Finite element exterior calculus, homological techniques, and applications

By Douglas N. Arnold, Richard S. Falk, Ragnar Winther. Acta Numerica, 2006

1,092 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/e0c19e68ab83798e6bee80779ef619c5405c3f21

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

A discrete de Rham complex with enhanced smoothness (2006, positive): “In order to show that elements v ∈ V (T ) andw ∈ W (T ) are uniquely determined by the degrees of freedom specified by (12), or by (14) and (15), respectively, we need preliminary results in two dimensions.”
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows (2015, background): “This problem was considered in the domain Ω = (0, 4)×(0, 2)\[2, 4]×[0, 1], see Figure 6.”
Finite element approximation of eigenvalue problems (2010, methodology): “…∈ L2(Ω,Λk), 1 The external derivative we are going to use can be naturally defined as follows: given ω ∈ C∞(Ω, Λk), dkωx(v1, . . . , vk+1) = Pk+1 j=1 (−1)j+1∂vj ωx(v1, . . . , v̂j , . . . , vk+1) (Arnold et al. 2006b, p. 15) where denotes the Hodge star operator mapping k-forms to (n − k)forms.”
Notes on Projective Differential Geometry (2008, methodology): “More precisely, the complex (6.2)is written in [ 1 ] as”
Structure-preserving discretization of incompressible fluids (2009, background): “Unlike previous discrete exterior calculus approaches, mostly based on chains and cochains (see [17, 7, 4] and references therein), we clearly distinguish between discrete vector fields and discrete forms acting on them.”
Unified form language: A domain-specific language for weak formulations of partial differential equations (2012, background): “1985], which again coincide with the P 2(T ) family on tetrahedra [Arnold et al. 2006].”
Divergence-conforming discretization for Stokes problem on locally refined meshes using LR B-splines (2015, background): “7 Ξ = [1 , 3/2 2, 3 ] Ξ = [ 3/2, 2, 3, 3] Ξ = [1 , 2, 3, 3]”
Finite Element Methods for Linear Elasticity (2008, methodology): “Using a generalization of an approach first developed in [8] in two dimensions and [6] in three dimensions, and then expanded further in [ 9 ], we establish a systematic way to obtain stable finite element approximation schemes.”
The divDiv-complex and applications to biharmonic equations (2018, background): “We refer to [8] for a presentation of some results of this section from a numerical analysis perspective.”
A review of numerical methods for nonlinear partial differential equations (2012, background): “There is a large catalog [5] of such finite-element basis functions which provide approximations to any desired order.”
Mimetic finite difference method (2014, methodology): “Finite element techniques have been recently recasted in the framework of the Whitney forms and formalized in the finite element exterior calculus, cf. [15] [16].”