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Solvability of a first order system in three-dimensional non-smooth domains

Michal Křížek, Pekka Neittaanmäki (1985)

Aplikace matematiky

A system of first order partial differential equations is studied which is defined by the divergence and rotation operators in a bounded nonsmooth domain Ω 𝐑 3 . On the boundary δ Ω , the vanishing normal component is prescribed. A variational formulation is given and its solvability is investigated.

Some initial boundary problems in electrodynamics for canonical domains in quaternions

Erhard V. Meister, L. Meister (2001)

Mathematica Bohemica

The initial boundary-transmission problems for electromagnetic fields in homogeneous and anisotropic media for canonical semi-infinite domains, like halfspaces, wedges and the exterior of half- and quarter-plane obstacles are formulated with the use of complex quaternions. The time-harmonic case was studied by A. Passow in his Darmstadt thesis 1998 in which he treated also the case of an homogeneous and isotropic layer in free space and above an ideally conducting plane. For thin layers and free...

Some new results in multiphase geometrical optics

Olof Runborg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta...

Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements

Salvatore Caorsi, Paolo Fernandes, Mirco Raffetto (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec’s edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proof is new, instead, and shows how families of spurious-free elements can...

Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements

Salvatore Caorsi, Paolo Fernandes, Mirco Raffetto (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proof is new, instead, and shows how families of spurious-free elements...

Stabilization methods in relaxed micromagnetism

Stefan A. Funken, Andreas Prohl (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potential u and magnetization 𝐦 . In [C. Carstensen and A. Prohl, Numer. Math. 90 (2001) 65–99], the conforming P 1 - ( P 0 ) d -element in d = 2 , 3 spatial dimensions is shown to...

Stabilization methods in relaxed micromagnetism

Stefan A. Funken, Andreas Prohl (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potential u and magnetization m. In [C. Carstensen and A. Prohl, Numer. Math.90 (2001) 65–99], the conforming P1 - (P0)d-element in d=2,3 spatial dimensions...

Stabilized Galerkin methods for magnetic advection

Holger Heumann, Ralf Hiptmair (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Taking the cue from stabilized Galerkin methods for scalar advection problems, we adapt the technique to boundary value problems modeling the advection of magnetic fields. We provide rigorous a priori error estimates for both fully discontinuous piecewise polynomial trial functions and -conforming finite elements.

Stable upwind schemes for the magnetic induction equation

Franz G. Fuchs, Kenneth H. Karlsen, Siddharta Mishra, Nils H. Risebro (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be...

Static electromagnetic fields in monotone media

Rainer Picard (1992)

Banach Center Publications

The paper considers the static Maxwell system for a Lipschitz domain with perfectly conducting boundary. Electric and magnetic permeability ε and μ are allowed to be monotone and Lipschitz continuous functions of the electromagnetic field. The existence theory is developed in the framework of the theory of monotone operators.

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