Skip to main content Accessibility help

Residual based a posteriori error estimators for eddy current computation

  • Rudi Beck (a1), Ralf Hiptmair (a2), Ronald H.W. Hoppe (a3) and Barbara Wohlmuth (a3)


We consider H(curl;Ω)-elliptic problems that have been discretized by means of Nédélec'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 refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.



Hide All
[1] Achchab, B., Agouzal, A., Baranger, J. and Maitre, J., Estimateur d'erreur a posteriori hiérarchique. Application aux éléments finis mixtes. IMPACT Comput. Sci. Engrg. 1 (1995) 3-35.
[2] Albanese, R. and Rubinacci, G., 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.
[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.
[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).
[6] Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864.
[7] D. Arnold, A. Mukherjee and L. Pouly, Locally adapted tetrahedral meshes using bisection. SIAM J. on Sci. Compt (submitted).
[8] Babuska, I. and Rheinboldt, W., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736-754.
[9] Babuska, I. and Rheinboldt, W., A posteriori error estimates for the finite element method. Internet. J. Numer. Methods Engrg. 12 (1978) 1597-1615.
[10] R. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, User's Guide 6.0. SIAM, Philadelphia (1990).
[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] Bank, R. and Weiser, A., some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985) 283-301.
[13] Bänsch, E., Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3 (1991) 181-191.
[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.
[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).
[16] Tetrahedral, J. Bey grid refinement. Computing 55 (1995) 355-378.
[17] Bornemann, F., An adaptive multilevel approach to parabolic equations I. General theory and 1D-implementation. IMPACT Comput. Sci. Engrg. 2 (1990) 279-317.
[18] Bornemann, F., 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.
[19] Bornemann, F., Erdmann, B. and Kornhuber, R., A posteriori error estimates for elliptic problems in two and three spaces dimensions. SIAM J. Numer. Anal. 33 (1996) 1188-1204.
[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.
[21] Bossavit, A., A rationale for edge elements in 3D field computations. IEEE Trans. Mag. 24 (1988) 74-79.
[22] Bossavit, A., 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 Academic Press Electromagnetism Series, no. 2 Academic Press, San Diego (1998).
[25] Braess, D. and Verfürth, R., A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431-2445.
[26] Carstensen, C., A posteriori error estimate for the mixed finite element method. Math. Comp. 66 (1997) 465-476.
[27] P. Ciarlet, The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, Vol. 4 North-Holland, Amsterdam (1978).
[28] Clemens, M., Schuhmann, R., van Rienen, U. and Weiland, T., 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.
[31] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Tech. Rep. 97-19, IRMAR, Rennes, France (1997).
[32] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, Tech. Rep. 98-24, IRMAR, Rennes, France (1998).
[33] Dirks, H., 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] Erikson, K., Estep, D., Hansbo, P. and Johnson, C., Introduction to adaptive methods for differential equations. Acta Numerica 4 (1995) 105-158.
[36] Eriksson, K. and Johnson, C., An adaptive finite element method for linear elliptic problems. Math. Comp. 50 (1988) 361-383.
[37] V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, Berlin (1986).
[38] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Heidelberg, New York (1991).
[39] R. Hiptmair, Multigrid method for Maxwell's equations. Tech. Rep. 374, Institut für Mathematik, Universität Augsburg (1997).
[40] Hiptmair, R., Canonical construction of finite elements. Math. Comp. 68 (1999) 1325-1346.
[41] Hoppe, R. and Wohlmuth, B., Adaptive multilevel iterative techniques for nonconforming finite element discretizations. East-West J. Numer. Math. 3 (1995) 179-197.
[42] Hoppe, R. and Wohlmuth, B., A comparison of a posteriori error estimators for mixed finite elements. Math. Comp. 68 (1999) 1347-1378.
[43] Hoppe, R. and Wohlmuth, B., Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. Model. Math. Anal. Numér. 30 (1996) 237-263.
[44] Hoppe, R. and Wohlmuth, B., Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems. SIAM J. Numer. Anal. 34 (1997) 1658-1687.
[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.
[46] Maubach, J., Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Stat. Comp. 16 (1995) 210-227.
[47] Monk, P., A mixed method for approximating Maxwell's equations. SIAM J. Numer. Anal. 28 (1991) 1610-1634.
[48] Monk, P., Analysis of a finite element method for Maxwell's equations. SIAM J. Numer. Anal. 29 (1992) 714-729.
[49] Nédélec, J., Mixed finite elements in R 3, Numer. Math. 35 (1980) 315-341.
[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).
[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).
[53] Scott, L. R. and Zhang, Z., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493.
[54] Verfürth, R., A posteriori error estimators for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62 (1994) 445-475.
[55] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, Stuttgart (1996).
[56] H. Whitney, Geometric Integration Theory. Princeton Univ. Press, Princeton (1957).
[57] Zhu, J. and Zienkiewicz, O., Adaptive techniques in the finite element method. Commun. Appl. Numer. Methods 4 (1988) 197-204.


Related content

Powered by UNSILO

Residual based a posteriori error estimators for eddy current computation

  • Rudi Beck (a1), Ralf Hiptmair (a2), Ronald H.W. Hoppe (a3) and Barbara Wohlmuth (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.