Discontinuous Galerkin Approximation of the Maxwell Eigenproblem

A theoretical framework for the analysis of discontinuous Galerkin approximations of the Maxwell eigenproblem with discontinuous coefficients is presented. Necessary and sufficient conditions for a spurious-free approximation are established, and it is shown that, at least on conformal meshes, basically all the discontinuous Galerkin methods in the literature actually fit into this framework. Relations with the classical theory for conforming approximations are also discussed.

[1]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[2]  Ilaria Perugia,et al.  Interior penalty method for the indefinite time-harmonic Maxwell equations , 2005, Numerische Mathematik.

[3]  J. Schwartz,et al.  Spectral theory : self adjoint operators in Hilbert space , 1963 .

[4]  Shan Zhao,et al.  DSC time-domain solution of Maxwell's equations , 2003 .

[5]  Ilaria Perugia,et al.  The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations , 2003, Math. Comput..

[6]  Klaus-Jürgen Bathe,et al.  On Mixed Elements for Acoustic Fluid-Structure Interactions , 1997 .

[7]  JEAN DESCLOUX,et al.  On spectral approximation. Part 2. Error estimates for the Galerkin method , 1978 .

[8]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

[9]  Daniele Boffi,et al.  Analysis of Finite Element Approximation of Evolution Problems in Mixed Form , 2004, SIAM J. Numer. Anal..

[10]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[11]  Annalisa Buffa,et al.  Remarks on the Discretization of Some Noncoercive Operator with Applications to Heterogeneous Maxwell Equations , 2005, SIAM J. Numer. Anal..

[12]  F. Kikuchi,et al.  Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism , 1987 .

[13]  D. Schötzau,et al.  Stabilized interior penalty methods for the time-harmonic Maxwell equations , 2002 .

[14]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[15]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[16]  Mark Embree,et al.  The role of the penalty in the local discontinuous Galerkin method for Maxwell’s eigenvalue problem , 2006 .

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

[18]  A. Buffa,et al.  Discontinuous Galerkin approximation of the Laplace eigenproblem , 2006 .

[19]  J. Hesthaven,et al.  High-Order Accurate Methods for Time-domain Electromagnetics , 2004 .

[20]  Clint Dawson,et al.  Some Extensions Of The Local Discontinuous Galerkin Method For Convection-Diffusion Equations In Mul , 1999 .

[21]  A. Bermúdez,et al.  Finite element vibration analysis of fluid-solid systems without spurious modes , 1995 .

[22]  Endre Süli,et al.  hp-Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. , 2002 .

[23]  P. Houston,et al.  Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes , 2007 .

[24]  彰 五十嵐.  N. Dunford and J. T. Schwartz (with the assistance of W. G. Bade and R. G. Bartle): Linear Operators. : Part II. Spectral Theoty. Self Adjoint Operators in Hilbert Space. Interscience. 1963. X+1065+7頁, 16×23.5cm, 14,000円。 , 1964 .

[25]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[26]  Gianni Gilardi,et al.  Magnetostatic and Electrostatic Problems in Inhomogeneous Anisotropic Media with Irregular Boundary and Mixed Boundary Conditions , 1997 .

[27]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[28]  Chi-Wang Shu,et al.  Locally divergence-free discontinuous Galerkin methods for the Maxwell equations , 2004, Journal of Computational Physics.

[29]  Martin Costabel,et al.  Discrete Compactness for the hp Version of Rectangular Edge Finite Elements , 2006, SIAM J. Numer. Anal..

[30]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[31]  J. Rappaz,et al.  On spectral approximation. Part 1. The problem of convergence , 1978 .

[32]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[33]  Alberto Valli,et al.  A domain decomposition approach for heterogeneous time-harmonic Maxwell equations , 1997 .

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

[35]  Peter Monk,et al.  Discrete compactness and the approximation of Maxwell's equations in R3 , 2001, Math. Comput..

[36]  J. Hesthaven,et al.  High–order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[37]  Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptical Problems Part I. Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptic Problems , 1999 .

[38]  Ohannes A. Karakashian,et al.  Piecewise solenoidal vector fields and the Stokes problem , 1990 .

[39]  Ilaria Perugia,et al.  An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems , 2002, J. Sci. Comput..

[40]  Jean-Claude Nédélec,et al.  Éléments finis mixtes incompressibles pour l'équation de Stokes dans ℝ3 , 1982 .

[41]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[42]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[43]  Mary F. Wheeler,et al.  Compatible algorithms for coupled flow and transport , 2004 .

[44]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[45]  Daniele Boffi,et al.  Fortin operator and discrete compactness for edge elements , 2000, Numerische Mathematik.

[46]  Ilaria Perugia,et al.  Mixed Discontinuous Galerkin Approximation of the Maxwell Operator: Non-Stabilized Formulation , 2005, J. Sci. Comput..

[47]  D. Boffi,et al.  Computational Models of Electromagnetic Resonators: Analysis of Edge Element Approximation , 1999 .

[48]  M. Shashkov,et al.  CONVERGENCE OF MIMETIC FINITE DIFFERENCE METHOD FOR DIFFUSION PROBLEMS ON POLYHEDRAL MESHES WITH CURVED FACES , 2006 .

[49]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[50]  Salvatore Caorsi,et al.  Mathematical Modelling and Numerical Analysis Spurious-free Approximations of Electromagnetic Eigenproblems by Means of Nedelec-type Elements , 2022 .

[51]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

[52]  M. Costabel,et al.  Singularities of Maxwell interface problems , 1999 .

[53]  Salvatore Caorsi,et al.  On the Convergence of Galerkin Finite Element Approximations of Electromagnetic Eigenproblems , 2000, SIAM J. Numer. Anal..

[54]  Daniele Boffi,et al.  On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form , 2000, Math. Comput..

[55]  Ilaria Perugia,et al.  Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids , 2001, SIAM J. Numer. Anal..

[56]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .