Existence, uniqueness and finite element approximation of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials

Abstract

Purpose

To provide sufficient conditions for existence, uniqueness and finite element approximability of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials.

Design/methodology/approach

The objectives are achieved by analysing the most simple conditions under which radiation, scattering and cavity problems are well posed and can be reliably solved by the finite element method. The above “most simple conditions” refer to the hypotheses allowing the exploitation of the simplest mathematical tools dealing with the well posedness of variationally formulated problems, i.e. Lax‐Milgram and first Strang lemmas.

Findings

The results of interest are found to hold true whenever the effective dielectric permittivity is uniformly positive definite on the regions where no losses are modelled in it and, moreover, the effective magnetic permeability is uniformly negative definite on the regions where no losses are modelled in it. The same good features hold true if “positive” is replaced by “negative” and vice versa in the previous sentence.

Research limitations/implications

It is a priori known that more sophisticated mathematical tools, like Fredholm alternative and compactness results, can provide more general results. However this would require a more complicated analysis and could be considered in a future research.

Practical implications

The design of practical devices involving metamaterials requires the use of reliable electromagnetic simulators. The finite element method is shown to be reliable even when metamaterials are involved, provided some simple conditions are satisfied.

Originality/value

For the first time to the best of authors' knowledge a numerical method is shown to be reliable in problems involving metamaterials.

Citation

Fernandes, P. and Raffetto, M. (2005), "Existence, uniqueness and finite element approximation of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 24 No. 4, pp. 1450-1469. https://doi.org/10.1108/03321640510615724