Finite elements in computational electromagnetism

By Ralf Hiptmair. Acta Numerica, 2002

972 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/144c5f54eea7e8fa487c9398068a0a45ddf11f51

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In Computational Mechanics

Citations in the corpus

Block preconditioning of real-valued iterative algorithms for complex linear systems (2007, methodology): “…integration of the time-dependent Schrödinger equation by implicit methods) (van Dijk & Toyama, 2007), electromagnetism (Maxwell's equations) (Hiptmair, 2002), molecular scattering (Poirier, 2000), structural dynamics (frequency response analysis of mechanical systems) (Feriani et al.,…”
Mathematical analysis of variational isogeometric methods* (2014, methodology): “R. Hiptmair (2002), Finite elements in computational electromagnetism.”
Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system (2008, neutral): “The following approximation property is shown in [18], Lemma 4.”
Finite Element Methods for Maxwell's Equations (2003, methodology): “ degrees:ı 1 volume(K) ∫ K pqdV for all q∈P k−3; we can dene an interpolation operator ˇ h. Then for suciently smooth scalar functions p ∇(ˇ hp)=r h(∇p): The recognition by Bossavit [11] (see also [31]) that this, and a similar equality for H(curl; ) and H(div; ) interpolation, constitute a discrete”
Finite Element Methods for Maxwell's Equations (2019, background): “An important work on the mathematical foundations of edge elements and the use of these elements for Maxwell’s equations is due to Hiptmair [31], and for a different approach see [44].”
Numerical methods for Hamiltonian PDEs (2006, methodology): “A discretization of (35) is carried out by using discrete differential forms [12, 63, 37] or difference forms [97].”
On preconditioned iteration methods for complex linear systems (2015, background): “This class of complex linear systems arises inmany important applications such as computational electrodynamics [1–3], time-dependent Schrödinger equations [4,5], acoustic scattering problems [6–8], numerical solution of (stiff)”
Finite element exterior calculus, homological techniques, and applications (2006, methodology): “Besides the work of Whitney and Bossavit already mentioned, we signal the contributions of Hiptmair (1999a, 2001, 2002), especially to the aspects of the theory that are relevant for electromagnetic problems.”
ROBUST SUBSPACE CORRECTION METHODS FOR NEARLY SINGULAR SYSTEMS (2007, neutral): “ples can also obtained from finite element methods for H(grad), H(div) and H(curl) systems [ 12 , 2].”