Title :
Theoretical Limitations of Discrete Exterior Calculus in the Context of Computational Electromagnetics
Author :
Kotiuga, P. Robert
Author_Institution :
Dept. of Electr. & Comput. Eng., Boston Univ., Boston, MA
fDate :
6/1/2008 12:00:00 AM
Abstract :
Discrete exterior calculus (DEC) attempts to mimic basic operations on differential forms in a discrete setting. This paper considers two concrete instances of how the completion of the DEC program is unachievable, and outlines the practical implications for computational electromagnetics. The two problems are the Commutative Cochain Problem (CCP) and the Discrete Star Localization Problem (DSLP). This paper elaborates on how the CCP is related to problems in computational magnetohydrodynamics, and how DSPL is related to material modeling and constructing discrete Hodge star operators whose matrix representations are maximally sparse in some concrete sense. A final section elaborates on the problem of Poincare duality on the level of cochains, and how the metric-free Whitney form finite element discretization of helicity functionals provides a model of how some of these theoretical obstacles can be side-stepped in three dimensions and, more generally, in dimensions.
Keywords :
computational electromagnetics; finite element analysis; magnetohydrodynamics; Poincare duality; Whitney form finite element discretization; commutative cochain problem; computational electromagnetics; computational magnetohydrodynamics; discrete Hodge star operators; discrete exterior calculus; discrete star localization problem; helicity functionals; material modeling; Magnetohydrodynamics;
Journal_Title :
Magnetics, IEEE Transactions on
DOI :
10.1109/TMAG.2007.915998