Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-20T02:12:52.264Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of two-level domain decomposition preconditioners based on aggregation

Published online by Cambridge University Press:  15 October 2004

Marzio Sala
Marzio Sala*
Affiliation:
CMCS/SB/EPFL, 1015 Lausanne, Switzerland. msala@sandia.gov.
Get access

Abstract

In this paper we present two-level overlapping domain decomposition preconditioners for the finite-element discretisation of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction is added. We present an algebraic way to define the coarse space, based on the concept of aggregation. This employs a (smoothed) aggregation technique and does not require the introduction of a coarse grid. We consider a set of assumptions on the coarse basis functions, to ensure bound for the resulting preconditioned system. These assumptions only involve geometrical quantities associated to the aggregates, namely their diameter and the overlap. A condition number which depends on the product of the relative overlap among the subdomains and the relative overlap among the aggregates is proved. Numerical experiments on a model problem are reported to illustrate the performance of the proposed preconditioners.

Keywords

Elliptic equationsdomain decompositionSchwarz methodsaggregation coarse corrections.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 5 , September 2004 , pp. 765 - 780
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Brezina, Robust iterative method on unstructured meshes. Ph.D. Thesis, University of Colorado at Denver (1997).
Brezina, M. and Vaněk, P., A black–box iterative solver based on a two–level Schwarz method. Computing 63 (1999) 233–263. CrossRef
Broeker, O., Grote, M.J., Mayer, C. and Reusken, A., Robust parallel smoothing for multigrid via sparse approximate inverses. SIAM J. Sci. Comput. 23 (2001) 1396–1417. CrossRef
Dryja, M. and Widlund, O.B., Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604–620. CrossRef
Dryja, M. and Widlund, O.B., Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48 (1995) 121–155. CrossRef
Formaggia, L., Scheinine, A. and Quarteroni, A., A numerical investigation of Schwarz domain decomposition techniques for elliptic problems on unstructured grids. Math. Comput. Simulations 44 (1994) 313–330. CrossRef
G.H. Golub and C.F. van Loan, Matrix Computations. The Johns Hopkins University Press, Baltimore, Maryland (1983).
L. Jenkins, T. Kelley, C.T. Miller and C.E. Kees, An aggregation-based domain decomposition preconditioner for groundwater flow. Technical Report TR00–13, Department of Mathematics, North Carolina State University (2000).
C.E. Kees, C.T. Miller, E.W. Jenkins and C.T. Kelley, Versatile multilevel Schwarz preconditioners for multiphase flow. Technical Report CRSC-TR01-32, Center for Research in Scientific Computation, North Carolina State University (2001).
Lasser, C. and Toselli, A., An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Technical Report 810, Dept. of Computer Science, Courant Institute (2000). Math. Comput. 72 (2003) 1215–1238. CrossRef
C. Lasser and A. Toselli, Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces. Technical Report TUM-M0109, Technische Universität München (2001).
W. Leontief, The structure of the American Economy. Oxford University Press, New York (1951).
Mandel, J. and Sekerka, B., A local convergence proof for the iterative aggregation method. Linear Algebra Appl. 51 (1983) 163–172. CrossRef
L. Paglieri, A. Scheinine, L. Formaggia and A. Quarteroni, Parallel conjugate gradient with Schwarz preconditioner applied to fluid dynamics problems, in Parallel Computational Fluid Dynamics, Algorithms and Results using Advanced Computer, Proceedings of Parallel CFD'96, P. Schiano et al., Eds. (1997) 21–30.
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994).
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999).
M. Sala and L. Formaggia, Parallel Schur and Schwarz based preconditioners and agglomeration coarse corrections for CFD problems. Technical Report 15, DMA-EPFL (2001).
M. Sala and L. Formaggia, Algebraic coarse grid operators for domain decomposition based preconditioners, in Parallel Computational Fluid Dynamics – Practice and Theory, P. Wilders, A. Ecer, J. Periaux, N. Satofuka and P. Fox, Eds., Elsevier Science, The Netherlands (2002) 119–126.
B.F. Smith, P. Bjorstad and W.D. Gropp, Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambrige (1996).
P. Le Tallec, Domain decomposition methods in computational mechanics, in Computational Mechanics Advances, J.T. Oden, Ed., North-Holland 1 (1994) 121–220.
R.S. Tuminaro and C. Tong, Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines, in SuperComputing 2000 Proceedings, J. Donnelley, Ed. (2000).
Vanek, P., Brezina, M. and Tezaur, R., Two-grid method for linear elasticity on unstructured meshes. SIAM J. Sci. Comput. 21 (1999) 900–923. CrossRef
Vanek, P., Brezina, M. and Mandel, J., Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88 (2001) 559–579.