Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

A NON-OVERLAPPING DOMAIN DECOMPOSITION METHOD FOR A DISCONTINUOUS GALERKIN METHOD: A NUMERICAL STUDY

  • Eun-Hee Park (Department of Artificial Intelligence and Software, Kangwon National University)
  • Received : 2023.09.03
  • Accepted : 2023.10.19
  • Published : 2023.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we propose an iterative method for a symmetric interior penalty Galerkin method for heterogeneous elliptic problems. The iterative method consists mainly of two parts based on a non-overlapping domain decomposition approach. One is an intermediate preconditioner constructed by understanding the properties of the discontinuous finite element functions and the other is a preconditioning related to the dual-primal finite element tearing and interconnecting (FETI-DP) methodology. Numerical results for the proposed method are presented, which demonstrate the performance of the iterative method in terms of various parameters associated with the elliptic model problem, the finite element discretization, and non-overlapping subdomain decomposition.

Keywords

Acknowledgement

The author would like to sincerely thank Professor Susanne C. Brenner and Professor Li-yeng Sung for taking the time to have a valuable discussion.

References

  1. P. F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case, ESIAM:M2AN. 41 (2007), 21-54. https://doi.org/10.1051/m2an:2007006
  2. D. Arnold, F. Brezzi, B. Cockburn, and L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), 1749-1779. https://doi.org/10.1137/S0036142901384162
  3. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), 742-760. https://doi.org/10.1137/0719052
  4. S. Brenner, E.-H. Park, and L.-Y. Sung, A BDDC preconditioner for a weakly over-penalized symmetric interior penalty method, Numer. Linear Alg. Appl. 20 (2013), 472-491. https://doi.org/10.1002/nla.1838
  5. S. C. Brenner, E.-H. Park, and L.-Y. Sung, A BDDC preconditioners for a symmetric interior penalty Galerkin method, Electron. Trans. Numer. Anal. 46 (2017), 190-214.
  6. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods (Third Edition), Springer-Verlag, New York, 2008.
  7. S. C. Brenner and L.-Y. Sung, BDDC and FETI-DP without matrices or vectors, Comput. Methods Appl. Mech. Engrg. 196 (2007), 1429-1435. https://doi.org/10.1016/j.cma.2006.03.012
  8. C. Canuto, L. F. Pavarino, and A. B. Pieri, BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree, IMA J. Numer. Anal. 34 (2014), 879-903. https://doi.org/10.1093/imanum/drt037
  9. L. T. Diosady and D. L. Darmofal, A unified analysis of balancing domain decomposition by constraints for discontinuous Galerkin discretizations, SIAM J. Numer. Anal. 50 (2012), 1695-1712. https://doi.org/10.1137/100812434
  10. M. Dryja, On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, Comput. Methods Appl. Math. 3 (2003), 76-85. https://doi.org/10.2478/cmam-2003-0007
  11. M. Dryja, J. Galvis, and M. Sarkis, BDDC methods for discontinuous Galerkin discretization of elliptic problems, J. of Complexity 23 (2007), 715-739. https://doi.org/10.1016/j.jco.2007.02.003
  12. M. Dryja, J. Galvis, and M. Sarkis, The analysis of a FETI-DP preconditioner for a full DG discretization of elliptic problems in two dimensions, Numer. Math. 131 (2015), 737-770. https://doi.org/10.1007/s00211-015-0705-x
  13. C. Farhat, M. Lesoinne, P. Le Tallec, and K. Pierson, FETI-DP: A dual-primal unified FETI method - part I: A faster alternative to the two-level FETI method, Int. J. Numer. Methods Engrg. 50 (2001), 1523-154.
  14. C. Farhat, M. Lesoinne, and K. Pierson, A scalable dual-primal domain decomposition method, Numer. Lin. Alg. Appl. 7 (2000), 687-714. https://doi.org/10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S
  15. X. Feng and O. A. Karakashian, Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation, J. Sci. Comput. 22 (2005), 289-314. https://doi.org/10.1007/s10915-004-4141-9
  16. D. Z. Kalchev and P. Vassilevski, Auxiliary space preconditioning of finite element equations using a nonconforming interior penalty reformulation and static condensation, SIAM J. Sci. Comput. 42 (2020), 1741-1764. https://doi.org/10.1137/19M1286815
  17. H. H. Kim, E. T. Chung, and C. S. Lee, FETI-DP preconditioners for a staggered discontinuous Galerkin formulation of the two-dimensional Stokes problem, Comput. Math. Appl. 68 (2014), 2233-2250. https://doi.org/10.1016/j.camwa.2014.07.031
  18. A. Klawonn, O. B. Widlund, and M. Dryja, Dual-Primal FETI methods for three dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal. 40 (2002), 159-176. https://doi.org/10.1137/S0036142901388081
  19. J. Li and O. B. Widlund, FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg. 66 (2006), 250-271. https://doi.org/10.1002/nme.1553
  20. J. Mandel and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl. 10 (2003), 639-659. https://doi.org/10.1002/nla.341
  21. J. Mandel and R. Tezaur, On the convergence of a dual-primal substructuring method, Numer. Math. 88 (2001), 543-558. https://doi.org/10.1007/s211-001-8014-1
  22. J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences, Lecture Notes in Phys., 58, Springer, Berlin, 1976.
  23. B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations - Theory and Implementation, SIAM, Philadelphia, 2008.
  24. A. Toselli and O. B. Widlund, Domain Decomposition Methods - Algorithms and Theory, Springer-Verlag, Berlin, 2005.
  25. M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), 152-161. https://doi.org/10.1137/0715010