de Rham currents in discrete electromagnetism

Abstract

Purpose

To describe and extend existing concepts of discrete electromagnetism in a unified formalism; to give examples for the usefulness of the presented ideas for our theoretical work, especially with regard to energy.

Design/methodology/approach

After a concise introduction to the mathematical concepts of discrete electromagnetism, we introduce continuous de Rham currents and give their discrete counterpart. We define operators acting upon discrete currents, and apply the theory to electromagnetism.

Findings

de Rham current theory yields a mathematical framework for the discussion of discrete electromagnetic problems: The focus is on energy‐balance equations; a discrete Lagrangian can be defined for various modeling problems; the Galerkin approach fits nicely into the proposed formalism; boundary terms in discrete formulations are an implicit feature to the theory.

Research limitations/implications

In this paper, we use the interpolation of discrete fields by Whitney forms on a simplicial cell complex. The resulting discrete formulation is identical to a Galerkin finite‐element method. Other numerical techniques that do not resort to Whitney‐form interpolation can equally be discussed in de Rham‐current terminology.

Originality/value

Rather than a novel numerical technique, the paper presents a unified mathematical framework for the discussion of different practical approaches. We advocate a canonical treatment of energy‐related quantities and of boundary terms in discrete formulations.

Citation

Auchmann, B. and Kurz, S. (2007), "de Rham currents in discrete electromagnetism", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 26 No. 3, pp. 743-757. https://doi.org/10.1108/03321640710751190

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