Finite Element Approximation of the Modified Maxwell's Stekloff Eigenvalues

The modified Maxwell's Stekloff eigenvalue problem arises recently from the inverse electromagnetic scattering theory for inhomogeneous media. This paper contains a rigorous analysis of both the eigenvalue problem and the associated source problem on Lipschitz polyhedra. A new finite element method is proposed to compute Stekloff eigenvalues. By applying the Babuska-Osborn theory, we prove an error estimate without additional regularity assumptions. Numerical results are presented for validation.

[1]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[2]  F. Kikuchi.  On a discrete compactness property for the Nedelec finite elements , 1989 .

[3]  M. Costabel.  A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains , 1990 .

[4]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[5]  Alberto Valli,et al.  An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations , 1999, Math. Comput..

[6]  Patrick Ciarlet,et al.  On traces for functional spaces related to Maxwell's equations Part I: An integration by parts formula in Lipschitz polyhedra , 2001 .

[7]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[8]  Ralf Hiptmair,et al.  Natural Boundary Element Methods for the Electric Field Integral Equation on Polyhedra , 2002, SIAM J. Numer. Anal..

[9]  Peter Monk,et al.  Error Analysis of a Finite Element-Integral Equation Scheme for Approximating the Time-Harmonic Maxwell System , 2002, SIAM J. Numer. Anal..

[10]  A. Buffa,et al.  On traces for H(curl,Ω) in Lipschitz domains , 2002 .

[11]  C. Schwab,et al.  Boundary element methods for Maxwell's equations on non-smooth domains , 2002, Numerische Mathematik.

[12]  Alfredo Bermúdez,et al.  Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers , 2004, Math. Comput..

[13]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[14]  G. Gatica,et al.  Finite element analysis of a time harmonic Maxwell problem with an impedance boundary condition , 2012 .

[15]  Fioralba Cakoni,et al.  Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids , 2014 .

[16]  Peter Monk,et al.  Stekloff Eigenvalues in Inverse Scattering , 2016, SIAM J. Appl. Math..

[17]  Aihui Zhou,et al.  Finite Element Methods for Eigenvalue Problems , 2016 .

[18]  Patrick Ciarlet,et al.  On the approximation of electromagnetic fields by edge finite elements. Part 1: Sharp interpolation results for low-regularity fields , 2016, Comput. Math. Appl..

[19]  Peter Monk,et al.  Electromagnetic Stekloff Eigenvalues in Inverse Scattering , 2017, SIAM J. Math. Anal..

[20]  Morten Hjorth-Jensen.  Eigenvalue Problems , 2021, Explorations in Numerical Analysis.

[21]  Jiguang Sun,et al.  Spectral Indicator Method for a Non-selfadjoint Steklov Eigenvalue Problem , 2018, Journal of Scientific Computing.

[22]  Hehu Xie,et al.  Guaranteed Eigenvalue Bounds for the Steklov Eigenvalue Problem , 2018, SIAM J. Numer. Anal..

[23]  Peter Monk,et al.  Finite Element Methods for Maxwell's Equations , 2003 .

[24]  A new finite element approach for the Dirichlet eigenvalue problem , 2020, Appl. Math. Lett..

[25]  Martin Halla,et al.  Electromagnetic Stekloff eigenvalues: approximation analysis , 2019, ESAIM: Mathematical Modelling and Numerical Analysis.