Residual based a posteriori error estimators for eddy current computation
ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 1, pp. 159-182. 
@article{M2AN_2000__34_1_159_0,
     author = {Beck, Rudi and Hiptmair, Ralf and Hoppe, Ronald H. W. and Wohlmuth, Barbara},
     title = {Residual based a posteriori error estimators for eddy current computation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {159--182},
     publisher = {Dunod},
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     number = {1},
     year = {2000},
     mrnumber = {1735971},
     zbl = {0949.65113},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2000__34_1_159_0/}
}
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Beck, Rudi; Hiptmair, Ralf; Hoppe, Ronald H. W.; Wohlmuth, Barbara. Residual based a posteriori error estimators for eddy current computation. ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 1, pp. 159-182. http://www.numdam.org/item/M2AN_2000__34_1_159_0/

[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. A 137 (1990) 16-22.

[3] Analysis of three dimensional electromagnetic fileds using edge elements. J. Comp. Phys. 108 (1993) 236-245. | Zbl

[4] A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H (rot; Ω) and the construction of an extension operator. Manuscripta math. 89 (1996) 159-178. | MR | Zbl

[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

[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. | MR | Zbl

[7] D. Arnold, A. Mukherjee and L. Pouly, Locally adapted tetrahedral meshes using bisection. SIAM J. on Sci. Compt (submitted). | MR | Zbl

[8] I. Babuška and W. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736-754. | MR | Zbl

[9] I. Babuška and W. Rheinboldt, A posteriori error estimates for the finite element method. Internet. J. Numer. Methods Engrg. 12 (1978) 1597-1615. | Zbl

[10] R. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, User's Guide 6.0. SIAM, Philadelphia (1990). | MR | Zbl

[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. | MR

[12] R. Bank and A. Weiser, some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985) 283-301. | MR | Zbl

[13] E. Bänsch, Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3 (1991) 181-191. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[16] J. Bey, Tetrahedral grid refinement. Computing 55 (1995) 355-378. | MR | Zbl

[17] F. Bornemann, An adaptive multilevel approach to parabolic equations I. General theory and lD-implementation. IMPACT Comput. Sci, Engrg. 2 (1990) 279-317. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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, Electromagnétisme, en vue de la modélisation. Springer-Verlag, Paris (1993). | MR | Zbl

[24] A. Bossavit, Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements. in Academic Press Electromagnetism Series, no. 2 Academic Press, San Diego (1998). | MR | Zbl

[25] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431-2445. | MR | Zbl

[26] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comp. 66 (1997) 465-476. | MR | Zbl

[27] P. Ciarlet, The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, Vol. 4 North-Holland, Amsterdam (1978). | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[31] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Tech. Rep. 97-19, IRMAR, Rennes, France (1997). | Zbl

[32] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, Tech. Rep. 98-24, IRMAR, Rennes, France (1998).

[33] H. Dirks, Quasi-stationary fields for microelectronic applications. Electrical Engineering 79 (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 Numerica 4 (1995) 105-158. | MR | Zbl

[36] K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems. Math. Comp. 50 (1988) 361-383. | MR | Zbl

[37] V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, Berlin (1986). | MR | Zbl

[38] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Heidelberg, New York (1991). | MR | Zbl

[39] R. Hiptmair, Multigrid method for Maxwell's equations. Tech. Rep. 374, Institut für Mathematik, Universität Augsburg (1997). | Zbl

[40] R. Hiptmair, Canonical construction of finite elements. Math. Comp. 68 (1999) 1325-1346. | MR | Zbl

[41] R. Hoppe and B. Wohlmuth, Adaptive multilevel iterative techniques for nonconforming finite element discretizations. East- West J. Numer. Math. 3 (1995) 179-197. | MR | Zbl

[42] R. Hoppe and B. Wohlmuth, A comparison of a posteriori error estimators for mixed finite elements. Math. Comp. 68 (1999) 1347-1378. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[46] J. Maubach, Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Stat. Comp. 16 (1995) 210-227. | MR | Zbl

[47] P. Monk, A mixed method for approximating Maxwell's equations. SIAM J. Numer. Anal. 28 (1991) 1610-1634. | MR | Zbl

[48] P. Monk, Analysis of a finite element method for Maxwell's equations. SIAM J. Numer. Anal. 29 (1992) 714-729. | MR | Zbl

[49] J. Nédélec, Mixed finite elements in R3, Numer. Math. 35 (1980) 315-341. | EuDML | MR | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

[53] L. R. Scott and Z. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | MR | Zbl

[54] R. Verfürth, A posteriori error estimators for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62 (1994) 445-475. | MR | Zbl

[55] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, Stuttgart (1996). | Zbl

[56] H. Whitney, Geometric Integration Theory. Princeton Univ. Press, Princeton (1957). | MR | Zbl

[57] J. Zhu and O. Zienkiewicz, Adaptive techniques in the finite element method. Commun. Appl. Numer. Methods 4 (1988) 197-204. | Zbl