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Sep 23, 2013
09/13

by
Dirk Pauly

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We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.

Source: http://arxiv.org/abs/1105.4073v1

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28

Sep 23, 2013
09/13

by
Dirk Pauly

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We discuss the radiation problem of total reflection for a time-harmonic generalized Maxwell system in a non-smooth exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the identity. By means of the limiting absorption principle, a Fredholm alternative holds true and the eigensolutions decay polynomially resp. exponentially at infinity. We prove that the corresponding eigenvalues do not accumulate even at zero. Next, we...

Source: http://arxiv.org/abs/1105.4066v1

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Sep 23, 2013
09/13

by
Dirk Pauly

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We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions which are up to a finite sum of special generalized spherical harmonics in another appropriately weighted Sobolev space. As a byproduct we prove a generalized spherical harmonics...

Source: http://arxiv.org/abs/1105.4070v1

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Sep 23, 2013
09/13

by
Dirk Pauly

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We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the identity. As a canonical application we show that the corresponding eigen-values do not accumulate in R \ {0} and that by means of Eidus' limiting absorption principle a Fredholm...

Source: http://arxiv.org/abs/1105.4105v1

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24

Sep 23, 2013
09/13

by
Dirk Pauly

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We continue the study of the operator of generalized Maxwell equations and completely discover the behavior of the solutions of the time-harmonic equations as the frequency tends to zero. Thereby, we identify degenerate operators in terms of special 'polynomially growing' solutions of a corresponding static problem, which must be added to the 'usual' Neumann series in order to describe the low frequency asymptotic adequately.

Source: http://arxiv.org/abs/1105.4088v1

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Sep 23, 2013
09/13

by
Peter Kuhn; Dirk Pauly

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We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.

Source: http://arxiv.org/abs/1105.4091v1

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Sep 23, 2013
09/13

by
Dirk Pauly; Tuomo Rossi

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We present both, theory and an algorithm for solving time-harmonic wave problems in a general setting. The time-harmonic solutions will be achieved by computing time-periodic solutions of the original wave equations. Thus, an exact controllability technique is proposed to solve the time-dependent wave equations. We discuss a first order Maxwell type system, which will be formulated in the framework of alternating differential forms. This enables us to investigate different kinds of classical...

Source: http://arxiv.org/abs/1105.4095v1

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Sep 23, 2013
09/13

by
Dirk Pauly; Sergey Repin

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This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between the exact and the approximate solution of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such...

Source: http://arxiv.org/abs/1105.4107v1

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Sep 23, 2013
09/13

by
Dirk Pauly; Sergey Repin

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This paper is concerned with the derivation of computable and guaranteed upper bounds of the difference between the exact and the approximate solution of an exterior domain boundary value problem for a linear elliptic equation. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy...

Source: http://arxiv.org/abs/1105.4101v1

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69

Sep 23, 2013
09/13

by
Dirk Pauly; Sergey Repin; Tuomo Rossi

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In this paper, we consider an initial boundary value problem for Maxwell's equations. For this hyperbolic type problem, we derive guaranteed and computable upper bounds for the difference between the exact solution and any pair of vector fields in the space-time cylinder that belongs to the corresponding admissible energy class. For this purpose, we use a method suggested earlier for the wave equation.

Source: http://arxiv.org/abs/1105.4110v1