Multilevel solution of the time‐harmonic Maxwell's equations based on edge elements

A widely used approach for the computation of time-harmonic electromagnetic fields is based on the well-known double-curl equation for either E or H, where edge elements are an appealing choice for finite element discretizations. Yet, the nullspace of the curl-operator comprises a considerable part of all spectral modes on the finite element grid. Thus standard multilevel solvers are rendered inefficient, as they essentially hinge on smoothing procedures like Gauss–Seidel relaxation, which cannot provide a satisfactory error reduction for modes with small or even negative eigenvalues. We propose to remedy this situation by an extended multilevel algorithm which relies on corrections in the space of discrete scalar potentials. After every standard V-cycle with respect to the canonical basis of edge elements, error components in the nullspace are removed by an additional projection step. Furthermore, a simple criterion for the coarsest mesh is derived to guarantee both stability and efficiency of the iterative multilevel solver. For the whole scheme we observe convergence rates independent of the refinement level of the mesh. The sequence of nested meshes required for our multilevel techniques is constructed by adaptive refinement. To this end we have devised an a posteriori error indicator based on stress recovery. Copyright © 1999 John Wiley & Sons, Ltd.

[1]  M. L. Barton,et al.  New vector finite elements for three‐dimensional magnetic field computation , 1987 .

[2]  Peter Monk,et al.  A finite element method for approximating the time-harmonic Maxwell equations , 1992 .

[3]  Theodoros D. Tsiboukis,et al.  Efficient mode analysis with edge elements and 3-D adaptive refinement , 1994 .

[4]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[5]  Claes Johnson,et al.  Adaptive finite element methods in computational mechanics , 1992 .

[6]  D. Griffel,et al.  Initial Boundary Value Problems in Mathematical Physics , 1986 .

[7]  B. M. Dillon,et al.  A comparison of formulations for the vector finite element analysis of waveguides , 1994 .

[8]  R. Skeel,et al.  Unified Multilevel Adaptive Finite Element Methods for Elliptic Problems , 1988 .

[9]  A. Wexler,et al.  Self-Adjoint Variational Formulation of Problems Having Non-Self-Adjoint Operators , 1978 .

[10]  Pierre Degond,et al.  On a finite-element method for solving the three-dimensional Maxwell equations , 1993 .

[11]  S. Ramo,et al.  Fields and Waves in Communication Electronics , 1966 .

[12]  A. Bossavit.  A rationale for 'edge-elements' in 3-D fields computations , 1988 .

[13]  H. V. D. Vorst,et al.  A Petrov-Galerkin type method for solving Axk=b, where A is symmetric complex , 1990 .

[14]  R. Hiptmair.  Multigrid Method for Maxwell's Equations , 1998 .

[15]  A. Bossavit.  Solving Maxwell equations in a closed cavity, and the question of 'spurious modes' , 1990 .

[16]  Electromagnetic boundary value problem in the presence of a partly lossy dielectric: considerations about the uniqueness of the solution , 1996 .

[17]  Arnulf Kost,et al.  Numerische Methoden in der Berechnung elektromagnetischer Felder , 1994 .

[18]  A. Bossavit.  Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism , 1988 .

[19]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[20]  C. Chen,et al.  The Variational Principle for Non-Self-Adjoint Electromagnetic Problems , 1980 .

[21]  H. Whitney.  Geometric Integration Theory , 1957 .

[22]  Harry Yserentant,et al.  Preconditioning indefinite discretization matrices , 1989 .

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

[24]  D. Baldomir,et al.  Differential forms and electromagnetism in 3-dimensional Euclidean space R3 , 1986 .

[25]  Eberhard Bänsch,et al.  Local mesh refinement in 2 and 3 dimensions , 1991, IMPACT Comput. Sci. Eng..

[26]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[27]  Roland W. Freund,et al.  Conjugate Gradient-Type Methods for Linear Systems with Complex Symmetric Coefficient Matrices , 1992, SIAM J. Sci. Comput..

[28]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

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

[30]  J. Pasciak,et al.  Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems , 1994 .

[31]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[32]  R. Roitzsch,et al.  KASKADE 3.0 - An Object Oriented Adaptive Finite Element Code , 1995 .

[33]  O. Zienkiewicz,et al.  Analysis of the Zienkiewicz–Zhu a‐posteriori error estimator in the finite element method , 1989 .

[34]  Andrew F. Peterson,et al.  Higher-order vector finite elements for tetrahedral cells , 1996 .

[35]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[36]  Peter Monk,et al.  Multigrid computation of vector potentials , 1995 .

[37]  K. Paulsen,et al.  Finite element computations of specific absorption rates in anatomically conforming full-body models for hyperthermia treatment analysis , 1993, IEEE Transactions on Biomedical Engineering.

[38]  Ralf Hiptmair,et al.  Canonical construction of finite elements , 1999, Math. Comput..

[39]  Wolfgang Hackbusch,et al.  Theorie und Numerik elliptischer Differentialgleichungen , 1986, Teubner Studienbücher.

[40]  Nathan Marcuvitz.  Waveguide Handbook , 1951 .

[41]  Peter P. Silvester,et al.  Finite elements for electrical engineers: Finite Elements for Electrical Engineers , 1996 .

[42]  Ralf Hiptmair,et al.  Overlapping Schwarz Methods for Vector Valued Elliptic Problems in Three Dimensions , 1997 .