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Multigrid in a weighted space arising from axisymmetric electromagnetics

Authors: Dylan M. Copeland, Jayadeep Gopalakrishnan and Minah Oh
Journal: Math. Comp. 79 (2010), 2033-2058
MSC (2010): Primary 65M55, 65N55, 65F10, 65N30, 78M10
Published electronically: May 24, 2010
MathSciNet review: 2684354
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Abstract: Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex, then the multigrid V-cycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a dual mixed method in weighted spaces. The uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments.

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  • 1. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Preconditioning in $ H({\rm div})$ and applications, Math. Comp., 66 (1997), pp. 957-984. MR 1401938 (97i:65177)
  • 2. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Multigrid in $ \mathbf{H}(\mathrm{div})$ and $ \mathbf{H}(\mathbf{curl})$, Numer. Math., 85 (2000), pp. 197-217. MR 1754719 (2001d:65161)
  • 3. F. ASSOUS, P. CIARLET, JR., AND S. LABRUNIE, Theoretical tools to solve the axisymmetric Maxwell equations, Math. Methods Appl. Sci., 25 (2002), pp. 49-78. MR 1874449 (2002j:78008)
  • 4. Z. BELHACHMI, C. BERNARDI, AND S. DEPARIS, Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem, Numer. Math., 105 (2006), pp. 217-247. MR 2262757 (2008c:65310)
  • 5. A. BERMúDEZ, C. REALES, R. RODRíGUEZ, AND P. SALGADO, Numerical analysis of a finite element method for the axisymmetric eddy current model of an induction furnace, IMA J. Numer. Anal., (To appear).
  • 6. C. BERNARDI, M. DAUGE, AND Y. MADAY, Spectral methods for axisymmetric domains, vol. 3 of Series in Applied Mathematics (Paris), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1999.
    Numerical algorithms and tests due to Mejdi Azaïez. MR 1693480 (2000h:65002)
  • 7. D. BOFFI, P. FERNANDES, L. GASTALDI, AND I. PERUGIA, Computational models of electromagnetic resonators: Analysis of edge element approximation, SIAM J. Numer. Anal., 36 (1999), pp. 1264-1290 (electronic). MR 1701792 (2000g:65112)
  • 8. S. BöRM AND R. HIPTMAIR, Multigrid computation of axisymmetric electromagnetic fields, Adv. Comput. Math., 16 (2002), pp. 331-356. MR 1894928 (2003d:78042)
  • 9. D. BRAESS AND W. HACKBUSCH, A new convergence proof for the multigrid method including the $ V$-cycle, SIAM J. Numer. Anal., 20 (1983), pp. 967-975. MR 714691 (85h:65233)
  • 10. J. H. BRAMBLE, Multigrid Methods, no. 294 in Pitman research notes in mathematics series, Longman Scientific & Technical, Harlow, UK, 1993. MR 1247694 (95b:65002)
  • 11. J. H. BRAMBLE AND J. XU, Some estimates for a weighted $ L^2$ projection, Math. Comp., 56 (1991), pp. 463-476. MR 1066830 (91k:65140)
  • 12. J. H. BRAMBLE AND X. ZHANG, The analysis of multigrid methods, in Handbook of Numerical Analysis, Vol. VII, North-Holland, Amsterdam, 2000, pp. 173-415. MR 1804746 (2001m:65183)
  • 13. F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, vol. 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 14. O. CHINELLATO, The Complex-Symmetric Jacobi-Davidson Algorithm and its Application to the Computation of some Resonance Frequencies of Anisotropic Lossy Axisymmetric Cavities, Dissertation ETH No. 16243, Swiss Federal Institute of Technology, Zürich, 2005.
  • 15. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam, 1978. MR 0520174 (58:25001)
  • 16. B. COCKBURN AND J. GOPALAKRISHNAN, Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comp., 74 (2005), pp. 1653-1677 (electronic). MR 2164091 (2006e:65215)
  • 17. D. M. COPELAND, J. GOPALAKRISHNAN, AND J. E. PASCIAK, A mixed method for axisymmetric div-curl systems, Math. Comp., 77 (2008), pp. 1941-1965. MR 2429870 (2009e:65171)
  • 18. D. M. COPELAND AND J. E. PASCIAK, A least-squares method for axisymmetric div-curl systems, Numer. Linear Algebra Appl., 13 (2006), pp. 733-752. MR 2266104 (2008c:78013)
  • 19. R. FALK AND J. OSBORN, Error estimates for mixed methods, RAIRO, 14 (1980), pp. 249-277. MR 592753 (82j:65076)
  • 20. V. GIRAULT AND P.-A. RAVIART, Finite element methods for Navier-Stokes equations, Theory and algorithms. vol. 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
  • 21. J. GOPALAKRISHNAN AND J. E. PASCIAK, The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations, Math. Comp., 75 (2006), pp. 1697-1719 (electronic). MR 2240631 (2007g:65116)
  • 22. R. HIPTMAIR, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36 (1999), pp. 204-225 (electronic). MR 1654571 (99j:65229)
  • 23. A. KUFNER, O. JOHN, AND S. FUČíK, Function Spaces, Monographs and textbooks on mechanics of solids and fluids, Noordhoff International Publishing, Leyden, The Netherlands, 1977. MR 0482102 (58:2189)
  • 24. P. LACOSTE AND J. GAY, A new family of finite elements for Maxwell-Fourier's equations, in Mathematical and numerical aspects of wave propagation phenomena (Strasbourg, 1991), SIAM, Philadelphia, PA, 1991, pp. 746-749. MR 1106041 (92c:65135)
  • 25. J.-C. NéDéLEC, Mixed Finite Elements in $ {\mathbb{R}}^3$, Numer. Math., 35 (1980), pp. 315-341. MR 592160 (81k:65125)
  • 26. J. NITSCHE, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math., 11 (1968), pp. 346-348. MR 0233502 (38:1823)
  • 27. P.-A. RAVIART AND J. M. THOMAS, Primal hybrid finite element methods for $ 2$nd order elliptic equations, Math. Comp., 31 (1977), pp. 391-413. MR 0431752 (55:4747)

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Additional Information

Dylan M. Copeland
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105

Minah Oh
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105

Keywords: Multigrid, axisymmetric, weighted Sobolev spaces, Maxwell equations, V-cycle, duality, superconvergence, mixed method, finite element
Received by editor(s): November 17, 2008
Received by editor(s) in revised form: June 11, 2009
Published electronically: May 24, 2010
Additional Notes: This work was supported in part by the National Science Foundation under grants DMS-0713833 and SCREMS-0619080.
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society