References
- R. Adams, Sobolev Spaces, Academic Press Inc, New York, 1975.
- 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
- 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
- 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
- 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
- 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
- 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
- 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. https://doi.org/10.1090/S0025-5718-08-02172-8
- 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.
- 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.
- 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
- V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986.
- 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.
- 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
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- W. F. Mitchell, A parallel multigrid method using the full domain partition, Electron. Trans. Numer. Anal. 6 (1997), 224-233.
- 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
- J. E. Pasciak, Two domain decomposition techniques for Stokes problems, Domain decomposition methods (Los Angeles, CA, 1988), 419-430. SIAM, Philadelphia, 1989.
- 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. https://doi.org/10.1137/S0036142998333092
- 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
- A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Spring-Verlag, Berlin, 1994.
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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.
- 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. https://doi.org/10.1090/S0025-5718-99-01149-7
- 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