Analysis of multilevel methods for eddy current problems
HTML articles powered by AMS MathViewer
- by R. Hiptmair PDF
- Math. Comp. 72 (2003), 1281-1303 Request permission
Abstract:
In papers by Arnold, Falk, and Winther, and by Hiptmair, novel multigrid methods for discrete $\mathbfit {H}(\mathbf {curl};\Omega )$-elliptic boundary value problems have been proposed. Such problems frequently occur in computational electromagnetism, particularly in the context of eddy current simulation.
This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods. It provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains. In particular, asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems.
The principal idea is to use approximate Helmholtz-decompositions of the function space $\mathbfit {H}(\mathbf {curl};\Omega )$ into an $H^1(\Omega )$-regular subspace and gradients. The main results of standard multilevel theory for $H^1(\Omega )$-elliptic problems can then be applied to both subspaces. This yields preliminary decompositions still outside the edge element spaces. Judicious alterations can cure this.
References
- R. Albanese and G. Rubinacci, Analysis of three dimensional electromagnetic fileds using edge elements, J. Comp. Phys., 108 (1993), pp. 236–245.
- H. Ammari, A. Buffa, and J.-C. Nédélec, A justification of eddy currents model for the Maxwell equations, SIAM J. Appl. Math. 60 (2000), no. 5, 1805–1823. MR 1761772, DOI 10.1137/S0036139998348979
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
- Eberhard Bänsch, Local mesh refinement in $2$ and $3$ dimensions, Impact Comput. Sci. Engrg. 3 (1991), no. 3, 181–191. MR 1141298, DOI 10.1016/0899-8248(91)90006-G
- Rudolf Beck, Peter Deuflhard, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell’s equations, Surveys Math. Indust. 8 (1999), no. 3-4, 271–312. MR 1737416
- Rudolf Beck and Ralf Hiptmair, Multilevel solution of the time-harmonic Maxwell’s equations based on edge elements, Internat. J. Numer. Methods Engrg. 45 (1999), no. 7, 901–920. MR 1696118, DOI 10.1002/(SICI)1097-0207(19990710)45:7<901::AID-NME611>3.0.CO;2-4
- J. Bey, Tetrahedral grid refinement, Computing 55 (1995), no. 4, 355–378 (English, with English and German summaries). MR 1370107, DOI 10.1007/BF02238487
- M. Sh. Birman and M. Z. Solomyak, $L_2$-theory of the Maxwell operator in arbitrary domains, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 61–76, 247 (Russian). MR 933995
- Anne-Sophie Bonnet-Ben Dhia, Christophe Hazard, and Stephanie Lohrengel, A singular field method for the solution of Maxwell’s equations in polyhedral domains, SIAM J. Appl. Math. 59 (1999), no. 6, 2028–2044. MR 1709795, DOI 10.1137/S0036139997323383
- F. Bornemann, A sharpened condition number estimate for the BPX-preconditioner of elliptic finite element problems on highly non-uniform triangulations, Tech. Rep. SC 91-9, ZIB, Berlin, Germany, September 1991.
- A. Bossavit, A rationale for edge elements in ${3}{D}$ field computations, IEEE Trans. Mag., 24 (1988), pp. 74–79.
- —, Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism, IEEE Proc. A, 135 (1988), pp. 493–500.
- —, Solving Maxwell’s equations in a closed cavity and the question of spurious modes, IEEE Trans. Mag., 26 (1990), pp. 702–705.
- —, A new viewpoint on mixed elements, Meccanica, 27 (1992), pp. 3–11.
- D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the $V$-cycle, SIAM J. Numer. Anal. 20 (1983), no. 5, 967–975. MR 714691, DOI 10.1137/0720066
- James H. Bramble, Multigrid methods, Pitman Research Notes in Mathematics Series, vol. 294, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1247694
- James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 57 (1991), no. 195, 23–45. MR 1079008, DOI 10.1090/S0025-5718-1991-1079008-4
- James H. Bramble and Jinchao Xu, Some estimates for a weighted $L^2$ projection, Math. Comp. 56 (1991), no. 194, 463–476. MR 1066830, DOI 10.1090/S0025-5718-1991-1066830-3
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- P. Ciarlet Jr. and Jun Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math. 82 (1999), no. 2, 193–219 (English, with English and French summaries). MR 1685459, DOI 10.1007/s002110050417
- M. Clemens and T. Weiland, Transient eddy current calculation with the FI-method, IEEE Trans. Magnetics, 35 (1999), pp. 1163–1166.
- Gary Cohen and Peter Monk, Mur-Nédélec finite element schemes for Maxwell’s equations, Comput. Methods Appl. Mech. Engrg. 169 (1999), no. 3-4, 197–217. MR 1675684, DOI 10.1016/S0045-7825(98)00154-6
- M. Costabel and M. Dauge, Singularities of Maxwell’s equations on polyhedral domains, in Analysis, Numerics and Applications of Differential and Integral Equations, M. Bach, ed., vol. 379 of Longman Pitman Res. Notes Math. Ser., Addison Wesley, Harlow, 1998, pp. 69–76.
- Martin Costabel, Monique Dauge, and Serge Nicaise, Singularities of Maxwell interface problems, M2AN Math. Model. Numer. Anal. 33 (1999), no. 3, 627–649. MR 1713241, DOI 10.1051/m2an:1999155
- 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), pp. 2980–2983.
- V. Girault, Curl-conforming finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $\textbf {R}^3$, The Navier-Stokes equations (Oberwolfach, 1988) Lecture Notes in Math., vol. 1431, Springer, Berlin, 1990, pp. 201–218. MR 1072191, DOI 10.1007/BFb0086071
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
- J. Gopalakrishnan and J. Pasciak, Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell Equations, Tech. Rep., Department of Mathematics, Texas A&M University, June 2000. Submitted to Math. Comp.
- V. Gradinaru and R. Hiptmair, Whitney elements on pyramids, Electron. Trans. Numer. Anal. 8 (1999), 154–168. MR 1744532
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- R. Hiptmair, Multigrid method for $\mathbf H(\textrm {div})$ in three dimensions, Electron. Trans. Numer. Anal. 6 (1997), no. Dec., 133–152. Special issue on multilevel methods (Copper Mountain, CO, 1997). MR 1615161
- R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325–1346. MR 1665954, DOI 10.1090/S0025-5718-99-01166-7
- R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
- Ralf Hiptmair and Ronald H. W. Hoppe, Multilevel methods for mixed finite elements in three dimensions, Numer. Math. 82 (1999), no. 2, 253–279. MR 1685461, DOI 10.1007/s002110050419
- R. Hiptmair and K. Neymeyr, Multilevel method for mixed eigenproblems, Report 159, SFB 382, Universität Tübingen, Tübingen, Germany, 2001. To appear in SIAM J. Sci. Comp.
- R. Hiptmair, T. Schiekofer, and B. Wohlmuth, Multilevel preconditioned augmented Lagrangian techniques for 2nd order mixed problems, Computing 57 (1996), no. 1, 25–48 (English, with English and German summaries). MR 1398269, DOI 10.1007/BF02238356
- Ralf Hiptmair and Andrea Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, Parallel solution of partial differential equations (Minneapolis, MN, 1997) IMA Vol. Math. Appl., vol. 120, Springer, New York, 2000, pp. 181–208. MR 1838270, DOI 10.1007/978-1-4612-1176-1_{8}
- M. Hochbruck and C. Lubich, Error analysis of Krylov methods in a nutshell, SIAM J. Sci. Comput., 19 (1998), pp. 695–701.
- P. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions, J. Appl. Phys., 61 (1987), pp. 3916–3918.
- P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR 972510
- Peter B. Monk, A mixed method for approximating Maxwell’s equations, SIAM J. Numer. Anal. 28 (1991), no. 6, 1610–1634. MR 1135758, DOI 10.1137/0728081
- Peter Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal. 29 (1992), no. 3, 714–729. MR 1163353, DOI 10.1137/0729045
- P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in ${\Bbb R}^3$, Math. Comp. 70 (2001), no. 234, 507–523. MR 1709155, DOI 10.1090/S0025-5718-00-01229-1
- J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
- J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
- P. Oswald, On function spaces related to finite element approximation theory, Z. Anal. Anwendungen 9 (1990), no. 1, 43–64 (English, with German and Russian summaries). MR 1063242, DOI 10.4171/ZAA/380
- —, Two remarks on multilevel preconditioners, Tech. Rep. 91/1, Methematisches Institut, FSU Jena, 1991.
- —, On discrete norm estimates related to multilevel preconditioners in the finite element method, in Constructive Theory of Functions, Proc. Int. Conf. Varna 1991, K. Ivanov, P. Petrushev, and B. Sendov, eds., Bulg. Acad. Sci., 1992, pp. 203–214.
- Peter Oswald, On the robustness of the BPX-preconditioner with respect to jumps in the coefficients, Math. Comp. 68 (1999), no. 226, 633–650. MR 1620239, DOI 10.1090/S0025-5718-99-01041-8
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
- L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
- Barry F. Smith, Petter E. Bjørstad, and William D. Gropp, Domain decomposition, Cambridge University Press, Cambridge, 1996. Parallel multilevel methods for elliptic partial differential equations. MR 1410757
- Andrea Toselli, Overlapping Schwarz methods for Maxwell’s equations in three dimensions, Numer. Math. 86 (2000), no. 4, 733–752. MR 1794350, DOI 10.1007/PL00005417
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
- Jinchao Xu, An introduction to multilevel methods, Wavelets, multilevel methods and elliptic PDEs (Leicester, 1996) Numer. Math. Sci. Comput., Oxford Univ. Press, New York, 1997, pp. 213–302. MR 1600688
- J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space, Report AM 223, Department of Mathematics, PennState University, College Park, PA, June 2000. Submitted.
- Harry Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), no. 4, 379–412. MR 853662, DOI 10.1007/BF01389538
- Harry Yserentant, Old and new convergence proofs for multigrid methods, Acta numerica, 1993, Acta Numer., Cambridge Univ. Press, Cambridge, 1993, pp. 285–326. MR 1224685, DOI 10.1017/S0962492900002385
- Xuejun Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), no. 4, 521–539. MR 1189535, DOI 10.1007/BF01385873
Additional Information
- R. Hiptmair
- Affiliation: Sonderforschungsbereich 382, Universität Tübingen, 72076 Tübingen, Germany
- Address at time of publication: Seminar für Angewandte Mathematik, ETH Zürich, CH-8092 Zürich, Switzerland
- Email: hiptmair@na.uni-tuebingen.de, ralf@hiptmair.de
- Received by editor(s): November 6, 2000
- Received by editor(s) in revised form: August 13, 2001, and September 19, 2001
- Published electronically: October 18, 2002
- Additional Notes: This work was supported by DFG as part of SFB 382
- © Copyright 2002 American Mathematical Society
- Journal: Math. Comp. 72 (2003), 1281-1303
- MSC (2000): Primary 65N55, 65N30, 35Q60
- DOI: https://doi.org/10.1090/S0025-5718-02-01468-0
- MathSciNet review: 1972736