This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.
%0 Journal Article
%1 hiptmair2002finite
%A Hiptmair, R.
%B Acta Numerica
%D 2002
%I Cambridge University Press
%K 65n30-pdes-bvps-finite-elements 78m10-optics-electromagnetism-finite-element-method
%P 237-339
%R 10.1017/S0962492902000041
%T Finite elements in computational electromagnetism
%U https://www.cambridge.org/core/article/finite-elements-in-computational-electromagnetism/C145D69E9F4109563E8EFFB9DB963C09
%V 11
%X This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.
@article{hiptmair2002finite,
abstract = {This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.},
added-at = {2022-02-21T03:47:09.000+0100},
author = {Hiptmair, R.},
biburl = {https://www.bibsonomy.org/bibtex/2ade982472a3a802ee1382a657b19af35/gdmcbain},
booktitle = {Acta Numerica},
doi = {10.1017/S0962492902000041},
interhash = {af11f35ac0b8cd7b6878443e7638f9bd},
intrahash = {ade982472a3a802ee1382a657b19af35},
issn = {09624929},
keywords = {65n30-pdes-bvps-finite-elements 78m10-optics-electromagnetism-finite-element-method},
pages = {237-339},
publisher = {Cambridge University Press},
timestamp = {2022-02-21T03:47:09.000+0100},
title = {Finite elements in computational electromagnetism},
url = {https://www.cambridge.org/core/article/finite-elements-in-computational-electromagnetism/C145D69E9F4109563E8EFFB9DB963C09},
volume = 11,
year = 2002
}