References
- P.F. Antonietti, B. Ayuso, Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. ESAIM: Math. Model. Numer. Anal. (ESAIM: M2AN) 42 (2008), pp. 443–469. doi: 10.1051/m2an:2008012
- L. Badea, X. Tai, and J. Wang, Convergence rate analysis of a multiplicative Schwarz method for variational inequalities, SIAM J. Numer. Anal. 41 (2003), pp. 1052–1073. doi: 10.1137/S0036142901393607
- Z.-Z. Bai, G.H. Golub, and M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003), pp. 603–626. doi: 10.1137/S0895479801395458
- J.H. Bramble, J.E. Pasciak, J. Wang, and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), pp. 1–21. doi: 10.1090/S0025-5718-1991-1090464-8
- M. Cai, Modeling and numerical simulation for the coupling of surface flow with subsurface flow, Ph.D. diss., Hong Kong University of Science and Technology, 2008.
- X.-C. Cai, Multiplicative Schwarz methods for parabolic problems, SIAM J. Sci. Comput. 15 (1994), pp. 587–603. doi: 10.1137/0915039
- Z. Cai and C.-Y.G. Lai, Convergence estimates of multilevel additive and multiplicative algorithms for non-symmetric and indefinite problems, Numer. Linear Algebra Appl. 3 (1996), pp. 205–220. doi: 10.1002/(SICI)1099-1506(199605/06)3:3<205::AID-NLA79>3.0.CO;2-K
- M. Cai and L.F. Pavarino, Hybrid and multiplicative overlapping Schwarz algorithms with standard coarse spaces for mixed linear elasticity and Stokes problems, Commun. Comput. Phys. 20 (2016), pp. 989–1015. doi: 10.4208/cicp.020815.080316a
- X.-C. Cai and O.B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Statist. Comput. 13 (1992), pp. 243–258. doi: 10.1137/0913013
- X.-C. Cai and O.B. Widlund, Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems, SIAM J. Numer. Anal. 30 (1993), pp. 936–952. doi: 10.1137/0730049
- M. Cai, L.F. Pavarino, and O.B. Widlund, Overlapping Schwarz methods with a standard coarse space for almost incompressible linear elasticity, SIAM J. Sci. Comput. 37 (2015), pp. A811–A830. doi: 10.1137/140981861
- H. Chen and X. Xu, Local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems, SIAM J. Numer. Anal. 47 (2010), pp. 4492–4516. doi: 10.1137/090755849
- C.R. Dohrmann and O.B. Widlund, An overlapping Schwarz algorithm for almost incompressible elasticity, SIAM J. Numer. Anal. 47 (2009), pp. 2897–2923. doi: 10.1137/080724320
- M. Dryja, An additive Schwarz algorithm for two- and three- dimensional finite element elliptic problems, In: T. Chan, R. Glowinski, G.A. Meurant, J. Périaux, O. Widlund, eds., Domain Decomposition Methods for Partial Differential Equations II. Philadelphia, 1989.
- M. Dryja, O.B. Widlund, Some domain decomposition algorithms for elliptic problems, In: L. Hayes, D. Kincaid, ed., Iterative Methods for Large Linear Systems, Academic Press, San Diego California, 1989.
- M. Dryja and O.B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), SIAM, Philadelphia, 1990, pp. 3–21.
- C. Echeverría, J. Liesen, D.B. Szyld, and P. Tichý, Convergence of the multiplicative Schwarz method for singularly perturbed convection-diffusion problems discretized on a Shishkin mesh, Electron. Trans. Numer. Anal. 48 (2018), pp. 40–62. doi: 10.1553/etna_vol48s40
- S.C. Eisenstat, H.C. Elman, and M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 20 (1983), pp. 345–357. doi: 10.1137/0720023
- R. Ernst, B. Flemisch, and B. Wohlmuth, A multiplicative Schwarzmethod and its application to nonlinear acoustic-structure interaction, M2AN Math. Model. Numer. Anal. 43 (2009), pp. 487–506. doi: 10.1051/m2an/2009010
- S. Giani and P. Houston, Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows, Numer. Math. Theory Methods Appl. 7 (2014), pp. 123–148. doi: 10.4208/nmtma.2014.1311nm
- R. Haferssas, P. Jolivet, and F. Nataf, An additive Schwarz method type theory for Lions's algorithm and a symmetrized optimized restricted additive Schwarz method, SIAM J. Sci. Comput. 39 (2017), pp. A1345–A1365. doi: 10.1137/16M1060066
- Y. Jiang and J. Zeng, A multiplicative Schwarz algorithm for the nonlinear complementarity problem with an M-function, Bull. Aust. Math. Soc. 82 (2010), pp. 353–366. doi: 10.1017/S0004972710000389
- T.V. Kolev, J. Xu, and Y. Zhu, Multilevel preconditioners for reaction-diffusion problems with discontinuous coefficients, J. Sci. Comput. 67 (2016), pp. 324–350. doi: 10.1007/s10915-015-0083-7
- S. Li and X.-C. Cai, Convergence analysis of two-level space-time additive Schwarz method for parabolic equations, SIAM J. Numer. Anal. 53 (2015), pp. 2727–2751. doi: 10.1137/140993776
- L. Li, T.-Z. Huang, and X.-P. Liu, Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems, Comput. Math. Appl. 54 (2007), pp. 147–159. doi: 10.1016/j.camwa.2006.12.024
- S. Li, X. Shao, and X.-C. Cai, Multilevel space-time additive Schwarz methods for parabolic equation, SIAM J. Sci. Comput. 40 (2018), pp. A3012–A3037. doi: 10.1137/17M113808X
- L. Marcinkowski, T. Rahman, A. Loneland, and J. Valdman, Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems, BIT Numer. Math. 56 (2016), pp. 967–993. doi: 10.1007/s10543-015-0581-x
- A.M. Matsokin and S.V. Nepomnyaschikh, A Schwarz alternating method in a subspace, Soviet Math.29 (1985), pp. 78–84.
- G. Migliorati and A. Quarteroni, Multilevel Schwarz methods for elliptic partial differential equations, Comput. Methods Appl. Mech. Eng. 200 (2011), pp. 2282–2296. doi: 10.1016/j.cma.2011.03.017
- R. Nabben, Comparisons between multiplicative and additive Schwarz iterations in domain decomposition methods, Numer. Math. 95 (2003), pp. 145–162. doi: 10.1007/s00211-002-0444-7
- L.F. Pavarino and S. Scacchi, Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system, SIAM J. Sci. Comput. 31 (2008), pp. 420–443. doi: 10.1137/070706148
- H. Rui, Multiplicative Schwarz methods for parabolic problems, Appl. Math. Comput. 136 (2003), pp. 593–610. doi: 10.1016/S0096-3003(02)00085-1
- M. Sarkis and D.B. Szyld, Optimal left and right additive Schwarz preconditioning for minimal residual methods with Euclidean and energy norms, Comput. Methods Appl. Mech. Eng. 196 (2007), pp. 1612–1621. doi: 10.1016/j.cma.2006.03.027
- B. Smith, P. Bjorstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.
- A. Toselli and O.B. Widlund, Domain Decomposition Methods – Algorithms and Theory, Springer-Verlag, Berlin, Heidelberg, 2005.
- J.-P Wang, Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems, SIAM J. Numer. Anal. 30 (1993), pp. 275–285. doi: 10.1137/0730013
- J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), pp. 581–613. doi: 10.1137/1034116
- D. Yang, Non-iterative parallel Schwarz algorithms based on overlapping domain decomposition for parabolic partial differential equations, Math. Comp. 86 (2017), pp. 2687–2718. doi: 10.1090/mcom/3102
- H. Yang, Q. Li, and H. Xu, A multiplicative Schwarz iteration scheme for solving the linear complementarity problem with an H-matrix, Linear Algebra Appl. 430 (2009), pp. 1085–1098. doi: 10.1016/j.laa.2008.10.005
- X.-J. Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), pp. 521–539. doi: 10.1007/BF01385873