본문 바로가기
KERD
논문KCI 등재

Multigrid methods for 3$D$ $\HCurl$ problems with nonoverlapping domain decomposition smoothers

오덕순
대한수학회 (The Korean Mathematical Society)|2024.07|대한수학회지 (Journal of the KMS)|제61권 제4호|pp. 659-681|DOI 10.4134/JKMS.j230054

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

Multigrid methodnonoverlapping domain decomposition$\HCurl$N\'{e}d\'{e}lec finite element
주제분류수학

초록

We propose V--cycle multigrid methods for vector field problems arising from the lowest order hexahedral N\'{e}d\'{e}lec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of smoothing method is necessary. We introduce new smoothers based on substructuring with nonoverlapping domain decomposition methods. We provide the convergence analysis and numerical experiments that support our theory.

Abstract

We propose V--cycle multigrid methods for vector field problems arising from the lowest order hexahedral N\'{e}d\'{e}lec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of smoothing method is necessary. We introduce new smoothers based on substructuring with nonoverlapping domain decomposition methods. We provide the convergence analysis and numerical experiments that support our theory.

참고문헌 (34)

  1. R. Anderson, "MFEM : A modular finite element library", Comput. Math. Appl, 81, pp.42~74 (2021)[DOI]
  2. D. N. Arnold, "Preconditioning in H(div)and applications", Math. Comp, 66(219), pp.957~984 (1997)[DOI]
  3. D. N. Arnold, "Multigrid preconditioning in H(div)on non-convex polygons", Comput. Appl. Math, 17(3), pp.303~315 (1998)
  4. D. N. Arnold, "Multigrid in H(div)and H(curl)", Numer. Math, 85(2), pp.197~217 (2000)[DOI]
  5. A. Bossavit, "Discretization of electromagnetic problems: the generalized finite differences”approach", Handb. Numer. Anal, 13, pp.105~197 (2005)
  6. J. H. Bramble, "New convergence estimates for multigrid algorithms", Math. Comp, 49(180), pp.311~329 (1987)[DOI]
  7. S. C. Brenner, "Multigrid methods for H(div)in three dimensions with nonoverlapping domain decomposition smoothers", Numer. Linear Algebra Appl, 25(5), pp.14~ (2018)[DOI]
  8. S. C. Brenner, "A smoother based on nonoverlapping domain decomposition methods for H(div) problems: a numerical study, Domain decomposition methods in science and engineering XXIV", Lect. Notes Comput. Sci. Eng, 125, pp.523~531 (2018), Springer[DOI]
  9. S. C. Brenner, "Texts in Applied Mathematics vol. 15" (2008), Springer
  10. "Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree" (2022)[URL]
  11. J. G. Calvo, "A two-level overlapping Schwarz method for H(curl)in two dimensions with irregular subdomains", Electron. Trans. Numer. Anal, 44, pp.497~521 (2015)
  12. J. G. Calvo, "A new coarse space for overlapping Schwarz algorithms for H(curl)problems in three dimensions with irregular subdomains", Numer. Algorithms, 83(3), pp.885~899 (2020)[DOI]
  13. C. R. Dohrmann, "A BDDC algorithm with deluxe scaling for threedimensional H(curl)problems", Comm. Pure Appl. Math, 69(4), pp.745~770 (2016)[DOI]
  14. W. Hackbusch, "Springer Series in Computational Mathematics vol. 4" (1985), Springer-Verlag
  15. R. Hiptmair, "Multigrid method for H(div)in three dimensions", Electron. Trans. Numer. Anal, 6, pp.133~152 (1997)
  16. R. Hiptmair, "Multigrid method for Maxwell’s equations", SIAM J. Numer. Anal, 36(1), pp.204~225 (1998)[DOI]
  17. R. Hiptmair, "A review of regular decompositions of vector fields: continuous, discrete, and structure-preserving", Spectral and high order methods for partial differential equations—ICOSAHOM 2018, 134, pp.45~60 (2018), Springer
  18. R. Hiptmair, "Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, Parallel solution of partial differential equations", IMA Vol. Math. Appl, 120, pp.181~208 (2000), Springer
  19. R. Hiptmair, "Nodal auxiliary space preconditioning in H(curl)and H(div)spaces", SIAM J. Numer. Anal, 45(6), pp.2483~2509 (2007)[DOI]
  20. Q. Hu, "Convergence of the Hiptmair-Xu preconditioner for Maxwell’s equations with jump coefficients(I) : extensions of the regular decomposition", SIAM J. Numer. Anal, 59(5), pp.2500~2535 (2021)[DOI]
  21. Q. Hu, "A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions", SIAM J. Numer. Anal, 41(5), pp.1682~1708 (2003)[DOI]
  22. T. V. Kolev, "Parallel auxiliary space AMG for H(curl)problems", J. Comput. Math, 27(5), pp.604~623 (2009)[DOI]
  23. Y. -J. Lee, "Robust subspace correction methods for nearly singular systems", Math. Models Methods Appl. Sci, 17(11), pp.1937~1963 (2007)[DOI]
  24. S. F. McCormick, "Multigrid Methods, Frontiers in Applied Mathematics vol. 3" (1987), Society for Industrial and Applied Mathematics
  25. "MFEM: Modular finite element methods"[URL]
  26. P. B. Monk, "A mixed method for approximating Maxwell’s equations", SIAM J. Numer. Anal, 28(6), pp.1610~1634 (1991)[DOI]
  27. P. Monk, "Numerical Mathematics and Scientific Computation" (2003), Oxford University Press
  28. J. C. N´ed´elec, "Mixed finite elements in R3", Numer. Math, 35(3), pp.315~341 (1980)[DOI]
  29. D. -S. Oh, "Smoothers based on nonoverlapping domain decomposition methods for H(curl)problems : A numerical study", J. Korean Soc. Ind. Appl. Math, 26(4), pp.323~332 (2022)[DOI]
  30. A. Toselli, "Overlapping Schwarz methods for Maxwell’s equations in three dimensions", Numer. Math, 86(4), pp.733~752 (2000)[DOI]
  31. A. Toselli, "Dual-primal FETI algorithms for edge finite-element approximations in 3D", IMA J. Numer. Anal, 26(1), pp.96~130 (2006)[DOI]
  32. S. Zampini, "Adaptive BDDC deluxe methods for H(curl), Domain Decomposition Methods in Science and Engineering XXIII", Lect. Notes Comput. Sci. Eng, 116, pp.285~292 (2017), Springer, Cham[DOI]
  33. S. Zampini, "Balancing domain decomposition by constraints algorithms for curl-conforming spaces of arbitrary order, Domain Decomposition Methods in Science and Engineering XXIV", Lect. Notes Comput. Sci. Eng, 125, pp.103~116 (2018), Springer, Cham[DOI]
  34. L. T. Zikatanov, "Two-sided bounds on the convergence rate of two-level methods", Numer. Linear Algebra Appl, 15(5), pp.439~454 (2008)[DOI]

관련 논문AI추천

Mixed Finite Volume Method for Two-dimensional Maxwell's Equations곽도영, 김광연대한수학회 · 대한수학회지2025 · 제62권 제1호A non-overlapping domain decomposition method for a discontinuous Galerkin method: a numerical study박은희강원경기수학회 · 한국수학논문집2023 · 제31권 제4호Preconditioners for a coupled problem by a penalty term arisen in an augmented Lagrangian method박은희, 이창옥대한수학회 · 대한수학회지2020 · 제57권 제5호A DUAL ITERATIVE SUBSTRUCTURING METHOD WITH A SMALL PENALTY PARAMETER박은희, 이창옥대한수학회 · 대한수학회지2017 · 제54권 제2호Hierarchical error estimators for lowest-order mixed finite element methods김광연강원경기수학회 · 한국수학논문집2014 · 제22권 제3호A posteriori error estimator for N\'{e}d\'{e}lec edge element methods of two-dimensional time-harmonic Maxwell's equation김광연대한수학회 · 대한수학회보2025 · 제62권 제5호

이 간행물의 다른 논문 (8건)

Quadratic B-spline Galerkin scheme for the solution of a space-fractional Burgers' equationKhadidja Bouabid, Nasserdine Kechkarpp. 621-657Realizing a Fake Projective Plane as a Degree 25 Surface in $\P^5$Zachary Lihn, Lev Borisovpp. 683-692Analyzing the duration of success and failure in Markov-modulated Bernoulli processes김유라pp. 693-711Central limit theorems for conditionally strong mixing and conditionally strictly stationary sequences of random variablesXiao-Lin Zeng, De-Mei Yuanpp. 713-742An invariant forth-order curve flow in centro-affine geometryYanhua Yu, Yuanyuan Gongpp. 743-760
원문 보기