all be phrased in terms the de Rham complex. For Navier-Stokes, there may be possibilities too.

References

Finite element method {{applied-math-stub