scholarly journals Discrete Differential Forms for Computational Modeling

2008 ◽  
pp. 287-324 ◽  
Author(s):  
Mathieu Desbrun ◽  
Eva Kanso ◽  
Yiying Tong
Keyword(s):  
Computational Modeling ◽  
Differential Forms ◽  
Discrete Differential Forms

scholarly journals Spline discrete differential forms

2012 ◽  
Vol 35 ◽  
pp. 197-202 ◽  
Author(s):  
Aurore Back ◽  
Eric Sonnendrücker
Keyword(s):  
Differential Forms ◽  
Discrete Differential Forms

scholarly journals Discrete Differential Forms for $(1+1)$-Dimensional Cosmological Space-Times

2010 ◽  
Vol 32 (3) ◽  
pp. 1140-1158 ◽  
Author(s):  
Ronny Richter ◽  
Jörg Frauendiener
Keyword(s):  
Differential Forms ◽  
Discrete Differential Forms

scholarly journals Hierarchical B-spline complexes of discrete differential forms

2018 ◽  
Vol 40 (1) ◽  
pp. 422-473 ◽  
Author(s):  
John A Evans ◽  
Michael A Scott ◽  
Kendrick M Shepherd ◽  
Derek C Thomas ◽  
Rafael Vázquez Hernández
Keyword(s):  
Stokes Problem ◽  
Differential Forms ◽  
Spatial Dimension ◽  
Necessary Condition ◽  
Two Dimensional ◽  
B Spline ◽  
Discrete Differential Forms ◽  
The Stability ◽  
Local Exactness ◽  
Sufficient And Necessary Condition

Abstract In this paper we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

scholarly journals Kontinuierliche und diskrete Differenzialformen als Ausgangspunkt für numerische Methoden in der Elektrodynamik

2003 ◽  
Vol 1 ◽  
pp. 73-80 ◽  
Author(s):  
S. Kurz
Keyword(s):  
Numerical Methods ◽  
Constitutive Equations ◽  
Differential Forms ◽  
Integral Form ◽  
Discretization Scheme ◽  
Integration Technique ◽  
Finite Integration Technique ◽  
Discrete Differential Forms ◽  
Starting Point ◽  
Discrete Setting

Abstract. Die Grundgleichungen der Elektrodynamik werden häufig in integraler Form aufgestellt. Die Umwandlung in partielle Differenzialgleichungen geschieht dann durch Anwendung der Integralsätze von Gauss und Stokes. Beide Integralsätze besitzen eine große formale Ähnlichkeit. Formuliert man die Maxwellschen Gleichungen mit Hilfe von Differenzialformen, wird diese zunächst formale Analogie verständlich, als Konsequenz eines abstrakteren Konzeptes. Neben der damit einhergehenden übersichtlichen und eleganten Darstellung der Elektrodynamik erhält man einen gut geeigneten Ausgangspunkt für numerische Methoden. Differenzialformen besitzen natürliche Entsprechungen im Diskreten, die diskreten Differenzialformen. Das hieraus in niedrigster Ordnung resultierende Diskretisierungsschema entspricht der Allokation von Freiheitsgraden auf zueinander dualen Gittersystemen, wie sie von der FIT (= Finite Integration Technique) bekannt ist. Größere Freiheiten hat man bei der Diskretisierung der Materialbeziehungen, die auf diskrete Hodge-Operatoren führt. Je nach verwendetem Ansatz (orthogonale oder baryzentrische duale Gitter) erhält man unterschiedliche numerische Verfahren. Kontinuierliche und diskrete Differenzialformen können deshalb als allgemeiner Ausgangspunkt für numerische Methoden in der Elektrodynamik betrachtet werden. The fundamental laws of electrodynamics are often stated in integral form. The conversion to partial differential equations is conveyed by application of the integral theorems of Gauss and Stokes. Both theorems bear a strong formal resemblance. If Maxwell’s equations are reformulated in terms of differential forms this seemingly formal analogy will become obvious, as a consequence of a more abstract underlying concept. Besides the accompanying concise and elegant formulation of electrodynamics a useful starting point for numerical methods is obtained. Differential forms possess natural correspondents in the discrete setting, the so called discrete differential forms. The resulting discretization scheme of lowest order corresponds to the allocation of degerees of freedom on dual grid pairs, which is well known from the FIT (= Finite Integration Technique). There is more freedom when the constitutive equations are to be discretized, which yields discrete Hodge operators. Depending on the employed technique (orthogonal or barycentric dual grids) one ends up with different numerical schemes. Continuous and discrete differential forms can therefore be regarded as a general starting point for numerical methods in electrodynamics.

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