Papers
Topics
Authors
Recent
Search
2000 character limit reached

New Analysis of Overlapping Schwarz Methods for Vector Field Problems in Three Dimensions with Generally Shaped Domains

Published 18 Apr 2024 in math.NA and cs.NA | (2404.11986v1)

Abstract: This paper introduces a novel approach to analyzing overlapping Schwarz methods for N\'{e}d\'{e}lec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (45)
  1. Preconditioning in H⁢(div)𝐻divH({\rm div})italic_H ( roman_div ) and applications. Math. Comp., 66(219):957–984, 1997.
  2. Multigrid in H⁢(div)𝐻divH({\rm div})italic_H ( roman_div ) and H⁢(curl)𝐻curlH({\rm curl})italic_H ( roman_curl ). Numer. Math., 85(2):197–217, 2000.
  3. Alain Bossavit. Discretization of electromagnetic problems: the “generalized finite differences” approach. In Handbook of numerical analysis. Vol. XIII, Handb. Numer. Anal., XIII, pages 105–197. North-Holland, Amsterdam, 2005.
  4. Susanne C. Brenner. Lower bounds for two-level additive Schwarz preconditioners with small overlap. SIAM J. Sci. Comput., 21(5):1657–1669, 2000. Iterative methods for solving systems of algebraic equations (Copper Mountain, CO, 1998).
  5. Multigrid methods for H⁢(div)𝐻divH({\rm div})italic_H ( roman_div ) in three dimensions with nonoverlapping domain decomposition smoothers. Numer. Linear Algebra Appl., 25(5):e2191, 14, 2018.
  6. A smoother based on nonoverlapping domain decomposition methods for H⁢(div)𝐻divH({\rm div})italic_H ( roman_div ) problems: a numerical study. In Domain decomposition methods in science and engineering XXIV, volume 125 of Lect. Notes Comput. Sci. Eng., pages 523–531. Springer, Cham, 2018.
  7. Multigrid solvers for the de rham complex with optimal complexity in polynomial degree, 2022. arXiv:2211.14284.
  8. First-order system least squares for second-order partial differential equations. I. SIAM J. Numer. Anal., 31(6):1785–1799, 1994.
  9. Mixed finite element methods for incompressible flow: stationary Stokes equations. Numer. Methods Partial Differential Equations, 26(4):957–978, 2010.
  10. Juan G. Calvo. A BDDC algorithm with deluxe scaling for H⁢(curl)𝐻curlH({\rm curl})italic_H ( roman_curl ) in two dimensions with irregular subdomains. Math. Comp., 85(299):1085–1111, 2016.
  11. Juan G. Calvo. A new coarse space for overlapping Schwarz algorithms for H⁢(curl)𝐻curlH(\rm curl)italic_H ( roman_curl ) problems in three dimensions with irregular subdomains. Numer. Algorithms, 83(3):885–899, 2020.
  12. Some recent tools and a BDDC algorithm for 3D problems in H⁢(curl)𝐻curlH({\rm curl})italic_H ( roman_curl ). In Domain decomposition methods in science and engineering XX, volume 91 of Lect. Notes Comput. Sci. Eng., pages 15–25. Springer, Heidelberg, 2013.
  13. A BDDC algorithm with deluxe scaling for three-dimensional H⁢(𝐜𝐮𝐫𝐥)𝐻𝐜𝐮𝐫𝐥H({\bf curl})italic_H ( bold_curl ) problems. Comm. Pure Appl. Math., 69(4):745–770, 2016.
  14. Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput., 15(3):604–620, 1994. Iterative methods in numerical linear algebra (Copper Mountain Resort, CO, 1992).
  15. Local bounded cochain projections. Math. Comp., 83(290):2631–2656, 2014.
  16. R. Hiptmair. Multigrid method for 𝐇⁢(div)𝐇div\mathbf{H}({\rm div})bold_H ( roman_div ) in three dimensions. Electron. Trans. Numer. Anal., 6:133–152, 1997. Special issue on multilevel methods (Copper Mountain, CO, 1997).
  17. R. Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal., 36(1):204–225, 1999.
  18. Regular decompositions of vector fields - continuous, discrete, and structure-preserving, 2019.
  19. Discrete regular decompositions of tetrahedral discrete 1-forms. In Maxwell’s equations—analysis and numerics, volume 24 of Radon Ser. Comput. Appl. Math., pages 199–258. De Gruyter, Berlin, [2019] ©2019.
  20. A review of regular decompositions of vector fields: continuous, discrete, and structure-preserving. In Spectral and high order methods for partial differential equations—ICOSAHOM 2018, volume 134 of Lect. Notes Comput. Sci. Eng., pages 45–60. Springer, Cham, [2020] ©2020.
  21. Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions. In Parallel solution of partial differential equations (Minneapolis, MN, 1997), volume 120 of IMA Vol. Math. Appl., pages 181–208. Springer, New York, 2000.
  22. Nodal auxiliary space preconditioning in 𝐇⁢(𝐜𝐮𝐫𝐥)𝐇𝐜𝐮𝐫𝐥{\bf H}({\bf curl})bold_H ( bold_curl ) and 𝐇⁢(div)𝐇div{\bf H}({\rm div})bold_H ( roman_div ) spaces. SIAM J. Numer. Anal., 45(6):2483–2509, 2007.
  23. A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal., 41(5):1682–1708, 2003.
  24. Parallel auxiliary space AMG for H⁢(curl)𝐻curlH({\rm curl})italic_H ( roman_curl ) problems. J. Comput. Math., 27(5):604–623, 2009.
  25. Parallel auxiliary space AMG solver for H⁢(div)𝐻divH({\rm div})italic_H ( roman_div ) problems. SIAM J. Sci. Comput., 34(6):A3079–A3098, 2012.
  26. Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci., 17(11):1937–1963, 2007.
  27. On a sharp estimate of overlapping schwarz methods in H⁢(curl)Hcurl\mathrm{H}(\mathrm{curl})roman_H ( roman_curl ) and H⁢(div)Hdiv\mathrm{H}(\mathrm{div})roman_H ( roman_div ), 2023. arXiv:2304.10026.
  28. Ping Lin. A sequential regularization method for time-dependent incompressible Navier-Stokes equations. SIAM J. Numer. Anal., 34(3):1051–1071, 1997.
  29. Peter B. Monk. A mixed method for approximating Maxwell’s equations. SIAM J. Numer. Anal., 28(6):1610–1634, 1991.
  30. Duk-Soon Oh. An alternative coarse space method for overlapping Schwarz preconditioners for Raviart-Thomas vector fields. In Domain decomposition methods in science and engineering XX, volume 91 of Lect. Notes Comput. Sci. Eng., pages 361–367. Springer, Heidelberg, 2013.
  31. Duk-Soon Oh. An overlapping Schwarz algorithm for Raviart-Thomas vector fields with discontinuous coefficients. SIAM J. Numer. Anal., 51(1):297–321, 2013.
  32. Duk-Soon Oh. A BDDC preconditioner for problems posed in H⁢(div)𝐻divH({\rm div})italic_H ( roman_div ) with deluxe scaling. In Domain decomposition methods in science and engineering XXII, volume 104 of Lect. Notes Comput. Sci. Eng., pages 355–361. Springer, Cham, 2016.
  33. Duk-Soon Oh. Multigrid methods for 3D𝐷Ditalic_D H⁢(𝐜𝐮𝐫𝐥)𝐻𝐜𝐮𝐫𝐥{H}(\mathbf{curl})italic_H ( bold_curl ) problems with nonoverlapping domain decomposition smoothers, 2022. submitted, arXiv:2205.05840.
  34. Duk-Soon Oh. Smoothers based on nonoverlapping domain decomposition methods for H⁢(curl)𝐻curlH({\rm curl})italic_H ( roman_curl ) problems: A numerical study. J. Korean Soc. Ind. Appl. Math., 26(4):323–332, 2022.
  35. BDDC algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields. Math. Comp., 87(310):659–692, 2018.
  36. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54(190):483–493, 1990.
  37. Andrea Toselli. Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math., 86(4):733–752, 2000.
  38. Andrea Toselli. Domain decomposition methods of dual-primal FETI type for edge element approximations in three dimensions. C. R. Math. Acad. Sci. Paris, 339(9):673–678, 2004.
  39. Andrea Toselli. Dual-primal FETI algorithms for edge finite-element approximations in 3D. IMA J. Numer. Anal., 26(1):96–130, 2006.
  40. Robust and efficient FETI domain decomposition algorithms for edge element approximations. COMPEL, 24(2):396–407, 2005.
  41. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.
  42. An iterative substructuring method for Raviart-Thomas vector fields in three dimensions. SIAM J. Numer. Anal., 37(5):1657–1676, 2000.
  43. Jinchao Xu. Iterative methods by space decomposition and subspace correction. SIAM Rev., 34(4):581–613, 1992.
  44. Stefano Zampini. Adaptive BDDC deluxe methods for H⁢(curl)Hcurl\rm H(curl)roman_H ( roman_curl ). In Domain decomposition methods in science and engineering XXIII, volume 116 of Lect. Notes Comput. Sci. Eng., pages 285–292. Springer, Cham, 2017.
  45. Balancing domain decomposition by constraints algorithms for curl-conforming spaces of arbitrary order. In Domain decomposition methods in science and engineering XXIV, volume 125 of Lect. Notes Comput. Sci. Eng., pages 103–116. Springer, Cham, 2018.

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (2)

  1. Duk-Soon Oh 
  2. Shangyou Zhang 

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up