Multigrid in a weighted space arising from axisymmetric electromagnetics
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- by Dylan M. Copeland, Jayadeep Gopalakrishnan and Minah Oh PDF
- Math. Comp. 79 (2010), 2033-2058 Request permission
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.References
- Douglas N. Arnold, Richard S. Falk, and R. Winther, Preconditioning in $H(\textrm {div})$ and applications, Math. Comp. 66 (1997), no. 219, 957–984. MR 1401938, DOI 10.1090/S0025-5718-97-00826-0
- 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
- F. Assous, P. Ciarlet Jr., and S. Labrunie, Theoretical tools to solve the axisymmetric Maxwell equations, Math. Methods Appl. Sci. 25 (2002), no. 1, 49–78. MR 1874449, DOI 10.1002/mma.279
- Zakaria Belhachmi, Christine Bernardi, and Simone Deparis, Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem, Numer. Math. 105 (2006), no. 2, 217–247. MR 2262757, DOI 10.1007/s00211-006-0039-9
- 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).
- Christine Bernardi, Monique Dauge, and Yvon Maday, Spectral methods for axisymmetric domains, Series in Applied Mathematics (Paris), vol. 3, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris; North-Holland, Amsterdam, 1999. Numerical algorithms and tests due to Mejdi Azaïez. MR 1693480
- 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), no. 4, 1264–1290. MR 1701792, DOI 10.1137/S003614299731853X
- S. Börm and R. Hiptmair, Multigrid computation of axisymmetric electromagnetic fields, Adv. Comput. Math. 16 (2002), no. 4, 331–356. MR 1894928, DOI 10.1023/A:1014533409747
- 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 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
- James H. Bramble and Xuejun Zhang, The analysis of multigrid methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 173–415. MR 1804746
- 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
- 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.
- 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
- Bernardo Cockburn and Jayadeep Gopalakrishnan, Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comp. 74 (2005), no. 252, 1653–1677. MR 2164091, DOI 10.1090/S0025-5718-05-01741-2
- Dylan M. Copeland, Jayadeep Gopalakrishnan, and Joseph E. Pasciak, A mixed method for axisymmetric div-curl systems, Math. Comp. 77 (2008), no. 264, 1941–1965. MR 2429870, DOI 10.1090/S0025-5718-08-02102-9
- Dylan M. Copeland and Joseph E. Pasciak, A least-squares method for axisymmetric div-curl systems, Numer. Linear Algebra Appl. 13 (2006), no. 9, 733–752. MR 2266104, DOI 10.1002/nla.495
- R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753
- 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
- Jayadeep Gopalakrishnan and Joseph E. Pasciak, The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations, Math. Comp. 75 (2006), no. 256, 1697–1719. MR 2240631, DOI 10.1090/S0025-5718-06-01884-9
- R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
- Alois Kufner, Oldřich John, and Svatopluk Fučík, Function spaces, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. MR 0482102
- Patrick Lacoste and Jean Gay, A new family of finite elements for Maxwell-Fourier’s equations, Mathematical and numerical aspects of wave propagation phenomena (Strasbourg, 1991) SIAM, Philadelphia, PA, 1991, pp. 746–749. MR 1106041
- 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. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11 (1968), 346–348 (German). MR 233502, DOI 10.1007/BF02166687
- P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391–413. MR 431752, DOI 10.1090/S0025-5718-1977-0431752-8
Additional Information
- Dylan M. Copeland
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: copeland@math.tamu.edu
- Jayadeep Gopalakrishnan
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
- MR Author ID: 661361
- Email: jayg@ufl.edu
- Minah Oh
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
- Email: oh@ufl.edu
- 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.
- © Copyright 2010 American Mathematical Society
- Journal: Math. Comp. 79 (2010), 2033-2058
- MSC (2010): Primary 65M55, 65N55, 65F10, 65N30, 78M10
- DOI: https://doi.org/10.1090/S0025-5718-2010-02384-1
- MathSciNet review: 2684354