×

zbMATH — the first resource for mathematics

Residual based a posteriori error estimators for eddy current computation. (English) Zbl 0949.65113
From the authors’ abstract: We consider \(H (\text{curl};\Omega)\)-elliptic problems that have been discretized by means of Nedelec’s edge elements on tetrahedral meshes. Such problems occur in the numerical computation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive mesh refinement. The fundamental tool in the analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78A55 Technical applications of optics and electromagnetic theory
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
Software:
RODAS; PLTMG
PDF BibTeX XML Cite
Full Text: DOI Link EuDML
References:
[1] B. Achchab, A. Agouzal, J. Baranger and J. Maitre, Estimateur d’erreur a posteriori hiérarchique. Application aux éléments finis mixtes. IMPACT Comput. Sci. Engrg.1 (1995) 3-35.
[2] R. Albanese and G. Rubinacci, Formulation of the eddy-current problem. IEE Proc. A137 (1990) 16-22. · Zbl 0722.65071
[3] Analysis of three dimensional electromagnetic fileds using edge elements. J. Comp. Phys.108 (1993) 236-245. · Zbl 0791.65096
[4] A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H(rot;\Omega ) and the construction of an extension operator. Manuscripta math.89 (1996) 159-178. · Zbl 0856.46019
[5] H. Ammari, A. Buffa and J.-C. Nédélec, A justification of eddy currents model for the Maxwell equations. Tech. Rep., IAN, University of Pavia, Pavia, Italy (1998). · Zbl 0978.35070
[6] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci.21 (1998) 823-864. Zbl0914.35094 · Zbl 0914.35094
[7] D. Arnold, A. Mukherjee and L. Pouly, Locally adapted tetrahedral meshes using bisection. SIAM J. on Sci. Compt (submitted). · Zbl 0973.65116
[8] I. Babuska and W. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal.15 (1978) 736-754. · Zbl 0398.65069
[9] I. Babuska and W. Rheinboldt, A posteriori error estimates for the finite element method. Internet. J. Numer. Methods Engrg.12 (1978) 1597-1615. · Zbl 0396.65068
[10] R. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, User’s Guide 6.0. SIAM, Philadelphia (1990). · Zbl 0717.68001
[11] R. Bank, A. Sherman and A. Weiser, Refinement algorithm and data structures for regular local mesh refinement., in Scientific Computing, R. Stepleman et al., Ed., Vol. 44, IMACS North-Holland, Amsterdam (1983) 3-17.
[12] R. Bank and A. Weiser, some a posteriori error estimators for elliptic partial differential equations. Math. Comp.44 (1985) 283-301. Zbl0569.65079 · Zbl 0569.65079
[13] E. Bänsch, Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg.3 (1991) 181-191. · Zbl 0744.65074
[14] R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. Surveys for Mathematics in Industry. · Zbl 0939.65136
[15] R. Beck and R. Hiptmair, Multilevel solution of the time-harmonic Maxwell equations based on edge elements. Tech. Rep. SC-96-51, ZIB Berlin (1996). in Internat. J. Numer. Methods Engrg. (To appear). · Zbl 0930.35167
[16] J. Bey, Tetrahedral grid refinement. Computing55 (1995) 355-378. · Zbl 0839.65135
[17] F. Bornemann, An adaptive multilevel approach to parabolic equations I. General theory and 1D-implementation. IMPACT Comput. Sci. Engrg.2 (1990) 279-317. · Zbl 0722.65055
[18] F. Bornemann, An adaptive multilevel approach to parabolic equations II. Variable-order time discretization based on a multiplicative error correction. IMPACT Comput. Sci. Engrg.3 (1991) 93-122. · Zbl 0735.65066
[19] F. Bornemann, B. Erdmann and R. Kornhuber, A posteriori error estimates for elliptic problems in two and three spaces dimensions. SIAM J. Numer. Anal.33 (1996) 1188-1204. · Zbl 0863.65069
[20] A. Bossavit, Mixed finite elements and the complex of Whitney forms, in The Mathematics of Finite Elements and Applications VI J. Whiteman Ed., Academic Press, London (1988) 137-144. Zbl0692.65053 · Zbl 0692.65053
[21] A. Bossavit, A rationale for edge elements in 3D field computations. IEEE Trans. Mag.24 (1988) 74-79.
[22] A. Bossavit, Solving Maxwell’s equations in a closed cavity and the question of spurious modes. IEEE Trans. Mag.26 (1990) 702-705.
[23] A. Bossavit, Électromagnétisme, en vue de la modélisation. Springer-Verlag, Paris (1993).
[24] A. Bossavit, Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements.in Academic Press Electromagnetism Series, no. 2 Academic Press, San Diego (1998). · Zbl 0945.78001
[25] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal.33 (1996) 2431-2445. · Zbl 0866.65071
[26] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comp.66 (1997) 465-476. Zbl0864.65068 · Zbl 0864.65068
[27] P. Ciarlet, The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, Vol. 4 North-Holland, Amsterdam (1978). · Zbl 0383.65058
[28] M. Clemens, R. Schuhmann, U. van Rienen and T. Weiland, Modern Krylov subspace methods in electromagnetic field computation using the finite integration theory. ACES J. Appl. Math.11 (1996) 70-84.
[29] M. Clemens and T. Weiland, Transient eddy current calculation with the FI-method. in Proc. CEFC ’98, IEEE (1998); IEEE Trans. Mag. submitted
[30] P. Clément, Approximation by finite element functions using local regularization. Revue Franc. Automat. Inform. Rech. Operat.9, R-2 (1975) 77-84. Zbl0368.65008 · Zbl 0368.65008
[31] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Tech. Rep. 97-19, IRMAR, Rennes, France (1997). · Zbl 0968.35113
[32] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, Tech. Rep. 98-24, IRMAR, Rennes, France (1998). · Zbl 0937.78003
[33] H. Dirks, Quasi-stationary fields for microelectronic applications. Electrical Engineering79 (1996) 145-155.
[34] P. Dular, J.-Y. Hody, A. Nicolet, A. Genon and W. Legros, Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms. IEEE Trans Magnetics MAG-30 (1994) 2980-2983.
[35] K. Erikson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numerica4 (1995) 105-158. Zbl0829.65122 · Zbl 0829.65122
[36] K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems. Math. Comp.50 (1988) 361-383. Zbl0644.65080 · Zbl 0644.65080
[37] V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, Berlin (1986). · Zbl 0585.65077
[38] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Heidelberg, New York (1991). · Zbl 0729.65051
[39] R. Hiptmair, Multigrid method for Maxwell’s equations. Tech. Rep. 374, Institut für Mathematik, Universität Augsburg (1997). · Zbl 0897.65046
[40] R. Hiptmair, Canonical construction of finite elements. Math. Comp.68 (1999) 1325-1346. Zbl0938.65132 · Zbl 0938.65132
[41] R. Hoppe and B. Wohlmuth, Adaptive multilevel iterative techniques for nonconforming finite element discretizations. East-West J. Numer. Math.3 (1995) 179-197. Zbl0836.65127 · Zbl 0836.65127
[42] R. Hoppe and B. Wohlmuth, A comparison of a posteriori error estimators for mixed finite elements. Math. Comp.68 (1999) 1347-1378. · Zbl 0929.65094
[43] R. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. Model. Math. Anal. Numér.30 (1996) 237-263. · Zbl 0843.65075
[44] R. Hoppe and B. Wohlmuth, Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems. SIAM J. Numer. Anal.34 (1997) 1658-1687. Zbl0889.65124 · Zbl 0889.65124
[45] R. Hoppe and B. Wohlmuth, Hierarchical basis error estimators for Raviart-Thomas discretizations of arbitrary order, in Finite Element Methods: Superconvergence, Post-processing and A Posteriori Estimates, M. Krizck, P. Neittaanmäki and R. Stenberg Eds., Marcel Dekker, New York (1997) 155-167. Zbl0902.65051 · Zbl 0902.65051
[46] J. Maubach, Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Stat. Comp.16 (1995) 210-227. Zbl0816.65090 · Zbl 0816.65090
[47] P. Monk, A mixed method for approximating Maxwell’s equations. SIAM J. Numer. Anal.28 (1991) 1610-1634. · Zbl 0742.65091
[48] P. Monk, Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal.29 (1992) 714-729. · Zbl 0761.65097
[49] J. Nédélec, Mixed finite elements in R3, Numer. Math.35 (1980) 315-341. · Zbl 0419.65069
[50] E. Ong, Hierarchical basis preconditioners for second order elliptic problems in three dimensions. Ph.D. thesis, Dept. of Math., UCLA, Los Angeles, CA, USA (1990).
[51] P. Oswald, Multilevel finite element approximation. Teubner Skripten zur Numerik, B.G. Teubner, Stuttgart (1994). · Zbl 0830.65107
[52] J. P. Ciarlet and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell equations. Tech. Rep. TR MATH-96-31 (105), Department of Mathematics, The Chinese University of Hong Kong (1996). Num. Math. (to appear). · Zbl 1126.78310
[53] L. R. Scott and Z. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483-493. Zbl0696.65007 · Zbl 0696.65007
[54] R. Verfürth, A posteriori error estimators for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp.62 (1994) 445-475. · Zbl 0799.65112
[55] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, Stuttgart (1996). · Zbl 0853.65108
[56] H. Whitney, Geometric Integration Theory. Princeton Univ. Press, Princeton (1957). · Zbl 0083.28204
[57] J. Zhu and O. Zienkiewicz, Adaptive techniques in the finite element method. Commun. Appl. Numer. Methods4 (1988) 197-204. Zbl0633.73070 · Zbl 0633.73070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.