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Regularity and multigrid analysis for Laplace-type axisymmetric equations


Author: Hengguang Li
Journal: Math. Comp. 84 (2015), 1113-1144
MSC (2010): Primary 65N30, 65N55; Secondary 35J05
DOI: https://doi.org/10.1090/S0025-5718-2014-02879-2
Published electronically: September 8, 2014
MathSciNet review: 3315502
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Abstract | References | Similar Articles | Additional Information

Abstract: Consider axisymmetric equations associated with Laplace-type operators. We establish full regularity estimates in high-order Kondrat$ '$ve-type spaces for possible singular solutions due to the non-smoothness of the domain and to the singular coefficients in the operator. Then, we show suitable graded meshes can be used in high-order finite element methods to achieve the optimal convergence rate, even when the solution is singular. Using these results, we further propose multigrid V-cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes. We prove the multigrid algorithm is a contraction, with the contraction number independent of the mesh level. Numerical tests are provided to verify the theorem.


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  • [1] 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 (2002j:78008), https://doi.org/10.1002/mma.279
  • [2] Constantin Bacuta, Victor Nistor, and Ludmil T. Zikatanov, Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates, Numer. Funct. Anal. Optim. 26 (2005), no. 6, 613-639. MR 2187917 (2006i:35036), https://doi.org/10.1080/01630560500377295
  • [3] Constantin Băcuţă, Victor Nistor, and Ludmil T. Zikatanov, Improving the rate of convergence of `high order finite elements' on polygons and domains with cusps, Numer. Math. 100 (2005), no. 2, 165-184. MR 2135780 (2006d:65130), https://doi.org/10.1007/s00211-005-0588-3
  • [4] Constantin Bacuta, Victor Nistor, and Ludmil T. Zikatanov, Improving the rate of convergence of high-order finite elements on polyhedra. II. Mesh refinements and interpolation, Numer. Funct. Anal. Optim. 28 (2007), no. 7-8, 775-824. MR 2347683 (2008g:65153), https://doi.org/10.1080/01630560701493263
  • [5] 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 (2008c:65310), https://doi.org/10.1007/s00211-006-0039-9
  • [6] 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 (2000h:65002)
  • [7] 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 (85h:65233), https://doi.org/10.1137/0720066
  • [8] James H. Bramble and Joseph E. Pasciak, New estimates for multilevel algorithms including the $ V$-cycle, Math. Comp. 60 (1993), no. 202, 447-471. MR 1176705 (94a:65064), https://doi.org/10.2307/2153097
  • [9] 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 (91m:65158), https://doi.org/10.2307/2938661
  • [10] Susanne C. Brenner, Convergence of the multigrid $ V$-cycle algorithm for second-order boundary value problems without full elliptic regularity, Math. Comp. 71 (2002), no. 238, 507-525. MR 1885612 (2003b:65132), https://doi.org/10.1090/S0025-5718-01-01361-8
  • [11] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
  • [12] 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 (2009e:65171), https://doi.org/10.1090/S0025-5718-08-02102-9
  • [13] Monique Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439 (91a:35078)
  • [14] Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
  • [15] 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 (electronic). MR 2240631 (2007g:65116), https://doi.org/10.1090/S0025-5718-06-01884-9
  • [16] P. Grisvard, Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209 (93h:35004)
  • [17] Bernd Heinrich, The Fourier-finite-element method for Poisson's equation in axisymmetric domains with edges, SIAM J. Numer. Anal. 33 (1996), no. 5, 1885-1911. MR 1411853 (97j:65178), https://doi.org/10.1137/S0036142994266108
  • [18] V. A. Kondratev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209-292 (Russian). MR 0226187 (37 #1777)
  • [19] V. A. Kozlov, V. G. Mazya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR 1469972 (98f:35038)
  • [20] V. A. Kozlov, V. G. Mazya, and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR 1788991 (2001i:35069)
  • [21] Young-Ju Lee and Hengguang Li, On stability, accuracy, and fast solvers for finite element approximations of the axisymmetric Stokes problem by Hood-Taylor elements, SIAM J. Numer. Anal. 49 (2011), no. 2, 668-691. MR 2792390 (2012e:65278), https://doi.org/10.1137/100783339
  • [22] Young-Ju Lee and Hengguang Li, Axisymmetric Stokes equations in polygonal domains: regularity and finite element approximations, Comput. Math. Appl. 64 (2012), no. 11, 3500-3521. MR 2992530, https://doi.org/10.1016/j.camwa.2012.08.014
  • [23] Hengguang Li, Finite element analysis for the axisymmetric Laplace operator on polygonal domains, J. Comput. Appl. Math. 235 (2011), no. 17, 5155-5176. MR 2817318 (2012m:65430), https://doi.org/10.1016/j.cam.2011.05.003
  • [24] Hengguang Li, Anna Mazzucato, and Victor Nistor, Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains, Electron. Trans. Numer. Anal. 37 (2010), 41-69. MR 2777235 (2012c:65195)
  • [25] Hengguang Li and Victor Nistor, Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM, J. Comput. Appl. Math. 224 (2009), no. 1, 320-338. MR 2474235 (2009j:65327), https://doi.org/10.1016/j.cam.2008.05.009
  • [26] B. Mercier and G. Raugel, Résolution d'un problème aux limites dans un ouvert axisymétrique par éléments finis en $ r$, $ z$ et séries de Fourier en $ \theta $, RAIRO Anal. Numér. 16 (1982), no. 4, 405-461 (French, with English summary). MR 684832 (84g:65154)
  • [27] Boniface Nkemzi, The Poisson equation in axisymmetric domains with conical points, J. Comput. Appl. Math. 174 (2005), no. 2, 399-421. MR 2106447 (2005h:35068), https://doi.org/10.1016/j.cam.2004.05.006
  • [28] Jinchao Xu and Ludmil Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space, J. Amer. Math. Soc. 15 (2002), no. 3, 573-597. MR 1896233 (2003f:65095), https://doi.org/10.1090/S0894-0347-02-00398-3

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Additional Information

Hengguang Li
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: hli@math.wayne.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02879-2
Received by editor(s): February 13, 2013
Received by editor(s) in revised form: August 28, 2013
Published electronically: September 8, 2014
Additional Notes: The author was supported in part by the NSF grant DMS-1158839.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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