scholarly journals Whitney forms and their extensions

2021 ◽  
pp. 113520
Author(s):  
Jonni Lohi ◽  
Lauri Kettunen
Keyword(s):  
Whitney Forms

A CONSTRUCTION OF SPACES OF COMPATIBLE DIFFERENTIAL FORMS ON CELLULAR COMPLEXES

2008 ◽  
Vol 18 (05) ◽  
pp. 739-757 ◽  
Author(s):  
SNORRE H. CHRISTIANSEN
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Differential Forms ◽  
Cochain Complex ◽  
Two Dimensional ◽  
Exterior Derivative ◽  
Whitney Forms ◽  
Reconstruction Operator ◽  
Mimetic Finite Differences ◽  
Cellular Complex

Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.

scholarly journals Efficient finite element assembly of high order Whitney forms

2015 ◽  
Vol 9 (2) ◽  
pp. 204-210 ◽  
Author(s):  
Nicolas Marsic ◽  
Christophe Geuzaine
Keyword(s):  
Finite Element ◽  
High Order ◽  
Whitney Forms

Approximation of the time-dependent induction equation with advection using Whitney elements

2016 ◽  
Vol 35 (1) ◽  
pp. 326-338 ◽  
Author(s):  
Caroline Nore ◽  
Houda Zaidi ◽  
Frederic Bouillault ◽  
Alain Bossavit ◽  
Jean-Luc Guermond
Keyword(s):  
Velocity Field ◽  
Dynamo Action ◽  
Content Type ◽  
Kinematic Dynamo ◽  
Von Karman ◽  
Whitney Forms ◽  
Dependent Variables ◽  
Induction Equation ◽  
Von Kármán ◽  
Good Agreement

Purpose – The purpose of this paper is to present a new formulation for taking into account the convective term due to an imposed velocity field in the induction equation in a code based on Whitney elements called DOLMEN. Different Whitney forms are used to approximate the dependent variables. The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. DOLMEN is developed to investigate the dynamo action in non-axisymmetric domains like the impeller driven flow of the von Kármán Sodium (VKS) experiment. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field. Design/methodology/approach – Different Whitney forms are used to approximate the dependent variables. The vector potential is discretized using first-order edge elements of the first family. The velocity is approximated by using the first-order Raviart-Thomas elements. The time stepping is done by using the Crank-Nicolson scheme. Findings – The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field. Originality/value – The findings offer a basis to a scenario for the VKS dynamo.

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