Computation of Maxwell eigenvalues on curvilinear domains using hp -version Nédélec elements

In this paper we present and numerically verify theoretical bounds on the growth of the conditioning number for an H(curl)-conforming basis suitable for variable order approximation on curvilinear quadrilateral or hexahedral meshes. These bounds are given explicitly in terms of the maximum polynomial degree of approximation employed throughout the mesh. Additionally, numerical examples demonstrating the use of the basis in the context of electromagnetic eigenvalue problems on curved domains with reentrant comers are given. These examples also serve as a preliminary investigation of hp-refinement in computing eigenvalues corresponding to both singular and non-singular eigenfunctions on curvilinear domains.

[1]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[2]  I. Babuska,et al.  Finite Element Analysis , 2021 .

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

[4]  Ning Hu,et al.  Bounds for eigenvalues and condition numbers in the p-version of the finite element method , 1998, Math. Comput..

[5]  J. P. Webb.  Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements , 1999 .

[6]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[7]  Jian-Ming Jin,et al.  The Finite Element Method in Electromagnetics , 1993 .

[8]  Willard Miller,et al.  The IMA volumes in mathematics and its applications , 1986 .

[9]  Mark Ainsworth,et al.  hp-Approximation Theory for BDFM and RT Finite Elements on Quadrilaterals , 2002, SIAM J. Numer. Anal..

[10]  Larry L. Schumaker,et al.  Error bounds for minimal energy bivariate polynomial splines , 2002, Numerische Mathematik.

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

[12]  Mark Ainsworth,et al.  An adaptive refinement strategy for hp -finite element computations , 1998 .

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

[14]  Leszek Demkowicz,et al.  An hp‐adaptive finite element method for electromagnetics—part II: A 3D implementation , 2002 .

[15]  Mark Ainsworth,et al.  Conditioning of Hierarchic p-Version Nédélec Elements on Meshes of Curvilinear Quadrilaterals and Hexahedra , 2003, SIAM J. Numer. Anal..

[16]  L. Demkowicz,et al.  An hp-adaptive finite element method for electromagnetics: Part 1: Data structure and constrained approximation , 2000 .

[17]  Ralf Hiptmair,et al.  HIGHER ORDER WHITNEY FORMS , 2001 .

[18]  L. Demkowicz,et al.  Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements , 1998 .

[19]  Alain Bossavit,et al.  Edge-elements for scattering problems , 1989 .

[20]  Z. Cendes,et al.  Spurious modes in finite-element methods , 1995 .

[21]  R. Hiptmair.  HIGHER ORDER WHITNEY FORMS - Abstract , 2001 .

[22]  Bill Senior,et al.  hp-Finite Element Procedures on Non-Uniform Geometric Meshes: Adaptivity and Constrained Approximation , 1999 .

[23]  Mark Ainsworth,et al.  Hierarchic hp-edge element families for Maxwell's equations on hybrid quadrilateral/triangular meshes , 2001 .

[24]  Joseph E. Flaherty,et al.  Grid generation and adaptive algorithms , 1999 .

[25]  Peter Monk,et al.  An analysis of Ne´de´lec's method for the spatial discretization of Maxwell's equations , 1993 .

[26]  G. Mur.  Edge elements, their advantages and their disadvantages , 1994 .

[27]  J.-C. Verite,et al.  A mixed fem-biem method to solve 3-D eddy-current problems , 1982 .

[28]  L. Demkowicz,et al.  Maxwell eigenvalues and discrete compactness in two dimensions , 2000 .