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Multigrid Methods Based on Hodge Decomposition for a Quad-Curl Problem

  • Susanne C. Brenner EMAIL logo , Jintao Cui and Li-yeng Sung
Published/Copyright: March 12, 2019

Abstract

In this paper we investigate multigrid methods for a quad-curl problem on graded meshes. The approach is based on the Hodge decomposition. The solution for the quad-curl problem is approximated by solving standard second-order elliptic problems and optimal error estimates are obtained on graded meshes. We prove the uniform convergence of the multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results.

MSC 2010: 65N30; 65N15; 35Q60

Award Identifier / Grant number: 11771367

Award Identifier / Grant number: DMS-16-20273

Funding statement: The work of the first and third authors was supported in part by the National Science Foundation under Grant No. DMS-16-20273. The second author’s work is supported in part by the National Natural Science Foundation of China (NSFC) Grant no. 11771367 and Hong Kong PolyU General Research Grant G-YBM.

References

[1] 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. 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-BSearch in Google Scholar

[2] I. Babuška, R. B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math. 33 (1979), no. 4, 447–471. 10.1007/BF01399326Search in Google Scholar

[3] R. E. Bank, A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations, SIAM J. Numer. Anal. 18 (1981), no. 4, 724–743. 10.1137/0718048Search in Google Scholar

[4] R. E. Bank and T. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35–51. 10.1090/S0025-5718-1981-0595040-2Search in Google Scholar

[5] D. Biskamp, Magnetic Reconnection in Plasmas, Cambridge University Press, Cambridge, 2005. Search in Google Scholar

[6] J. H. Bramble, D. Y. Kwak and J. E. Pasciak, Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems, SIAM J. Numer. Anal. 31 (1994), no. 6, 1746–1763. 10.1137/0731089Search in Google Scholar

[7] J. H. Bramble, J. E. Pasciak and J. Xu, The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems, Math. Comp. 51 (1988), no. 184, 389–414. 10.1090/S0025-5718-1988-0930228-6Search in Google Scholar

[8] J. H. Bramble and X. Zhang, The analysis of multigrid methods, Handbook of Numerical Analysis. Vol. VII, North-Holland, Amsterdam (2000), 173–415. 10.1016/S1570-8659(00)07003-4Search in Google Scholar

[9] J. J. Brannick, H. Li and L. T. Zikatanov, Uniform convergence of the multigrid V-cycle on graded meshes for corner singularities, Numer. Linear Algebra Appl. 15 (2008), no. 2–3, 291–306. 10.1002/nla.574Search in Google Scholar

[10] S. C. Brenner, J. Cui, T. Gudi and L.-Y. Sung, Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes, Numer. Math. 119 (2011), no. 1, 21–47. 10.1007/s00211-011-0379-ySearch in Google Scholar

[11] S. C. Brenner, J. Cui, Z. Nan and L.-Y. Sung, Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations, Math. Comp. 81 (2012), no. 278, 643–659. 10.1090/S0025-5718-2011-02540-8Search in Google Scholar

[12] S. C. Brenner, J. Cui and L.-Y. Sung, Multigrid methods for the symmetric interior penalty method on graded meshes, Numer. Linear Algebra Appl. 16 (2009), no. 6, 481–501. 10.1002/nla.630Search in Google Scholar

[13] S. C. Brenner, J. Gedicke and L.-Y. Sung, Hodge decomposition for two-dimensional time-harmonic Maxwell’s equation: impedance boundary condition, Math. Methods Appl. Sci. 40 (2017), no. 2, 370–390. 10.1002/mma.3398Search in Google Scholar

[14] S. C. Brenner and L. Owens, A W-cycle algorithm for a weakly over-penalized interior penalty method, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37–40, 3823–3832. 10.1016/j.cma.2007.02.011Search in Google Scholar

[15] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[16] S. C. Brenner, J. Sun and L.-y. Sung, Hodge decomposition methods for a quad-curl problem on planar domains, J. Sci. Comput. 73 (2017), no. 2–3, 495–513. 10.1007/s10915-017-0449-0Search in Google Scholar

[17] F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems 26 (2010), no. 7, Article ID 074004. 10.1088/0266-5611/26/7/074004Search in Google Scholar

[18] L. Chacón, A. N. Simakov and A. Zocco, Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics, Phys. Rev. Lett. 99 (2007), Article ID 235001. 10.1103/PhysRevLett.99.235001Search in Google Scholar PubMed

[19] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[20] J. Cui, Multigrid methods for two-dimensional Maxwell’s equations on graded meshes, J. Comput. Appl. Math. 255 (2014), 231–247. 10.1016/j.cam.2013.05.007Search in Google Scholar

[21] M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Math. 1341, Springer, Berlin, 1988. 10.1007/BFb0086682Search in Google Scholar

[22] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar

[23] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Search in Google Scholar

[24] W. Hackbusch, Multigrid Methods and Applications, Springer Ser. Comput. Math. 4, Springer, Berlin, 1985. 10.1007/978-3-662-02427-0Search in Google Scholar

[25] Q. Hong, J. Hu, S. Shu and J. Xu, A discontinuous Galerkin method for the fourth-order curl problem, J. Comput. Math. 30 (2012), no. 6, 565–578. 10.4208/jcm.1206-m3572Search in Google Scholar

[26] V. A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292. Search in Google Scholar

[27] P. D. Lax, Functional Analysis, Pure Appl. Math. (N. Y.), Wiley-Interscience, New York, 2002. Search in Google Scholar

[28] P. Le Tallec, A mixed finite element approximation of the Navier-Stokes equations, Numer. Math. 35 (1980), no. 4, 381–404. 10.1007/BF01399007Search in Google Scholar

[29] S. F. McCormick, Multigrid Methods, Front. Appl. Math. 3, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1987. 10.1137/1.9781611971057Search in Google Scholar

[30] P. Monk and J. Sun, Finite element methods for Maxwell’s transmission eigenvalues, SIAM J. Sci. Comput. 34 (2012), no. 3, B247–B264. 10.1137/110839990Search in Google Scholar

[31] S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter Exp. Math. 13, Walter de Gruyter, Berlin, 1994. 10.1515/9783110848915Search in Google Scholar

[32] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg, 2012. 10.1007/978-3-642-10455-8Search in Google Scholar

[33] J.-C. Nédélec, A new family of mixed finite elements in 3, Numer. Math. 50 (1986), no. 1, 57–81. 10.1007/BF01389668Search in Google Scholar

[34] J. Sun, A mixed FEM for the quad-curl eigenvalue problem, Numer. Math. 132 (2016), no. 1, 185–200. 10.1007/s00211-015-0708-7Search in Google Scholar

[35] U. Trottenberg, C. W. Oosterlee and A. Schüller, Multigrid, Academic Press, San Diego, 2001. Search in Google Scholar

[36] Y.-J. Yon and D. Kwak, Nonconforming multigrid method for nonsymmetric and indefinite problems, Comput. Math. Appl. 30 (1995), 1–7. 10.1016/0898-1221(95)00159-VSearch in Google Scholar

[37] K. Yosida, Functional Analysis. Reprint of the sixth (1980) edition, Classics Math., Springer, Berlin, 1995. 10.1007/978-3-642-61859-8Search in Google Scholar

[38] H. Yserentant, The convergence of multilevel methods for solving finite-element equations in the presence of singularities, Math. Comp. 47 (1986), no. 176, 399–409. 10.1090/S0025-5718-1986-0856693-9Search in Google Scholar

[39] B. Zheng, Q. Hu and J. Xu, A nonconforming finite element method for fourth order curl equations in 3, Math. Comp. 80 (2011), no. 276, 1871–1886. 10.1090/S0025-5718-2011-02480-4Search in Google Scholar

Received: 2019-01-22
Accepted: 2019-01-22
Published Online: 2019-03-12
Published in Print: 2019-04-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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