Vector potential formulation of a quasi-static EM induction problem: existence, uniqueness and stability of the weak sol
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Vector potential formulation of a quasi-static EM induction problem: existence, uniqueness and stability of the weak solution O. Souˇcek · Z. Martinec · J. Velímský
Received: 5 April 2011 / Accepted: 10 May 2011 / Published online: 21 May 2011 © Springer-Verlag 2011
Abstract We deal with the electromagnetic induction in a conductor with 3D distribution of electric conductivity in quasi-static approximation with the focus on theoretical aspects related to the solvability of this problem. We formulate the initial, boundary-value problem of electromagnetic induction in terms of a magnetic vector potential only, first in differential and then in integral forms. We prove that the problem is well posed in the Hadamard sense, that a solution exists, is unique and continuously dependent on data. The fact that no electric scalar potential is employed in the formulation and no gauge condition is imposed on the magnetic vector potential makes the formulation attractive for numerical implementations. Keywords
Quasi-static EM induction · Vector potential · Existence and uniqueness
Mathematics Subject Classification (2000) 35A01 (Existence problems: global existence, local existence, non-existence) · 35A02 (Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness) · 78M99 (Optics, Electromagnetic Theory None of the above, but in this section) 1 Introduction The method of electromagnetic (EM) induction sounding of the Earth has been recently subject to rapid development on spatial scales ranging from local sub-surface studies O. Souˇcek (B) · Z. Martinec Department of Geophysics, School of Cosmic Physics, Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin, Ireland e-mail: [email protected] J. Velímský Department of Geophysics, Faculty of Mathematics and Physics, Charles University in Prague, V Holešoviˇckách 2, 180 00 Praha 8, Czech Republic
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to global problems solved in the spherical domain of the Earth. Motivated by the influx of new, high-quality data from large instrument arrays, seafloor sensors, or low-orbit satellites, and with increasing computational power available, the modelling community has been aiming at the development of fast and accurate methods capable of solving the EM induction equation in media with 3D distribution of conductivity, and complicated geometry. Various formulations in terms of electric and magnetic fields or potentials, and numerical methods, including finite differences on staggered grids, integral methods, finite elements, and spectral methods have been employed for this purpose. For a comprehensive overview, see Avdeev (2005), and references therein. In this paper we concentrate on the theoretical properties of a particular formulation employing the vector magnetic potential with implicit gauging, which is attractive for the implementation due to reduced number of unknowns. 1.1 Formulation of the problem We will explore the quasi-static approximation to Maxwell’s equations for a conductor occupying a regular
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