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On the existence and uniqueness of Maxwell's equations in bounded domains with application to magnetotellurics
Abstract
We analyze the solution of the time-harmonic Maxwell equations with vanishing elec-tric permittivity in bounded domains and subject to absorbing boundary conditions. The problem arises naturally in magnetotellurics when considering the propagation of electromagnetic waves within the earth's interior. Existence and uniqueness are shown under the assumption that the source functions are square integrable. In this case, the electric and magnetic fields belong to H(curl; Ω). If, in addition, the divergences of the source functions are square integrable and the coefficients are Lipschitz-continuous, a stronger regularity result is obtained. A decomposition of the space of square integrable vector functions and a new compact imbedding result are exploited.
... To conclude, let us prove the compact embedding of the space W N (ξ; Ω) into L 2 (Ω), extending the result of Theorem 2.4.10. A similar result may be found in [100] (in the connected boundary case). Proof. ...
... . Therefore, the function space in which (4.1) may hold is unclear. In the literature, it is generally assumed or stated without proper justification (see, e.g., [43,100,70,56]), that the condition (4.2) holds in L 2 t (Γ); and, in particular, that one should look for the solution of the associated time-harmonic Maxwell problem in the space ...
The numerical simulation of electromagnetic problems in complex physical settings is a trending topic which conveys many scientific and industrial applications, such as the design of optical metamaterials, or the study of cold plasmas. The mathematical and numerical analysis of Maxwell problems is wellknown in simple physical contexts, when the material parameters are isotropic. Some results in anisotropic media exist, but they generally tend to focus on the case where the material tensors are real symmetric (or complex) Hermitian) definite positive. However, problems in more complex media are not covered by the standard theory. Therefore, new mathematical tools need to be developped to analyse thses problems. This thesis aims at analysing time-harmonic electromagnetic problems for a general class of complex anisotropic material tensors. These are called ellopptic materials. We derive an extended functional framework well-suited for these anisotropic problems, generalizing well-known results. We study the well-posedness of Maxwell boundary value problems for Dirichlet, Neumann, and Robin boundary conditions. For the Robin case, the characterization of appropriate function spaces for Robin traces is addressed. The regularity of the solution and its curl is studied, and elements of numerical analysis for edge finite elements are provided. In the perspective of the use of Domain Decomposition Methods (DDM) for accelerated numerical computing, various decomposed formulations are proposed and studied, focusing on their right meaning in terms of function spaces and equivalence with the global problem. These results are complemented with some numerical DDM experimentations in anisotropic media.
... The existence and uniqueness of the solution for (1) is assured (cf. [25]). ...
... Substituting (24), (25) and (27) into (33), we easily have ...
In this paper, high-order rectangular edge elements are used to solve the two dimensional time-harmonic Maxwell’s equations. Superconvergence for the Nédélec interpolation at the Gauss points is proved for both the second and third order edge elements. Using this fact, we obtain the superconvergence results for the electric field , magnetic field H and in the discrete norm when the Maxwell’s equations are solved by both elements. Extensive numerical results are presented to justify our theoretical analysis.
... . Therefore, the function space in which the Robin condition of (2.6) may hold is unclear. In the literature, it is generally assumed or stated without proper justification (see, e.g., [19,34,24,20]), that the Robin condition of (2.6) holds in L 2 t (Γ A ); and, in particular, that one should look for the solution of the associated time-harmonic Maxwell problem in the space ...
... Also, assume that a is a real-valued, Lipschitz-continuous function on Γ such that 0 < a min ≤ a(x) for x ∈ Γ. A proof of the following existence and uniqueness results for (2.3)-(2.4) is given in 23 . 3 and ω = 0. ...
We present a nonconforming mixed finite element scheme for the approximate solution of the time-harmonic Maxwell's equations in a three-dimensional, bounded domain with absorbing boundary conditions on artificial boundaries. The numerical procedures are employed to solve the direct problem in magnetotellurics consisting in determining a scattered electromagnetic field in a model of the earth having bounded conductivity anomalies of arbitrary shapes. A domain-decomposition iterative algorithm which is naturally parallelizable and is based on a hybridization of the mixed method allows the solution of large three-dimensional models. Convergence of the approximation by the mixed method is proved, as well as the convergence of the iteration.
... Mathematical Problems in Engineering 3 where (⋅, ⋅) denotes the usual scalar product in the Hilbert space 2 (Ω) of complex-valued functions. The well-posedness of problem (6) is addressed by the following theorem; see also [19,22,23]. ...
The solution fields of Maxwell’s equations are known to exhibit singularities near corners, crack tips, edges, and so forth of the physical domain. The structures of the singular fields are well known up to some undetermined coefficients. In two-dimensional domains with corners and cracks, the unknown coefficients are real constants. However, in three-dimensional domains the unknown coefficients are functions defined along the corresponding edges. This paper proposes explicit formulas for the computation of these coefficients in the case of two-dimensional domains with corners and three-dimensional domains with straight edges. The coefficients of the singular fields along straight edges of three-dimensional domains are represented in terms of Fourier series. The formulas presented are aimed at the numerical approximation of the coefficients of the singular fields. They can also be used for the construction of adaptive
H
1
-nodal finite-element procedures for the efficient numerical treatment of Maxwell’s equations in nonsmooth domains.
... Results on existence and uniqueness of the solution of (2.4)-(2.5) have been given in [30]. ...
We present a collection of global and domain decomposed mixed finite element schemes for the approximate solution of the time-harmonic Maxwell equations in a three-dimensional bounded domain with absorbing boundary conditions on the artificial boundaries. The numerical procedures enable to solve efficiently the direct problem in magnetotellurics to find the electromagnetic scattered field in an earth model of arbitrary conductivity properties. The domain decomposition algorithm is a naturally parallelizable iterative procedure providing a necessary tool when dealing with large three-dimensional models. Convergence results for the numerical procedures are derived.
Multi-observable probabilistic inversion (e.g. Afonso et al., 2013a,b, 2016b) is a recent framework specifically designed to provide insights into the physico-chemical structure of the lithosphere and its complex interactions with the sublithospheric upper mantle. Of particular relevance is the inclusion of 3D magnetotelluric (MT) data, as it provides complementary information not only on the thermal structure but also on water content and fluid pathways; this is critical for understanding and
imaging the complex fluid-rock interactions responsible for mineralization events and water-assisted tectonism. However, in order to isolate the effect of fluids from other potential compositional and thermal 'background" effects, MT data needs to be informed by other data sets such as seismic and gravity data.
In order to include MT data into multi-observable probabilistic inversions, we first need to solve the problem of computational effciency when solving the MT equations in 3D. For this, we have combined probabilistic inversion methods, parallel MT solvers (Zyserman & Santos, 2000) and advanced reduced order modelling techniques to obtain fast, yet accurate, solutions to both the MT
inversion and the full 3D joint inversion of MT and surface wave data. Such a probabilistic formalism offers a natural framework to assess non-uniqueness and uncertainties affecting the inversion, which are otherwise hard to quantify using traditional inversion methods.
The outcomes of this thesis demonstrate the capabilities of the conceptual and numerical framework for 3D multi-observable probabilistic inversions and open up new exciting opportunities for integrated geophysical imaging of the Earth's interior.
In this paper, a nonconforming mixed finite element method (FEM) is presented to approximate time-dependent Maxwell's equations in a three-dimensional bounded domain with absorbing boundary conditions (ABC). By employing traditional variational formula, instead of adding penalty terms, we show that the discrete scheme is robust. Meanwhile, with the help of the element's typical properties and derivative transfer skills, the convergence analysis and error estimates for semidiscrete and backward Euler fully-discrete schemes are given, respectively. Numerical tests show the validity of the proposed method.
We present nonoverlapping domain decomposition methods for the approximation of both electromagnetic fields in a three-dimensional bounded domain satisfying absorbing boundary conditions. A Seidel-type domain decomposition iterative method is introduced based on a hybridization of a nonconforming mixed finite element method. Convergence results for the numerical procedure are proved by introducing a suitable pseudo-energy. The spectral radius of the iterative procedure is estimated and a method for choosing an optimal matching parameter is given. A red-black Seidel-type method which is readily parallelizable is also introduced and analyzed. Numerical experiments confirm that the presented algorithms are faster than the conventional Jacobi-type ones.
In this paper we study the motion of a magnetic field H in a conductive medium Ω⊂R3 under the influence of a system generator. By neglecting displacement currents, the magnetic field satisfies a nonlinear Maxwell's system: Ht+∇×[ρ(x,t)∇×H]=f(|H|)H, where f(|H|)H represents the magnetic currents depending upon the strength of H. We prove that under appropriate initial and boundary conditions, the system has a global solution and the solution is also unique. Moreover, we show that the solution H will blow up in finite time if f(s) satisfies certain growth conditions. Finally, we generalize the results to the problem associated with a nonlinear boundary condition.
We present a nonconforming mixed finite element scheme for the approximate solution of the time-harmonic Maxwell's equations in a three-dimensional, bounded domain with absorbing boundary conditions on artificial boundaries. The numerical procedures are employed to solve the direct problem in magnetotellurics consisting in determining a scattered electromagnetic field in a model of the earth having bounded conductivity anomalies of arbitrary shapes. A domain-decomposition iterative algorithm which is naturally parallelizable and is based on a hybridization of the mixed method allows the solution of large three-dimensional models. Convergence of the approximation by the mixed method is proved, as well as the convergence of the iteration.
We present a collection of global and domain decomposed mixed finite element schemes for the approximate solution of the time-harmonic Maxwell equations in a three-dimensional bounded domain with absorbing boundary conditions on the artificial boundaries. The numerical procedures enable to solve efficiently the direct problem in magnetotellurics to find the electromagnetic scattered field in an earth model of arbitrary conductivity properties. The domain decomposition algorithm is a naturally parallelizable iterative procedure providing a necessary tool when dealing with large three-dimensional models. Convergence results for the numerical procedures are derived.
We consider the Dirichlet Laplacian operator on a curved quantum guide in with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method.
Magnetostatic and electrostatic problems with mixed boundary conditions are studied. The medium can have a nonsmooth boundary and very irregular physical properties due to inhomogeneity and anisotropy. The topological assumptions are general enough to meet the requirements of the engineering applications. Necessary and sufficient conditions for solvability are found and the set of the solutions is characterized. Moreover, uniqueness is recovered by means of a finite number of supplementary conditions which are equivalent to prescribing a finite number of suitably chosen fluxes or potentials. A functional framework in which other important problems of electromagnetics naturally fit is developed.
In many cases, the numerical resolution of Maxwell's equations is very expensive in terms of computational cost. The Darwin model, an approximation of Maxwell's equations obtained by neglecting the divergence free part of the displacement current, can be used to compute the solution more economically. However, this model requires the electric field to be decomposed into two parts for which no straightforward boundary conditions can be derived. In this paper, we consider the case of a computational domain which is not simply connected. With the help of a functional framework, a decomposition of the fields is derived. It is then used to characterize mathematically the solutions of the Darwin model on such a domain.