## Finite element exterior calculus, homological techniques, and applications (2006)

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Venue: | ACTA NUMERICA |

Citations: | 73 - 13 self |

### BibTeX

@ARTICLE{Arnold06finiteelement,

author = {Douglas N. Arnold and Richard S. Falk and Ragnar Winther},

title = {Finite element exterior calculus, homological techniques, and applications},

journal = {ACTA NUMERICA},

year = {2006},

volume = {15},

pages = {1--155}

}

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### Abstract

### Citations

1770 | Perturbation Theory of Linear Operators - Kato - 1966 |

1467 |
The finite element method for elliptic problems
- Ciarlet
- 1978
(Show Context)
Citation Context ...s of scalar and vector functions. In the tables below, we summarize the correspondences between spaces of finite element differential forms and classical finite element spaces: the Lagrange elements (=-=Ciarlet 1978-=-), the Raviart–Thomas introduced in two dimensions by Raviart and Thomas (1977) and generalized to three dimensions by Nédélec (1980); the Brezzi–Douglas–Marini elements introduced by Brezzi, Douglas ... |

955 | Mixed and Hybrid Finite Element Methods - Brezzi, Fortin - 1991 |

906 | Representation Theory - Fulton, Harris - 1991 |

821 | The Mathematical Theory of Finite Element Methods - Brenner, Scott - 1994 |

757 | Finite Element Methods for Navier-Stokes Equations - Girault, Raviart - 1986 |

461 | Domain decomposition: parallel multilevel methods for elliptic partial differential equations - Bjørstad - 1996 |

444 | Differential Forms in Algebraic Topology - Bott, Tu - 1982 |

194 |
Approximation by finite element functions using local regularisation. RAIRO Modélisation Mathématique et Analyse Numérique
- Clément
- 1975
(Show Context)
Citation Context ...ham sequence are smooth enough that the projection operators are bounded on them, they are generally too smooth to include the finite element spaces. In finite element theory the Clément interpolant (=-=Clément 1975-=-) is often invoked to overcome problems of this sort. In our situation, the Clément interpolant would be defined by assigning to a given form in ω ∈ L 2 Λ k (Ω) a finite element differential form ˇ Πh... |

192 |
Cyclic Homology
- Loday
- 1998
(Show Context)
Citation Context ..., ω ∈HrΛ k (3.7) The operator κ maps the Koszul algebra PΛ toitself. Thereitiscalled the Koszul operator (Guillemin and Sternberg 1999, Chapter 3.1), and gives rise to the homogeneous Koszul complex (=-=Loday 1992-=-, Chapter 3.4.6), 0 →Hr−nΛ n κ −−→ Hr−n+1Λ n−1 κ −−→ ··· κ −−→ HrΛ 0 → 0. (3.8) We show below that this complex is exact for r>0. Adding over polynomial degrees, we obtain the Koszul complex (for any ... |

188 |
On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. R.A.I.R.O. analyse numérique R-2
- Brezzi
- 1974
(Show Context)
Citation Context ...2 Λ n (Ω; K), and the solution operator is a bounded operator L 2 Λ n (Ω; V) → HΛ n−1 (Ω; V) × L 2 Λ n (Ω; V) × L 2 Λ n (Ω; K). This will follow from the general theory of such saddle point problems (=-=Brezzi 1974-=-) once we establish two conditions: (W1) �τ� 2 HΛ ≤ c1〈Aτ, τ〉 whenever τ ∈ HΛ n−1 (Ω; V) satisfies 〈 dτ,v〉 =0 ∀v ∈ L 2 Λ n (Ω; V) and〈Sτ,q〉 =0∀q ∈ L 2 Λ n (Ω; K), (W2) for all nonzero (v, q) ∈ L 2 Λ n... |

187 | A mixed finite element method for 2nd order elliptic problems - Raviart, Thomas - 1977 |

181 | Finite Element Methods for Maxwell’s Equations - Monk - 2003 |

170 | Iterative solution of large sparse systems of equations - Hackbusch - 1994 |

159 |
Geometric integration theory
- Whitney
- 1957
(Show Context)
Citation Context ...sociated to f = fσ ∈ Δk(T )adifferentialk-form on T , which we shall call the Whitney form associated to the subsimplex f, given by k� φσ := (4.3) (−1) i=0 i λσ(i) dλσ(0) ∧···∧ �dλ σ(i) ∧···∧ dλσ(k) (=-=Whitney 1957-=-, equation (12), p. 139). Now if f ′ = fρ is a k-subsimplex different from f, thenforsomei, σ(i) /∈ R(ρ), and so the trace of λσ(i) and dλσ(i) both vanish on f ′ . Thus Trf ′ φσ = 0. On the other hand... |

157 |
Supersymmetry and equivariant de Rham theory
- Guillemin, Sternberg
- 1999
(Show Context)
Citation Context ...R n � �� � � �� � n + r n r + k n + r = = , n k r n − k and dim HrΛ k (R n )=dimPrΛ k (R n−1 ). The space of polynomial differential forms ∞� n� PΛ = HrΛ k r=0 k=0 (3.1) is called the Koszul algebra (=-=Guillemin and Sternberg 1999-=-, Chapter 3.1). For each polynomial degree r ≥ 0 we get a homogeneous polynomial subcomplex of the de Rham complex: 0 →HrΛ 0 d −−→ Hr−1Λ 1 d −−→ ··· d −−→ Hr−nΛ n → 0. (3.2) We shall verify below the ... |

157 | Mixed finite elements in R 3 - Nedelec - 1980 |

155 | Partial differential equations - Taylor - 1996 |

125 | The analysis of multigrid methods - Bramble, Zhang - 2000 |

122 | Partial differential equations I, Basic theory, volume 115 of Applied Mathematical Sciences - Taylor - 1995 |

112 | J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations for elliptic problems - Bramble, Pasciak - 1988 |

109 | U.: A course in homological algebra - Hilton, Stammbach - 1970 |

108 | A.K.: Survey lectures on the mathematical foundations of the finite element method - Babuska, Aziz - 1972 |

98 | Vector potentials in three-dimensional non-smooth domains - Amrouche, Bernardi, et al. - 1998 |

93 | Fast iterative solution of stabilised Stokes systems I: Using simple diagonal preconditioners - Wathen, Silvester - 1993 |

78 | Schwarz methods of Neumann-Neumann type for three dimensional elliptic finite element problems - DRYJA, WIDLUND - 1995 |

78 |
Multigrid method for Maxwell’s equations
- Hiptmair
- 1998
(Show Context)
Citation Context ...ation of finite element exterior calculus was developed by many people over a long period of time. Besides the work of Whitney and Bossavit already mentioned, we signal the contributions of Hiptmair (=-=Hiptmair 1999-=-a, Hiptmair 2001, Hiptmair 2002), especially to the aspects of the theory that are relevant for electromagnetic problems. Interest in the subject grew with the 2002 presentation at the International C... |

71 | A preconditioned iterative method for saddlepoint problems - Rusten, Winther - 1992 |

70 | Mixed finite elements for second order elliptic problems in three variables - Brezzi, Douglas, et al. - 1987 |

65 | Finite elements in computational electromagnetism - Hiptmair |

55 | A new convergence proof for the multigrid method including the V-cycle - Braess, Hackbusch - 1983 |

49 | Differential and Riemannian Manifolds - Lang - 1995 |

45 |
Global estimates for mixed methods for second order elliptic equations
- ROBERTS
- 1985
(Show Context)
Citation Context ...ev and Gunzburger (2005) for a more recent exposition. 7.7. Improved error estimates – basic bounds As is well known from the theory of mixed finite element methods (Falk and Osborn 1980, Douglas and =-=Roberts 1985-=-), it is sometimes possible to get improved error estimates for each term in the mixed formulation by decoupling them. In this subsection, we show how this more refined analysis can be carried out for... |

45 | A new family of mixed finite elements - Nédélec - 1986 |

42 |
Error estimates for mixed methods
- FALK, OSBORN
- 1980
(Show Context)
Citation Context ...mposition principle. See also Bochev and Gunzburger (2005) for a more recent exposition. 7.7. Improved error estimates – basic bounds As is well known from the theory of mixed finite element methods (=-=Falk and Osborn 1980-=-, Douglas and Roberts 1985), it is sometimes possible to get improved error estimates for each term in the mixed formulation by decoupling them. In this subsection, we show how this more refined analy... |

41 | Eigenvalue Problems - Babuˇska, Osborn - 1991 |

40 | An optimal preconditioner for a class of saddle point problems with a penalty term - Klawonn - 1998 |

39 | Iterative techniques for time dependent Stokes problems - Bramble, Pasciak - 1997 |

37 | forms: A class of finite elements for three dimensional computations - Bossavit, Whitney - 1988 |

35 | PEERS: A new mixed finite element for plane elasticity - Arnold, Brezzi, et al. - 1984 |

33 |
Canonical construction of finite elements
- Hiptmair
- 1999
(Show Context)
Citation Context ...ation of finite element exterior calculus was developed by many people over a long period of time. Besides the work of Whitney and Bossavit already mentioned, we signal the contributions of Hiptmair (=-=Hiptmair 1999-=-a, Hiptmair 2001, Hiptmair 2002), especially to the aspects of the theory that are relevant for electromagnetic problems. Interest in the subject grew with the 2002 presentation at the International C... |

32 | Multilevel iterative methods for mixed finite element discretizations of elliptic problems - VASSILEVSKI, WANG - 1992 |

29 | On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form - Boffi, Brezzi, et al. |

28 | Preconditioning discrete approximations of the ReissnerMindlin plate model - ARNOLD, FALK, et al. - 1997 |

26 | Differential complexes and numerical stability - Arnold |

26 | A family of mixed finite elements for the elasticity problem - Stenberg - 1988 |

25 | A coercive bilinear form for Maxwell’s equations - Costabel - 1991 |

22 |
Some equilibrium finite element methods for two-dimensional elasticity problems
- Johnson, Mercier
- 1978
(Show Context)
Citation Context ...gulation of the domain, while the approximate stress space consists of piecewise polynomials with respect to a different, more refined, triangulation (Fraeijs de Veubeke 1965, Watwood and Hartz 1968, =-=Johnson and Mercier 1978-=-, Arnold, Douglas and Gupta 1984b). Because of the lack of suitable mixed elasticity elements that strongly impose the symmetry of the stresses, a number of authors have developed approximation scheme... |

21 |
The harmonic operator for exterior differential forms
- Gaffney
- 1951
(Show Context)
Citation Context ...ctness and Poincaré’s inequality If Ω is a smoothly bounded oriented Riemannian manifold with boundary, then the space HΛ k (Ω)∩ ˚ H ∗ Λ(Ω) is a subspace of H 1 Λ k (Ω), and there holds the estimate (=-=Gaffney 1951-=-) �ω� H 1 ≤ c(� dω� + �δω� + �ω�). (A similar result holds for ˚ HΛ k (Ω) ∩ H ∗ Λ k (Ω).) We can then apply Rellich’s lemma to conclude that HΛ k (Ω) ∩ ˚ H ∗ Λ(Ω) is compactly embedded in L 2 Λ k (Ω).... |

21 | Multigrid method for H(div) in three dimensions, Electron - Hiptmair - 1997 |