Cai, Xiao-Chuan Additive Schwarz algorithms for parabolic convection-diffusion equations. (English) Zbl 0737.65078 Numer. Math. 60, No. 1, 41-62 (1991). See the preview in Zbl 0723.65075. Cited in 83 Documents MSC: 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling 35K15 Initial value problems for second-order parabolic equations Keywords:finite element; parabolic convection-diffusion equations; domain decomposition; convergence rates; minimal residual method; numerical result; Schwarz’s alternating method; finite elements; preconditioning Citations:Zbl 0723.65075 PDFBibTeX XMLCite \textit{X.-C. Cai}, Numer. Math. 60, No. 1, 41--62 (1991; Zbl 0737.65078) Full Text: DOI EuDML References: [1] Babu?ka, I. (1972): The mathematical foundations of the finite element method, with applications to partial differential equations, A.D. Aziz ed. Academic Press, New York London · Zbl 0268.65052 [2] Bj?rstad, P.E., Widlund, O.B. (1986): Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal.23, 1093-1120 · Zbl 0615.65113 [3] Bramble, J.H. (1966): A second order finite difference analogue of the first biharmonic boundary value problem. Numer. Math.9, 236-249 · Zbl 0154.41105 · doi:10.1007/BF02162087 [4] Bramble, J.H., Pasciak, J.E., Schatz, A.H. (1986): The construction of preconditioners for elliptic problems by substructuring, I. Math. Comput.47, 103-134 · Zbl 0615.65112 · doi:10.1090/S0025-5718-1986-0842125-3 [5] Bramble, J.H., Pasciak, J.E., Schatz, A.H. (1987): The construction of preconditioners for elliptic problems by substructuring, II. Math. Comput.49, 1-16 · Zbl 0623.65118 · doi:10.1090/S0025-5718-1987-0890250-4 [6] Bramble, J.H., Pasciak, J.E., Schatz, A.H. (1988): The construction of preconditioners for elliptic problems by substructuring, III. Math. Comput.51, 415-430 · Zbl 0701.65070 [7] Bramble, J.H., Pasciak, J.E., Shatz, A.H. (1989): The construction of preconditoners for elliptic problems by substructuring, IV. Math. Comput.53, 1-24 · Zbl 0668.65082 [8] Cai, X.-C. (1989): Some domain decomposition algorithms for nonselfadjoint elliptic and parabolic partial differential equations. Ph.D. thesis, Courant Institute [9] Cai, X.-C. (1990): An additive Schwarz algorithm for nonselfadjoint elliptic equations. In: T. Chan, R. Glowinski, J. P?riaux, O. Widlund, eds., Third International, Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia · Zbl 0701.65072 [10] Cai, X.-C., Widlund, O.B. (1990): Multiplicative Schwarz algorithms for nonsymmetric and indefinite elliptic and parabolic problems. Tech. Rep. CCS-90-7, Center for Comput. Sci., Univ. of Kentucky [11] Dawson, C., Du, Q. Dupont, T.F., (1989): A finite difference domain decomposition algorithm for numerical solution of the heat equation. Tech. Rep., 89-09, Univ. of Chicago [12] Dryja, M. (1989): An additive Schwarz algorithm for two-and three-dimensional finite element elliptic problems. In: T. Chan, R. Glowinski, G.A. Meurant, J. P?riaux, O. Widlund, eds., Domain Decomposition Methods for Partial Differential Equations II. Philadelphia · Zbl 0681.65075 [13] Dryja, M., Widlund, O.B. (1987): An additive variant of the Schwarz alternating method for the case of many subregions. Tech. Rep. 339, Dept. of Comp. Sci., Courant Institute [14] Dryja, M., Widlund, O.B. (1989): Some domain decomposition algorithms for elliptic problems. In: L. Hayes, D. Kincaid, eds., Iterative Methods for Large Linear Systems. Academic Press, San Diego California [15] Dryja, M., Widlund, O.B. (1990): Towards a unified theory of domain decomposition algorithms for elliptic problems. In: T. Chan, R. Glowinski, J. P?riaux, O. Widlund, eds., Third International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia · Zbl 0719.65084 [16] Eisenstat, S.C., Elman, H.C., Schultz, M.H. (1983): Variational iterative methods for nonsymmetric system of linear equations. SIAM J. Numer. Anal.20, 345-357 · Zbl 0524.65019 · doi:10.1137/0720023 [17] Ewing, R.E., Lazarov, R.D., Pasciak, J.E., Vassilevski, P.S. (1989): Finite element methods for parabolic problems with time steps variable in space. Tech. Rep. # 1989-05, Inst. for Sci. Comp., Univ. of Wyoming [18] Grisvard, P. (1985): Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA · Zbl 0695.35060 [19] Johnson, C. (1987): Numerical Solution of Partial Differential Equation by the Finite Element Method. Cambridge University Press, Cambridge · Zbl 0628.65098 [20] Kuznetsov, Yu.A. (1989): Domain decomposition methods for time-dependent problems. Preprint · Zbl 0705.65072 [21] Lions, P.L. (1988): On the Schwarz alternating method. I. In: R. Glowinski, G. H. Golub, G. A. Meurant, J. P?riaux, eds., Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia · Zbl 0658.65090 [22] Ne?as, J. (1964): Sur la Coercivit? des Formes Sesquilin?aires, Elliptiques. Rev. Roumaine Math. Pures Appl.9, 47-69 · Zbl 0196.40701 [23] Saad, Y., Schultz, M.H. (1986): GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comp.7, 865-869 · Zbl 0599.65018 [24] Yserentant, H. (1986): On the multi-level splitting of finite element spaces. Numer Math.49, 379-412 · Zbl 0608.65065 · doi:10.1007/BF01389538 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.