DOI QR코드

DOI QR Code

A PARALLEL FINITE ELEMENT ALGORITHM FOR SIMULATION OF THE GENERALIZED STOKES PROBLEM

  • Shang, Yueqiang
  • Received : 2015.05.21
  • Published : 2016.05.31

Abstract

Based on a particular overlapping domain decomposition technique, a parallel finite element discretization algorithm for the generalized Stokes equations is proposed and investigated. In this algorithm, each processor computes a local approximate solution in its own subdomain by solving a global problem on a mesh that is fine around its own subdomain and coarse elsewhere, and hence avoids communication with other processors in the process of computations. This algorithm has low communication complexity. It only requires the application of an existing sequential solver on the global meshes associated with each subdomain, and hence can reuse existing sequential software. Numerical results are given to demonstrate the effectiveness of the parallel algorithm.

Keywords

generalized Stokes problem;finite element;parallel algorithm;parallel computing;domain decomposition

References

  1. R. Adams, Sobolev Spaces, Academic Press Inc, New York, 1975.
  2. M. Ainsworth and S. Sherwin, Domain decomposition preconditioners for p and hp finite element approximations of Stokes equations, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 243-266. https://doi.org/10.1016/S0045-7825(98)00356-9
  3. R. E. Bank, Some variants of the Bank-Holst parallel adaptive meshing paradigm, Comput. Vis. Sci. 9 (2006), no. 3, 133-144. https://doi.org/10.1007/s00791-006-0029-6
  4. R. E. Bank and M. Holst, A new paradigm for parallel adaptive meshing algorithms, SIAM J. Sci. Comput. 22 (2000), no. 4, 1411-1443. https://doi.org/10.1137/S1064827599353701
  5. R. E. Bank and P. K. Jimack, A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations, Concurrency Computat.: Pract. Exper. 13 (2001), 327-350. https://doi.org/10.1002/cpe.569
  6. J. H. Bramble and J. E. Pasciak, A domain decomposition technique for Stokes problems, Appl. Numer. Math. 6 (1990), no. 4, 251-261. https://doi.org/10.1016/0168-9274(90)90019-C
  7. M. A. Casarin, Schwarz preconditioners for the spectral element discretization of the steady Stokes and Navier-Stokes equations, Numer. Math. 89 (2001), no. 2, 307-339. https://doi.org/10.1007/PL00005469
  8. V. Dolean, F. Nataf, and G. Rapin, Deriving a new domain decomposition method for the Stokes equations using the Smith factorization, Math. Comp. 78 (2009), no. 266, 789-814.
  9. H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, 2005.
  10. P. F. Fischer, N. I. Miller, and H. M. Tufo, An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, Parallel solution of partial differential equations (Minneapolis, MN, 1997), 159-180, IMA Vol. Math. Appl., 120, Springer, New York, 2000.
  11. D. K. Gartling, A test problem for outflow boundary conditions-flow over a backward-facing step, Internat. J. Numer. Methods Fluids 11 (1990), 953-967. https://doi.org/10.1002/fld.1650110704
  12. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986.
  13. Y. N. He, J. C. Xu, and A. H. Zhou, Local and parallel finite element algorithms for the Navier-Stokes problem, J. Comput. Math. 24 (2006), no. 3, 227-238.
  14. Y. N. He, J. C. Xu, A. H. Zhou, and J. Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math. 109 (2008), no. 3, 415-434. https://doi.org/10.1007/s00211-008-0141-2
  15. M. Q. Jiang and P. L. Dai, A parallel nonoverlapping domain decomposition method for Stokes problems, J. Comput. Math. 24 (2006), no. 2, 209-224.
  16. 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), no. 12, 2233-2250. https://doi.org/10.1016/j.camwa.2014.07.031
  17. H. H. Kim and C. O. Lee, A two-level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns, Internat. J. Numer. Methods Eng. 88 (2011), no. 13, 1390-1410. https://doi.org/10.1002/nme.3227
  18. H. H. Kim, C. O. Lee, and E. H. Park, A FETI-DP formulation for the Stokes problem without primal pressure components, SIAM J. Numer. Anal. 47 (2010), no. 6, 4142-4162. https://doi.org/10.1137/080731876
  19. A. Klawonn, An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput. 19 (1998), no. 2, 540-552. https://doi.org/10.1137/S1064827595279575
  20. A. Klawonn and L. F. Pavarino, Overlapping Schwarz methods for mixed linear elasticity and Stokes problems, Comput. Methods Appl. Mech. Engrg. 165 (1998), no. 1-4, 233-245. https://doi.org/10.1016/S0045-7825(98)00059-0
  21. A. Klawonn and L. F. Pavarino, A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems, Numer. Linear Algebra Appl. 7 (2000), no. 1, 1-25. https://doi.org/10.1002/(SICI)1099-1506(200001/02)7:1<1::AID-NLA183>3.0.CO;2-J
  22. J. Li, A dual-primal FETI method for incompressible Stokes equations, Numer. Math. 102 (2005), no. 2, 257-275. https://doi.org/10.1007/s00211-005-0653-y
  23. J. Li and X. Tu, A nonoverlapping domain decomposition method for incompressible Stokes equations with continuous pressures, SIAM J. Numer. Anal. 51 (2013), no. 2, 1235-1253. https://doi.org/10.1137/120861503
  24. J. Li and O. Widlund, BDDC algorithms for incompressible Stokes equations, SIAM J. Numer. Anal. 44 (2006), no. 6, 2432-2455. https://doi.org/10.1137/050628556
  25. W. F. Mitchell, A parallel multigrid method using the full domain partition, Electron. Trans. Numer. Anal. 6 (1997), 224-233.
  26. W. F. Mitchell, The full domain partition approach to distributing adaptive grids, Appl. Numer. Math. 26 (1998), no. 1-2, 265-275. https://doi.org/10.1016/S0168-9274(97)00095-0
  27. J. E. Pasciak, Two domain decomposition techniques for Stokes problems, Domain decomposition methods (Los Angeles, CA, 1988), 419-430. SIAM, Philadelphia, 1989.
  28. L. F. Pavarino and O. B. Widlund, Iterative substructuring methods for spectral element discretizations of elliptic systems II: Mixed methods for linear elasticity and Stokes flow, SIAM J. Numer. Anal. 37 (2000), no. 2, 375-402.
  29. L. F. Pavarino and O. B. Widlund, Balancing Neumann-Neumann methods for incompressible Stokes equations, Comm. Pure Appl. Math. 55 (2002), no. 3, 302-335. https://doi.org/10.1002/cpa.10020
  30. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Spring-Verlag, Berlin, 1994.
  31. E. M. Ronquist, Domain decomposition methods for the steady Stokes equations, Proceedings of the 11th International Conference on Domain Decomposition Methods in Greenwich, England. DDM.org, 1999. Available at www.ddm.org/DD11/index.html.
  32. Y. Q. Shang, A parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the Navier-Stokes equations, J. Math. Anal. Appl. 403 (2013), no. 2, 667-679. https://doi.org/10.1016/j.jmaa.2013.02.060
  33. Y. Q. Shang, Parallel defect-correction algorithms based on finite element discretization for the Navier-Stokes equations, Comput. & Fluids 79 (2013), 200-212. https://doi.org/10.1016/j.compfluid.2013.03.021
  34. Y. Q. Shang and Y. N. He, Parallel finite element algorithm based on full domain partition for stationary Stokes equations, Appl. Math. Mech. (English Ed.) 31 (2010), no. 5, 643-650. https://doi.org/10.1007/s10483-010-0512-x
  35. Y. Q. Shang and Y. N. He, Parallel iterative finite element algorithms based on full domain partition for the stationary Navier-Stokes equations, Appl. Numer. Math. 60 (2010), no. 7, 719-737. https://doi.org/10.1016/j.apnum.2010.03.013
  36. P. Le Tallec and A. Patra, Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure fields, Comput. Methods Appl. Mech. Engrg. 145 (1997), no. 3, 361-379. https://doi.org/10.1016/S0045-7825(96)01207-8
  37. X. Tu, A three-level BDDC algorithm for a saddle point problem, Numer. Math. 119 (2011), no. 1, 189-217. https://doi.org/10.1007/s00211-011-0375-2
  38. X. Tu and J. Li, A unified dual-primal finite element tearing and interconnecting approach for incompressible Stokes equations, Internat. J. Numer. Methods Eng. 94 (2013), no. 2, 128-149. https://doi.org/10.1002/nme.4439
  39. C. M. Wang, A preconditioner for FETI-DP method of Stokes problem with mortar-type discretization, Math. Prob. Eng. 2013 (2013), Art. ID 485628, 1-11.
  40. J. C. Xu and A. H. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000), no. 231, 881-909.
  41. J. C. Xu and A. H. Zhou, Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math. 14 (2001), no. 4, 293-327. https://doi.org/10.1023/A:1012284322811

Acknowledgement

Supported by : Natural Science Foundation of China, Central Universities, Science and Technology Foundation of Guizhou Province of China