Summary
We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.
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References
Bank, R.E., Dupont, T.F., Yserentant, H. (1988): The hierarchical basis multigrid method. Numer. Math.52, 427–458
Bjørstad, P.E., Moe, R., Skogen, M. (1990): Parallel domain decomposition and iterative refinement algorithms. In: W. Hackbusch, ed., Parallel Algorithms for PDEs. Proceedings of the 6th GAMM-Seminar held in Kiel, Germany, January 19–21, 1990. Vieweg, Braunschweig, Wiesbaden
Bjørstad, P.E., Skogen, M. (1992): Domain decomposition algorithms of Schwarz type, designed for massively parallel computers. In: T.F. Chan, D.E. Keyes, G.A. Meurang, J.S. Scroggs, R.G. Voigt, eds., Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA (to appear)
Bornemann, F., Yserentant, H. (1992): A basic norm equivalence for the theory of multilevel methods. Numer. Math. (in press)
Bramble, J.H., Pasciak, J.E. (1991): New estimates for multilever algorithms including the V-cycle. Tech. Report BNL-46730, Department of Applied Science, Brookhaven National Laboratory
Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J. (1991): Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput.57, 1–21
Bramble, J.H., Pasciak, J.E., Xu, J. (1990): Parallel multilever preconditioners. Math. Comput.55, 1–22
Dryja, M., Widlund, O.B. (1987): An additive variant of the Schwarz alternating method for the case of many subregions. Tech. Report 339, also Ultracomputer Note 131, Department of Computer Science, Courant Insitute
Dryja, M., Widlund, O.B. (1989): Some domain decomposition algorithms for alliptic problems. In: L. Hayes, D. Kincaid, eds., Iterative methods for large linear systems. Academic Press, San Diego, California, pp. 273–291. Proceeding of the Conference on Iterative Methods for Larger Linear Systems held in Austin, Texas, October 19–21, 1988, to celebrate the sixty-fifth birthday of David M. Young, Jr.
Dryja, M., Widlund, O.B. (1987): Multilevel additive methods for elliptic finite element problems. In: W. Hackbusch, ed., Parallel algorithms for partial differential equations. Proceedings of the Sixth GAMM-Seminar, Kiel, January 19–21, 1990. Vieweg, Braunschweig
Hardy, G.H., Littlewood, J.E., Pólya, G. (1934): Inequalities. Cambridge University Press, Cambridge
Hackbusch, W. (ed.) (1985): Multi-grid methods and applications. Springer, Berlin Heidelberg New York
McCormick, S.F. (ed) (1987): Multigrid methods. SIAM, Philadelphia, PA
Nepomnyaschikh, S.V. (1987): On the application of the method of bordering for elliptic mixed boundary value problems and on the difference norms ofW 1/22 (S). Tech. Report 106, Computing Center of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk [in Russian]
Nečas, J. (1967): Les méthodes directes in théorie des équations elliptiques. Academia, Prague
Oswald, P. (1992): On discrete norm estimates related to multilevel preconditioners in the finite element method. In: Proc. Int. Conf. Theory of Functions, Varna91 (to appear)
Widlund, O.B. (1987): An extension theorem for finite element spaces with three applications. In: W. Hackbusch, K. Witsch, eds., Numerical techniques in continuum mechanics. Notes on numerical fluid mechanics. vol. 16. Proceedings of the Second GAMM-Seminar, Kiel, January, 1986. Vieweg, Braunschweig Weisbaden, pp. 110–122
Widlund, O.B. (1993): Some Schwarz methods for symmetric and nonsymmetric elliptic problem. In: T.F. Chan, D.E. Keyes, G.A. Meurant, J.S. Scroggs, R.G. Voigt, eds., Fifth Conference on Domain Decomposition Methods for Partial Differential Equations (to appear)
Xu, J. (1989): Theory of Multilevel Methods. Ph.D. thesis, Cornell University
Xu, J. (1990): Iterative methods by space decomposition and subspace correction. Tech. report, Penn State University, University Park, PA (revised, 1992) (accepted by SIAM Rev.)
Yserentant, H. (1986): On the multi-level splitting of finite element spaces. Numer. Math.49, 412
Zhang, X. (1991): Multilevel additive Schwarz methods. Tech. Report 582, Courant Institute of Mathematical Sciences, Department of Computer Science
Zhang, X. (1991): Studies in domain decomposition: Multilevel methods and the biharmonic Dirichlet problem. PhD thesis, Courant Institute, New York University
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This work was supported in part by the National Science Foundation under Grants NSF-CCR-8903003 at Courant Institute of Mathematical Sciences, New York University and NSF-ASC-8958544 at Department of Computer Science, University of Maryland.