arXiv subjects

Martin Halla

Publications and source records attributed to Martin Halla.

21 records · Page 2Linked to original sources

arXiv2019

Electromagnetic Stekloff eigenvalues: existence and behavior in the selfadjoint case

In [Camano, Lackner, Monk, SIAM J. Math. Anal., Vol. 49, No. 6, pp. 4376-4401 (2017)] it was suggested to use Stekloff eigenvalues for Maxwell equations as target signature for nondestructive testing via inverse scattering. The authors recognized that in general the eigenvalues due not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The Fredholmness and the approximation of both problems were analyzed in [Halla, arXiv:1909.00689 (2019)]. The present work considers the original eigenvalue problem in the selfadjoint case. We report that apart for a countable set of particular frequencies, the spectrum consists of three disjoint parts: The essential spectrum consisting of the point zero, an infinite sequence of positive eigenvalues which accumulate only at infinity and an infinite sequence of negative eigenvalues which accumulate only at zero. The analysis is based on a representation of the operator as block operator. For small/big enough eigenvalue parameter the Schur-complements with respect to different components can be build. For each Schur-complement the existence of an infinite sequence of eigenvalues is proved via a fixed point technique similar to [Cakoni, Haddar, Applicable Analysis, 88:4, 475-493 (2009)]. The modified eigenvalue problem considered in the above references arises as limit of one of the Schur-complements.

math.SP↗

arXiv2019

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

We consider Galerkin approximations of holomorphic Fredholm operator eigenvalue problems for which the operator values don't have the structure "coercive+compact". In this case the regularity (in sense of [O. Karma, Numer. Funct. Anal. Optim. 17 (1996)]) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework immediately improves the results of [T. Hohage, L. Nannen, BIT 55(1) (2015)], is immediately applicable to analyze approximations of eigenvalue problems related to [A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Num. Math. 138(4) (2018)] and is already applied in [G. Unger, preprint (2017)].

math.NA↗

arXiv2015

Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems

We consider time-harmonic linear elasticity equations in domains containing two-dimensional semi-infinite strips. Since for such problems there exist modes with different signs of group and phase velocity, standard perfectly matched layer (PML) as well as standard Hardy space infinite element methods fail. We apply a recently developed infinite element method for a physically correct discretization of such waveguide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are certain Hardy spaces. The Hardy space is chosen such that the construction of a simple infinite element is possible. The method does not use a modal separation and works on intervals of frequencies. On those intervals the involved operators are frequency independent and hence lead to linear eigenvalue problems when computing resonances. Numerical experiments containing convergence tests and resonance problems are included.

math.NA↗