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Domain decomposition algorithms

Published online by Cambridge University Press:  07 November 2008

Tony F. Chan
Affiliation:
Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90024, USA Email: [email protected].
Tarek P. Mathew
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036, USA Email: [email protected].

Abstract

Domain decomposition refers to divide and conquer techniques for solving partial differential equations by iteratively solving subproblems defined on smaller subdomains. The principal advantages include enhancement of parallelism and localized treatment of complex and irregular geometries, singularities and anomalous regions. Additionally, domain decomposition can sometimes reduce the computational complexity of the underlying solution method.

In this article, we survey iterative domain decomposition techniques that have been developed in recent years for solving several kinds of partial differential equations, including elliptic, parabolic, and differential systems such as the Stokes problem and mixed formulations of elliptic problems. We focus on describing the salient features of the algorithms and describe them using easy to understand matrix notation. In the case of elliptic problems, we also provide an introduction to the convergence theory, which requires some knowledge of finite element spaces and elementary functional analysis.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

Agoshkov, V.I. (1988), ‘Poincaré-Steklov operators and domain decomposition methods in finite dimensional spaces’, in First Int. Symp. Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds), SIAM (Philadelphia, PA).Google Scholar
Agoshkov, V.I. and Lebedev, V.I. (1985), ‘The Poincaré–Steklov operators and the domain decomposition methods in variational problems’, in Computational Processes and Systems Nauka (Moscow) 173227. In Russian.Google Scholar
Arnold, D.N. and Brezzi, F. (1985), ‘Mixed and nonconforming finite element methods: Implementation, post processing and error estimates’, Math. Model. Numer. Anal. 19, 732.CrossRefGoogle Scholar
Ashby, S.F., Saylor, P.E. and Scroggs, J.S. (1992), ‘Physically motivated domain decomposition preconditioners’, in Proc. Second Copper Mountain Conf. on Iterative Methods, Vol. 1, Comput. Math. Group, University of Colorado at Denver.Google Scholar
Astrakhantsev, G.P. (1978), ‘Method of fictitious domains for a second-order elliptic equation with natural boundary conditions’, USSR Comput. Math. Math. Phys. 18, 114121.CrossRefGoogle Scholar
Atamian, C., Dinh, Q.V., Glowinski, R., He, J. and Périaux, J. (1991), ‘Control approach to fictitious-domain methods application to fluid dynamics and electromagnetics’, in Fourth Int. Symp. Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Axelsson, O. and Vassilevski, P. (1990), ‘Algebraic multilevel preconditioning methods, II’, SIAM J. Numer. Anal. 27, 15691590.CrossRefGoogle Scholar
Babuška, I. (1957), ‘Über Schwarzsche Algorithmen in partielle Differentialgleichungen der mathematischen Physik’, ZAMM 37 (7/8), 243245.CrossRefGoogle Scholar
Babuška, I., Craig, A., Mandel, J. and Pitkäranta, J. (1991), ‘Efficient preconditioning for the p-version finite element method in two dimensions’, SIAM J. Numer. Anal. 28(3), 624661.CrossRefGoogle Scholar
Bank, R.E., Dupont, T.F. and Yserentant, H. (1988), ‘The hierarchical basis multigrid method’, Numer. Math. 52, 427458.CrossRefGoogle Scholar
Berger, M. and Bokhari, S. (1987), ‘A partitioning strategy for nonuniform problems on multiprocessors’, IEEE Trans. Comput. 36, 570580.CrossRefGoogle Scholar
Bernardi, C. and Maday, Y. (1992), ‘Approximations spectral de problèmes aux limites elliptiques’, in Mathematiques et Applications, Vol. 10, Springer (Paris).Google Scholar
Bernardi, C., Maday, Y. and Patera, A. (1989), ‘A new nonconforming approach to domain decomposition: the mortar element method’, Nonlinear Partial Differential Equations and their Applications (Brezis, H. and Lions, J.L., eds), Pitman (London).Google Scholar
Bernardi, C., Debit, N. and Maday, Y. (1990), ‘Coupling finite element and spectral methods’, Math. Comput. 54, 2139.CrossRefGoogle Scholar
Bjørstad, P.E. and Skogen, M. (1992), ‘Domain decomposition algorithms of Schwarz type, designed for massively parallel computers’, Fifth Int. Symp. Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Bjørstad, P.E. and Widlund, O.B. (1984), ‘Solving elliptic problems on regions partitioned into substructures’, Elliptic Problem Solvers II (Birkhoff, G. and Schoenstadt, A., eds), Academic (London), 245256.CrossRefGoogle Scholar
Bjørstad, P.E. and Widlund, O.B. (1986), ‘Iterative methods for the solution of elliptic problems on regions partitioned into substructures’, SIAM J. Numer. Anal. 23 (6), 10931120.CrossRefGoogle Scholar
Bjørstad, P.E. and Widlund, O.B. (1989), ‘To overlap or not to overlap: A note on a domain decomposition method for elliptic problems’, SIAM J. Sci. Stat. Comput. 10 (5), 10531061.CrossRefGoogle Scholar
Börgers, C. (1989), ‘The Neumann–Dirichlet domain decomposition method with inexact solvers on the subdomains’, Numer. Math. 55, 123136.CrossRefGoogle Scholar
Börgers, C. and Widlund, O.B. (1990), ‘On finite element domain imbedding methods’, SIAM J. Numer. Anal. 27(4), 963978.CrossRefGoogle Scholar
Bornemann, F. and Yserentant, H. (1993), ‘A basic norm equivalence for the theory of multilevel methods’, Num. Math. 64, 455476.CrossRefGoogle Scholar
Bourgat, J.-F., Glowinski, R., Le Tallec, P. and Vidrascu, M. (1989), ‘Variational formulation and algorithm for trace operator in domain decomposition calculations’, in Second Int. Conf. on Domain Decomposition Methods (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Bramble, J.H. and Pasciak, J.E. (1988), ‘A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems’, Math. Comput. 50, 118.CrossRefGoogle Scholar
Bramble, J.H. and Xu, J. (1991), ‘Some estimates for a weighted L2 projection’, Math. Comput. 56, 463476.Google Scholar
Bramble, J.H., Ewing, R.E., Pasciak, J.E. and Schatz, A.H. (1988), ‘A preconditioning technique for the efficient solution of problems with local grid refinement’, Comput. Meth. Appl. Mech. Engrg 67, 149159.CrossRefGoogle Scholar
Bramble, J.H., Ewing, R.E., Parashkevov, R.R. and Pasciak, J.E. (1992), ‘Domain decomposition methods for problems with partial refinement’, SIAM J. Sci. Comput. 13(1), 397410.CrossRefGoogle Scholar
Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (1986a), ‘The construction of preconditioners for elliptic problems by substructuring, I’, Math. Comput. 47, 103134.CrossRefGoogle Scholar
Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (1986b), ‘An iterative method for elliptic problems on regions partitioned into substructures’, Math. Comput. 46(173), 361369.CrossRefGoogle Scholar
Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (1989), ‘The construction of preconditioners for elliptic problems by substructuring, IV’, Math. Comput. 53, 124.Google Scholar
Bramble, J.H., Pasciak, J.E. and Xu, J. (1990), ‘Parallel multilevel preconditionersMath. Comput. 55, 122.CrossRefGoogle Scholar
Bramble, J.H., Pasciak, J.E., Wang, J. and Xu, J. (1991), ‘Convergence estimates for product iterative methods with applications to domain decomposition’, Math. Comput. 57(195), 121.CrossRefGoogle Scholar
Brenner, S.C. (1993), ‘Two-level additive Schwarz preconditioners for nonconforming finite element methods’, in Proc. Seventh Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, to appear.Google Scholar
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer (Berlin).CrossRefGoogle Scholar
Buzbee, B.L., Dorr, F., George, J. and Golub, G. (1971), ‘The direct solution of the discrete poisson equation on irregular regions’, SIAM J. Numer. Anal. 11, 722736.CrossRefGoogle Scholar
Cai, X.-C. (1991), ‘Additive Schwarz algorithms for parabolic convection-diffusion equations’, Numer. Math. 60(1), 4161.CrossRefGoogle Scholar
Cai, X.-C. (1993), ‘Multiplicative Schwarz methods for parabolic problems’, SIAM J. Sci. Comput., to appear.Google Scholar
Cai, X.-C. and Widlund, O. (1992), ‘Domain decomposition algorithm for indefinite elliptic problems’, SIAM J. Sci. Comput. 13(1), 243258.CrossRefGoogle Scholar
Cai, X.-C. and Widlund, O. (1993), ‘Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems’, SIAM J. Numer. Anal. 30(4), 936952.CrossRefGoogle Scholar
Cai, X.-C., Gropp, W.D. and Keyes, D.E. (1992), ‘A comparison of some domain decomposition algorithms for nonsymmetric elliptic problems’, in Fifth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Canuto, C. and Funaro, D. (1988), ‘The Schwarz algorithm for spectral methods’, SIAM J. Numer. Anal. 25(1), 2440.CrossRefGoogle Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988), Spectral Methods in Fluid Dynamics, Springer (Berlin).CrossRefGoogle Scholar
Chan, T.F. (1987), ‘Analysis of preconditioners for domain decomposition’, SIAM J. Numer. Anal. 24(2), 382390.CrossRefGoogle Scholar
Chan, T.F. and Shao, J. (1993), ‘Optimal coarse grid size in domain decomposition’, Technical Report 93–24, UCLA CAM Report, Los Angeles, CA 90024–1555.Google Scholar
Chan, T.F. and Goovaerts, D. (1992), ‘On the relationship between overlapping and nonoverlapping domain decomposition methods’, SIAM J. Matrix Anal. Appl. 13(2), 663.CrossRefGoogle Scholar
Chan, T.F. and Hou, T.Y. (1991), ‘Eigendecompositions of domain decomposition interface operators for constant coefficient elliptic problems’, SIAM J. Sci. Comput. 12(6), 14711479.CrossRefGoogle Scholar
Chan, T.F. and Keyes, D.F. (1990), ‘Interface preconditioning for domain decomposed convection diffusion operators’, in Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Chan, T.F. and Mathew, T.P. (1992), ‘The interface probing technique in domain decomposition’, SIAM J. Matrix Anal. Appl. 13(1), 212238.CrossRefGoogle Scholar
Chan, T.F. and Mathew, T. (1993), ‘Domain decomposition preconditioners for convection diffusion problems’, in Domain Decomposition Methods for Partial Differential Equations (Quarteroni, A., ed.), American Mathematical Society (Providence, RI), to appear.Google Scholar
Chan, T.F. and Resasco, D.C. (1985), ‘A survey of preconditioners for domain decomposition’, Technical Report /DCS/RR-414, Yale University.Google Scholar
Chan, T.F. and Resasco, D.C. (1987), ‘Analysis of domain decomposition preconditioners on irregular regions’, in Advances in Computer Methods for Partial Differential Equations – VI (Vichnevetsky, R. and Stepleman, R., eds), IMACS, 317322.Google Scholar
Chan, T.F., Hou, T.Y. and Lions, P.L. (1991a), ‘Geometry related convergence results for domain decomposition algorithms’, SIAM J. Numer. Anal. 28(2), 378.CrossRefGoogle Scholar
Chan, T.F., Weinan, E. and Sun, J. (1991b), ‘Domain decomposition interface preconditioners for fourth order elliptic problems’, Appl. Numer. Math. 8, 317331.CrossRefGoogle Scholar
Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds (1992a), Fifth Conf. on Domain Decomposition Methods for Partial Differential Equations, SIAM (Philadelphia, PA).Google Scholar
Chan, T.F., Mathew, T.P. and Shao, J.-P. (1992b), ‘Efficient variants of the vertex space domain decomposition algorithm’, Technical Report CAM 92-07, Department of Mathematics, UCLA, to appear in SIAM J. Sci. Comput.Google Scholar
Chan, T.F., Glowinski, R., Périaux, J. and Widlund, O., eds (1989), Second Int. Conf. on Domain Decomposition Methods, SIAM (Philadelphia, PA).Google Scholar
Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds (1990), Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM (Philadelphia, PA).Google Scholar
Chin, R.C.Y., Hedstrom, G.W., McGraw, J.R. and Howes, F.A. (1986), ‘Parallel computation of multiple scale problems’, in New Computing Environments: Parallel, Vector and Systolic (Wouk, A., ed.) SIAM (Philadelphia, PA).Google Scholar
Courant, R. and Hilbert, D. (1962), Methods of Mathematical Physics, Vol. 2, Interscience (New York).Google Scholar
Cowsar, L.C. (1993), ‘Dual variable Schwarz methods for mixed finite elements’, Technical Report TR93-09, Department of Mathematical Sciences, Rice University.Google Scholar
Cowsar, L.C. and Wheeler, M.F. (1991), ‘Parallel domain decomposition method for mixed finite elements for elliptic partial differential equations’, in Fourth Int. Symp.on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Cowsar, L.C., Mandel, J. and Wheeler, M.F. (1993), ‘Balancing domain decomposition for mixed finite elements’, Technical Report TR93-08, Department of Mathematical Sciences, Rice University.Google Scholar
Dawson, C.N. and Du, Q. (1991), ‘A domain decomposition method for parabolic equations based on finite elements’, in Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A.Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Dawson, C., Du, Q. and Dupont, T.F., (1991), ‘A finite difference domain decomposition algorithm for numerical solution of the heat equation’, Math. Comput. 57, 195.CrossRefGoogle Scholar
De Roeck, Y.-H. (1989), ‘A local preconditioner in a domain-decomposed method’, Technical Report TR89/10, Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, Toulouse, France.Google Scholar
De Roeck, Y.-H. and Le Tallec, P. (1991), ‘Analysis and test of a local domain decomposition preconditioner’, in Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y., Meurant, G., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Despres, B. (1991), ‘Methodes de decomposition de domaines pour les problemes de propagation d'ondes en regime harmoniques’, PhD thesis, University of Paris IX, Dauphine.Google Scholar
Dinh, Q., Glowinski, R. and Périaux, J. (1984), ‘Solving elliptic problems by domain decomposition methods with applications’, in Elliptic Problem Solvers II (Birkhoff, G. and Schoenstadt, A., eds), Academic (New York), 395426.Google Scholar
Dryja, M. (1982), ‘A capacitance matrix method for Dirichlet problem on polygon region’, Numer. Math. 39, 5164.CrossRefGoogle Scholar
Dryja, M. (1984), ‘A finite element-capacitance method for elliptic problems on regions partitioned into subregions’, Numer. Math. 44, 153168.CrossRefGoogle Scholar
Dryja, M. (1988), ‘A method of domain decomposition for 3-D finite element problems’, in First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds) SIAM (Philadelphia, PA).Google Scholar
Dryja, M. (1989), ‘An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems’, in Second Int. Conf. on Domain Decomposition Methods (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Dryja, M. (1991), ‘Substructuring methods for parabolic problems’, in Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Dryja, M. and Widlund, O.B. (1987), ‘An additive variant of the Schwarz alternating method for the case of many subregions’, Technical Report 339, also Ultra-computer Note 131, Department of Computer Science, Courant Institute.Google Scholar
Dryja, M. and Widlund, O.B. (1989a), ‘On the optimality of an additive iterative refinement method’, in Proc. Fourth Copper Mountain Conf. on Multigrid Methods, SIAM (Philadelphia, PA) 161170.Google Scholar
Dryja, M. and Widlund, O.B. (1989b), ‘Some domain decomposition algorithms for elliptic problems’, in Iterative Methods for Large Linear Systems (Hayes, L. and Kincaid, D., eds), Academic (San Diego, CA), 273291.Google Scholar
Dryja, M. and Widlund, O.B. (1990), ‘Towards a unified theory of domain decomposition algorithms for elliptic problems’, in Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM(Philadelphia, PA).Google Scholar
Dryja, M. and Widlund, O.B. (1992a), ‘Additive Schwarz methods for elliptic finite element problems in three dimensions’, in Fifth Conf.on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Dryja, M. and Widlund, O.B. (1992b), ‘Domain decomposition algorithms with small overlap’, Technical Report 606, Department of Computer Science, Courant Institute, to appear in SIAM J. Sci. Comput.Google Scholar
Dryja, M. and Widlund, O.B. (1993a), ‘Schwarz methods of Neumann–Neumann type for three-dimensional elliptic finite element problems’, Technical Report 626, Department of Computer Science, Courant Institute.Google Scholar
Dryja, M. and Widlund, O.B. (1993b), ‘Some recent results on Schwarz type domain decomposition algorithms’, in Sixth Conf. on Domain Decomposition Methods for Partial Differential Equations (Quarteroni, A., ed.), American Mathematical Society (Providence, RI), toappear. Technical Report 615, Department of Computer Science, Courant Institute.CrossRefGoogle Scholar
Dryja, M., Smith, B.F. and Widlund, O.B. (1993), ‘Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions’, Technical Report 638, Department of Computer Science, Courant Institute. SIAM J. Numer. Anal., submitted.Google Scholar
Ernst, O. and Golub, G. (1992), ‘A domain decomposition approach to solving the helmholtz equation with a radiation boundary condition’, Technical Report 92-08, Stanford University, Computer Science Department, Numerical Analysis Project, Stanford, CA 94305.Google Scholar
Ewing, R.E. and Wang, J. (1991), ‘Analysis of the Schwarz algorithm for mixed finite element methods’, RAIRO, Math. Modell. Numer. Anal. 26(6), 739756.CrossRefGoogle Scholar
Farhat, C. and Lesoinne, M. (1993), ‘Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics’, Int. J. Numer. Meth. Engrg 36, 745764.CrossRefGoogle Scholar
Farhat, C. and Roux, F.X. (1992), ‘An unconventional domain decomposition method for an efficient parallel solution of large scale finite element systems’, SIAM J. Sci. Comput. 13, 379396.CrossRefGoogle Scholar
Finogenov, S.A. and Kuznetsov, Y.A. (1988), ‘Two-stage fictitious components method for solving the Dirichlet boundary value problem’, Sov. J. Numer. Anal. Math. Modell. 3(4), 301323.CrossRefGoogle Scholar
Fischer, P.F. and Rønquist, E.M. (1993), ‘Spectral element methods for large scale parallel Navier–Stokes calculations’, in Second Int. Conf. on Spectral and High Order Methods for PDE's. Proc. ICOSAHOM 92, a conference held in Montpellier, France, June 1992. To appear in Comput. Meth. Appl. Mech. Engrg.Google Scholar
Fortin, M. and Aboulaich, R. (1988), ‘Schwarz's decomposition method for incompressible flow problems’, in First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds), SIAM(Philadelphia, PA).Google Scholar
Fox, G. (1988), ‘A review of automatic load balancing and decomposition methods for the hypercube’, in Numerical Algorithms for Modern Parallel Computers (Schultz, M., ed.), Springer (Berlin), 6376.CrossRefGoogle Scholar
Freund, R.W., Golub, G.H. and Nachtigal, N. (1992), ‘Iterative solution of linear systems’, Acta Numerica, Cambridge University Press (Cambridge), 57100.Google Scholar
Funaro, D., Quarteroni, A. and Zanolli, P. (1988), ‘An iterative procedure with interface relaxation for domain decomposition methods’, SIAM J. Numer. Anal. 25, 12131236.CrossRefGoogle Scholar
Garbey, M. (1992), ‘Domain decomposition to solve layers and singular perturbation problems’, in Fifth Conf. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Gastaldi, F., Quarteroni, A. and Sacchi-Landriani, G. (1990), ‘On the coupling of two-dimensional hyperbolic and elliptic equations: analytical and numerical approach’, in Domain Decomposition Methods for Partial Differential Equations (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA) 2363.Google Scholar
Girault, V. and Raviart, P.-A. (1986), Finite Element Approximation of the Navier–Stokes Equations: Theory and Algorithms, Springer (Berlin).CrossRefGoogle Scholar
Glowinski, R. and Wheeler, M.F. (1988), ‘Domain decomposition and mixed finite element methods for elliptic problems’, in First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds), SIAM (Philadelphia, PA).Google Scholar
Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds (1988), Proc. First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM (Philadelphia, PA).Google Scholar
Glowinski, R., Kinton, W. and Wheeler, M.F. (1990a), ‘Acceleration of domain decomposition algorithms for mixed finite elements by multilevel methods’, in Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds (1991), Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM (Philadelphia, PA).Google Scholar
Glowinski, R., Périaux, J. and Terrasson, G. (1990b), ‘On the coupling of viscous and inviscid models for compressible fluid flows via domain decomposition’, in Domain Decomposition Methods for Partial Differential Equations (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Golub, G. and Mayers, D. (1984), ‘The use of preconditioning over irregular regions’, in Computing Methods in Applied Sciences and Engineering, VI (Glowinski, R. and Lions, J.L., eds), North-Holland (Amsterdam, New York, Oxford) 314.Google Scholar
Goovaerts, D. (1989), ‘Domain decomposition methods for ellipticpartial differential equations’, PhD thesis, Department of Computer Science, Catholic University of Leuven.Google Scholar
Goovaerts, D., Chan, T. and Piessens, R. (1991), ‘The eigenvalue spectrum of domain decomposed preconditioners’, Appl. Numer. Math. 8, 389410.CrossRefGoogle Scholar
Griebel, M. (1991), ‘Multilevel algorithms considered as iterative methods on indefinite systems’, Inst. für Informatik, Tech. Univ. Munchen, SFB 342/29/91A.Google Scholar
Griebel, M. and Oswald, P. (1993), ‘Remarks on the abstract theory of additive and multiplicative Schwarz algorithms’, Inst. für Informatik, Tech. Univ. Munchen, SFB 342/6/93A.Google Scholar
Gropp, W.D. (1992), ‘Parallel computing and domain decomposition’, in Fifth Conf. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Gropp, W. and Keyes, D. (1993), ‘Domain decomposition as a mechanism for using asymptotic methods’, in Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (Kaper, H. and Garbey, M., eds), Vol. 384, NATO ASI Series C, 93106.CrossRefGoogle Scholar
Gropp, W. and Smith, B. F. (1992), ‘Experiences with domain decomposition in three dimensions: overlapping Schwarz methods’, Mathematics and Computer Science Division, Argonne National Laboratory, to appear in Proc. Sixth Int. Symp. on Domain Decomposition Methods.Google Scholar
Hackbusch, W. (1993), Iterative Methods for Large Sparse Linear Systems, Springer (Heidelberg).Google Scholar
Hedstrom, G.W. and Howes, F.A. (1990), ‘Domain decomposition for a boundary value problem with a shock layer’, in Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
D'Hennezel, F. (1992), ‘Domain decomposition method with non-symmetric interface operator‘, Fifth Conf. on Domain Decomposition Methods for Partial Differential Equations (Chan, Tony F., Keyes, David E., Meurant, Gérard A., Scroggs, Jeffrey S. and Voigt, Robert G., eds), SIAM (Philadelphia, PA).Google Scholar
Hoffmann, K.-H. and Zou, J. (1992)’, Solution of biharmonic problems by the domain decomposition method‘, Technical Report No. 387, DFG-SPP, Technical University of Munich.Google Scholar
Kang, L. S. (1987), Parallel Algorithms and Domain Decomposition, Wuhan University Press (China). In Chinese.CrossRefGoogle Scholar
Kernighan, B. and Lin, S. (1970), ‘An efficient heuristic procedure for partitioning graphs’, Bell Systems Tech. J. 29, 291307.CrossRefGoogle Scholar
Keyes, D.E. and Gropp, W.D. (1987), ‘A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation’, SIAM J. Sci. Comput. 8 (2), s166s202.CrossRefGoogle Scholar
Keyes, D.E. and Gropp, W.D. (1989), ‘Domain decomposition with local mesh refinement’, Technical Report YALEU/DCS/RR-726, Yale University.Google Scholar
Kron, G. (1953), ‘A set of principles to interconnect the solutions of physical systems’, J. Appl. Phys. 24(8), 965.CrossRefGoogle Scholar
Kuznetsov, Y.A. (1988), ‘New algorithms for approximate realization of implicit difference schemes’, Sov. J. Numer. Anal. Math. Modell. 3, 99114.CrossRefGoogle Scholar
Kuznetsov, Y.A. (1990), ‘Domain decomposition methods for unsteady convection-diffusion problems’, in IXth Int. Conf. on Computing Methods in Applied Science and Engineering (Glowinski, R. and Lions, J.L., eds), INRIA (Paris) 327344.Google Scholar
Kuznetsov, Y.A. (1991), ‘Overlapping domain decomposition methods for feproblems with elliptic singular perturbed operators’, in Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Laevsky, Yu. M. (1992), ‘Direct domain decomposition method for solving parabolic equations’, Preprint no. 940, Novosibirsk, Computing Center Siberian Branch Academy of Sciences. In Russian.Google Scholar
Laevsky, Yu. M. (1993), ‘On the domain decomposition method for parabolic problems’, Bull. Novosibirsk Comput. Center 1, 4162.Google Scholar
Le Tallec, P. (1994), ‘Domain decomposition methods in computational mechanics’, J. Comput. Mech. Adv., to appear.Google Scholar
Le Tallec, P., De Roeck, Y.-H. and Vidrascu, M. (1991), ‘Domain-decomposition methods for large linearly elliptic three-dimensional problems’, J. Comput. Appl. Math. 34.CrossRefGoogle Scholar
Lebedev, V.I. (1986), Composition Methods, USSR Academy of Sciences, Moscow. In Russian.Google Scholar
Lions, P.L. (1988), ‘On the Schwarz alternating method. I.’, in First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds), SIAM (Philadelphia, PA).Google Scholar
Lions, P.L. (1989), ‘On the Schwarz alternating method. II, in Second Int. Conf. on Domain Decomposition Methods (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Lions, J.-L. and Magenes, E. (1972), Nonhomogeneous Boundary Value Problems and Applications, Vol. I, Springer (New York, Heidelberg, Berlin).Google Scholar
Tao, Lu, Shih, T. and Liem, C. (1992), Domain Decomposition Methods: New Numerical Techniques for Solving PDE, Science Publishers (Beijing, China).Google Scholar
Maday, Y. and Patera, A.T. (1989), ‘Spectral element methods for the Navier–Stokes equations’, in State of the Art Surveys in Computational Mechanics (Noor, A.K. and Oden, J.T., eds), ASME (NewYork).Google Scholar
Mandel, J. (1989a), ‘Efficient domain decomposition preconditioning for the p-version finite element method in three dimensions’, Technical Report, Computational Mathematics Group, University of Colorado at Denver.Google Scholar
Mandel, J. (1989b), ‘Two-level domain decomposition preconditioning for the p-version finite element version in three dimensions’, Int. J. Numer. Meth. Engrg, 29, 10951108.CrossRefGoogle Scholar
Mandel, J. (1990), ‘Hierarchical preconditioning and partial orthogonalization for the p-version finite element method’, in Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Mandel, J. (1992), ‘Balancing domain decomposition’, Commun. Numer. Meth. Engrg 9, 233241.CrossRefGoogle Scholar
Mandel, J. and Brezina, M. (1992), ‘Balancing domain decomposition: Theory and computations in two and three dimensions’, Technical Report, Computational Mathematics Group, University of Colorado at Denver.Google Scholar
Mandel, J. and McCormick, S. (1989), ‘Iterative solution of elliptic equations with refinement: The model multi-level case’, in Second Int. Conf, on Domain Decomposition Methods (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Marchuk, G.I., Kuznetsov, Y.A. and Matsokin, A.M. (1986), ‘Fictitious domain and domain decomposition methods’, Sov. J. Numer. Anal. Math. Modell. 1, 361.CrossRefGoogle Scholar
Marini, L.D. and Quarteroni, A. (1989), ‘A relaxation procedure for domain decomposition methods using finite elements’, Numer. Math. 56, 575598.CrossRefGoogle Scholar
Mathew, T.P. (1989), ‘Domain decomposition and iterative refinement methods for mixed finite element discretisations of elliptic problems’, PhD thesis, Courant Institute of Mathematical Sciences. Technical Report 463, Department of Computer Science, Courant Institute.Google Scholar
Mathew, T.P. (1993a), ‘Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results’, Num. Math. 65(4), 445468.CrossRefGoogle Scholar
Mathew, T.P. (1993b), ‘Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part II: Theory’, Num. Math. 65(4), 469492.CrossRefGoogle Scholar
Matsokin, A.M. and Nepomnyaschikh, S.V. (1985), ‘A Schwarz alternating method in a subspace’, Sov. Math. 29(10), 7884.Google Scholar
McCormick, S. (1984), ‘Fast adaptive composite grid (FAC) methods’, in Defect Correction Methods: Theory and Applications (Böhmer, K. and Stetter, H.J., eds), Computing Supplement 5, Springe (Wien), 115121.CrossRefGoogle Scholar
McCormick, S.F. (1989), Multilevel Adaptive Methods for Partial Differential Equations, SIAM (Philadelphia, PA).CrossRefGoogle Scholar
Meddahi, S. (1993), ‘The Schwarz algorithm for a Raviart–Thomas mixed method’, preprint, to appear.Google Scholar
Meurant, G.A. (1991), ‘Numerical experiments with a domain decomposition method for parabolic problems on parallel computers’, in Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Miller, K. (1965), ‘Numerical analogs of the Schwarz alternating procedure’, Numer. Math. 7, 91103.CrossRefGoogle Scholar
Morchoisne, Y. (1984), ‘Inhomogeneous flow calculations by spectral methods: Mono-domain and multi-domain techniques’, in Spectral Methods for Partial Differential Equations (Voigt, R.G., Gottlieb, D. and Hussaini, M.Y., eds), SIAM-CBMS, 181208.Google Scholar
Morgenstern, D. (1956), ‘Begründung des alternierenden Verfahrens durch Orthogonalprojektion’, ZAMM 36, 78.CrossRefGoogle Scholar
Nataf, F. and Rogier, F. (1993), ‘Factorization of the advection-diffusion operator and domain decomposition method’, in Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (Kaper, H. and Garbey, M., eds), Vol. 384, NATO ASI Series C, 123133.CrossRefGoogle Scholar
Nečas, J. (1967), Les Méthodes Directes en Théorie des Équations Elliptiques, Academia (Prague).Google Scholar
Nepomnyaschikh, S.V. (1984), ‘On the application of the method of bordering for elliptic mixed boundary value problems and on the difference norms of . In Russian.Google Scholar
Nepomnyaschikh, S.V. (1986), ‘Domain decomposition and Schwarz methods in a subspace for the approximate solution of elliptic boundary value problems’, Computing Center of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk, USSR.Google Scholar
O'Leary, D.P. and Widlund, O.B. (1979), ‘Capacitance matrix methods for the helmholtz equation on general three-dimensional regions’, Math. Comput. 33, 849879.CrossRefGoogle Scholar
Ong, M. (1989), ‘Hierarchical basis preconditioners for second-order elliptic problems in three dimensions’, Technical Report 89–3, Department of Applied Maths, University of Washington, Seattle.Google Scholar
Oswald, P. (1991), ‘On discrete norm estimates related to multilevel preconditioners in the finite element method’, in Proc. Int. Conf. Theory of Functions, Varna 91, to appear.Google Scholar
Pahl, S. (1993), ‘Domain decomposition for the Stokes problem’, Master's thesis, University of Witwatersrand, Johannesburg.Google Scholar
Pavarino, L.F. (1992), Domain decomposition algorithms for the p-version finite element method for elliptic problems, PhD thesis, Courant Institute of Mathematical Sciences, Department of Mathematics.Google Scholar
Pavarino, L.F. (1993a), ‘Schwarz methods with local refinement for the p-version finite element method’, Technical Report TR93-01, Rice University, Department of Computational and Applied Mathematics, submitted to Numer. Math.Google Scholar
Pavarino, L.F. (1993b), ‘Some Schwarz algorithms for the spectral element method’, in Sixth Conf. on Domain Decomposition Methods for Partial Differential Equations (Quarteroni, A., ed.), American Mathematical Society (Providence, RI), to appear. Technical Report 614, Department of Computer Science, Courant Institute.Google Scholar
Pavarino, L.F. and Widlund, O.B. (1993), ‘Iterative substructuring methods for p-version finite elements in three dimensions’, Technical Report, Courant Institute of Mathematical Sciences, Department of Computer Science, to appear.Google Scholar
Phillips, T.N. (1992), ‘Pseudospectral domain decomposition techniques for the Navier–Stokes equations’, in Fifth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Pothen, A., Simon, H. and Liou, K. (1990), ‘Partitioning sparse matrices with eigenvector of graphs’, SIAM J. Math. Anal. Appl. 11(3), 430452.CrossRefGoogle Scholar
Proskurowski, W. and Vassilevski, P. (1992), ‘Preconditioning nonsymmetric and indefinite capacitance matrix problems in domain imbedding’, Technical Report, UCLA, CAM Report 92-48, University of California, Los Angeles. SIAM J. Sci. Comput., to appear.Google Scholar
Proskurowski, W. and Vassilevski, P. (1994), ‘Preconditioning capacitance matrix problems in domain imbedding’, SIAM J. Sci. Comput., to appear.CrossRefGoogle Scholar
Proskurowski, W. and Widlund, O.B. (1976), ‘On the numerical solution of helmholtz's equation by the capacitance matrix method’, Math. Comput. 30, 433468.Google Scholar
Przemieniecki, J.S. (1963), ‘Matrix structural analysis of substructures’, Amer. Inst. Aero. Astro. J. 1(1), 138147.Google Scholar
Quarteroni, A. (1989), ‘Domain decomposition algorithms for the Stokes equations’, in Second Int. Conf. on Domain Decomposition Methods (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Quarteroni, A., ed. (1993), Domain Decomposition Methods in Science and Engineering, American Mathematical Society (Providence, RI).Google Scholar
Quarteroni, A. and Valli, A. (1990), ‘Theory and applications of Steklov–Poincaré operators for boundary-value problems: the heterogeneous operator case’, in Proc. 4th Int. Conf. on Domain Decomposition Methods, Moscow (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Resasco, D.C. (1990), ‘Domain decomposition algorithms for elliptic partial differential equations’, PhD thesis, Department of Computer Science, Yale University.Google Scholar
Rusten, T. (1991), ‘Iterative methods for mixed finite element systems’, University of Oslo.Google Scholar
Rusten, T. and Winther, R. (1992), ‘A preconditioned iterative method for saddle point problems’, SIAM J. Matrix Anal. 13(3), 887.CrossRefGoogle Scholar
Sarkis, M. (1993), ‘Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients’, Technical Report 629, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University.Google Scholar
Scapini, F. (1990), ‘The alternating Schwarz method applied to some biharmonic variational inequalities’, Calcolo 27, 5772.CrossRefGoogle Scholar
Scapini, F. (1991), ‘A decomposition method for some biharmonic problems’, J. Comput. Math. 9, 291300.Google Scholar
Schwarz, H.A. (1890), Gesammelte Mathematische Abhandlungen, Vol. 2, Springer (Berlin) 133143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Vol. 15, 1870, 272–286.CrossRefGoogle Scholar
Scroggs, J.S. (1989), ‘A parallel algorithm for nonlinear convection diffusion equations’, in Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, SIAM (Philadelphia, PA).Google Scholar
Simon, H. (1991), ‘Partitioning of unstructured problems for parallel processing’, Comput. Sys. Engrg 2(2/3), 135148.CrossRefGoogle Scholar
Smith, B.F. (1990), ‘Domain decomposition algorithms for the partial differential equations of linear elasticity’, PhD thesis, Courant Institute of Mathematical Sciences. Technical Report 517, Department of Computer Science, Courant Institute.Google Scholar
Smith, B.F. (1991), ‘A domain decomposition algorithm for elliptic problems in three dimensions’, Numer. Math. 60(2), 219234.CrossRefGoogle Scholar
Smith, B.F. (1992), ‘An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems’, SIAM J. Sci. Comput. 13(1), 364378.CrossRefGoogle Scholar
Smith, B.F. (1993), ‘A parallel implementation of an iterative substructuring algorithm for problems in three dimensions’, SIAM J. Sci. Comput. 14(2), 406423.CrossRefGoogle Scholar
Smith, B.F. and Widlund, O.B. (1990), ‘A domain decomposition algorithm using a hierarchical basis’, SIAM J. Sci. Comput. 11(6), 12121220.CrossRefGoogle Scholar
Smith, B.F., Bjørstad, P. and Gropp, W.D. (1994), ‘Domain decomposition: algorithms, implementations and a little theory’, to appear.Google Scholar
Sobolev, S.L. (1936), ‘L'algorithme de Schwarz dans la théorie de l'elasticité’, C. R. (Dokl.) Acad. Sci. URSS IV((XIII) 6), 243246.Google Scholar
Sun, J. and Zou, J. (1991), ‘Ddm preconditioner for 4th order problems by using B-spline finite element method’, in Fourth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Tang, W. P. (1988), ‘Schwarz splitting and template operators’, Department of Computer Science, Stanford University.Google Scholar
Tong, C.H., Chan, T.F. and Kuo, C.C.J. (1991)’, A domain decomposition preconditioner based on a change to a multilevel nodal basis’, SIAM J. Sci. Comput. 12, 14861495.CrossRefGoogle Scholar
Tsui, W. (1991), ‘Domain decomposition of biharmonic and Navier–Stokes equations’, PhD thesis, Department of Mathematics, University of California, Los Angeles.Google Scholar
Wang, J. (1991), ‘Convergence analysis of Schwarz algorithms and multilevel decomposition iterative methods: Part I’, Proc. IMACS Int. Symp. on Iterative Methods in Linear Algebra (Beauwens, R. and De Groen, P., eds), (Belgium).Google Scholar
Wang, J. (1993), ‘Convergence analysis of schwarz algorithms and multilevel decomposition iterative methods: Part II’, SIAM J. Numer. Anal. 30(4), 953970.CrossRefGoogle Scholar
Wheeler, M.F. and Gonzalez, R. (1984), ‘Mixed finite element methods for petroleum reservoir engineering problems’, in Computing Methods in Applied Sciences and Engineering, VI (Glowinski, R. and Lions, J.L., eds), North-Holland (New York)639658.Google Scholar
Widlund, O.B. (1987), ‘An extension theorem for finite element spaces with three applications’, in Numerical Techniques in Continuum Mechanics: Notes on Numerical Fluid Mechanics, Vol. 16, (Hackbusch, W. and Witsch, K., eds), Friedr. Vieweg und Sohn (Braunschweig/Wiesbaden) 110122. Proc. Second GAMM Seminar, Kiel, January, 1986.CrossRefGoogle Scholar
Widlund, O.B. (1988), ‘Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane’, in First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (Glowinski, R., Golub, G.H., Meurant, G.A. and Périaux, J., eds), SIAM (Philadelphia, PA).Google Scholar
Widlund, O.B. (1989a), ‘Iterative solution of elliptic finite element problems on locally refined meshes’, in Finite Element Analysis in Fluids (Chung, T.J. and Karr, G.R., eds), University of Alabama in Huntsville Press (Huntsville, Alabama), 462467.Google Scholar
Widlund, O.B. (1989b), ‘Optimal iterative refinement methods’, in Second Int. Conf. on Domain Decomposition Methods (Chan, T., Glowinski, R., Périaux, J. and Widlund, O., eds), SIAM (Philadelphia, PA).Google Scholar
Williams, R. (1991), ‘Performance of dynamic load balancing algorithms for unstructured mesh calculations’, Concurrency 3, 457481.CrossRefGoogle Scholar
Xu, J. (1989), ‘Theory of multilevel methods’, PhD thesis, Cornell University.Google Scholar
Xu, J. (1992a), ‘Iterative methods by space decomposition and subspace correction’, SIAM Rev. 34, 581613.CrossRefGoogle Scholar
Xu, J. (1992b), ‘A new class of iterative methods for nonselfadjoint or indefinite problems’, SIAM J. Numer. Anal. 29(2), 303319.CrossRefGoogle Scholar
Xu, J. (1992c), ‘Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems’, Fifth Conf. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds) SIAM (Philadelphia, PA).Google Scholar
Xu, J. and Cai, X.-C. (1992), ‘A preconditioned GMRES method for nonsymmetric or indefinite problems’, Math. Comput. 59, 311319.CrossRefGoogle Scholar
Yserentant, H. (1986), ‘On the multi-level splitting of finite element spaces’, Numer. Math. 49, 379412.CrossRefGoogle Scholar
Zhang, X. (1991), ‘Studies in domain decomposition: Multilevel methods and the biharmonic Dirichlet problem’, PhD thesis, Courant Institute, New York University.Google Scholar
Zhang, X. (1992a), ‘Domain decomposition algorithms for the biharmonic Dirichlet problem’, in Fifth Conf. on Domain Decomposition Methods for Partial Differential Equations (Chan, T.F., Keyes, D.E., Meurant, G.A., Scroggs, J.S. and Voigt, R.G., eds), SIAM (Philadelphia, PA).Google Scholar
Zhang, X. (1992b), ‘Multilevel Schwarz methods’, Num. Math. 63(4), 521539.CrossRefGoogle Scholar
Zhang, X. (1992c), ‘Multilevel Schwarz methods for the biharmonic dirichlet problem’, Technical Report CS-TR2907 (UMIACS-TR-92-60), University of Maryland, Department of Computer Science, submittedto SIAM J. Sci. Comput.Google Scholar