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.

Classification: 65M55, 65Y05

Keywords: elliptic equations, domain decomposition, Schwarz methods, aggregation coarse corrections

@article{M2AN_2004__38_5_765_0, author = {Sala, Marzio}, title = {Analysis of two-level domain decomposition preconditioners based on aggregation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {38}, number = {5}, year = {2004}, pages = {765-780}, doi = {10.1051/m2an:2004038}, zbl = {1078.65120}, mrnumber = {2104428}, language = {en}, url = {http://www.numdam.org/item/M2AN_2004__38_5_765_0} }

Sala, Marzio. Analysis of two-level domain decomposition preconditioners based on aggregation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 5, pp. 765-780. doi : 10.1051/m2an:2004038. http://www.numdam.org/item/M2AN_2004__38_5_765_0/

[1] Robust iterative method on unstructured meshes. Ph.D. Thesis, University of Colorado at Denver (1997).

,[2] A black-box iterative solver based on a two-level Schwarz method. Computing 63 (1999) 233-263. | Zbl 0951.65133

and ,[3] Robust parallel smoothing for multigrid via sparse approximate inverses. SIAM J. Sci. Comput. 23 (2001) 1396-1417. | Zbl 1004.65043

, , and ,[4] Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604-620. | Zbl 0802.65119

and ,[5] Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48 (1995) 121-155. | Zbl 0824.65106

and ,[6] A numerical investigation of Schwarz domain decomposition techniques for elliptic problems on unstructured grids. Math. Comput. Simulations 44 (1994) 313-330. | Zbl 1017.65524

, and ,[7] Matrix Computations. The Johns Hopkins University Press, Baltimore, Maryland (1983). | MR 733103 | Zbl 0559.65011

and ,[8] An aggregation-based domain decomposition preconditioner for groundwater flow. Technical Report TR00-13, Department of Mathematics, North Carolina State University (2000). | Zbl 1036.65109

, , and ,[9] Versatile multilevel Schwarz preconditioners for multiphase flow. Technical Report CRSC-TR01-32, Center for Research in Scientific Computation, North Carolina State University (2001).

, , and ,[10] 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. | Zbl 1038.65135

and ,[11] Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces. Technical Report TUM-M0109, Technische Universität München (2001). | MR 1962683 | Zbl 1007.65082

and ,[12] The structure of the American Economy. Oxford University Press, New York (1951).

,[13] A local convergence proof for the iterative aggregation method. Linear Algebra Appl. 51 (1983) 163-172. | Zbl 0494.65014

and ,[14] 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.

, , and ,[15] Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994). | MR 1299729 | Zbl 0803.65088

and ,[16] Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999). | MR 1857663 | Zbl 0931.65118

and ,[17] Parallel Schur and Schwarz based preconditioners and agglomeration coarse corrections for CFD problems. Technical Report 15, DMA-EPFL (2001).

and ,[18] 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.

and ,[19] Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambrige (1996). | MR 1410757 | Zbl 0857.65126

, and ,[20] Domain decomposition methods in computational mechanics, in Computational Mechanics Advances, J.T. Oden, Ed., North-Holland 1 (1994) 121-220. | Zbl 0802.73079

,[21] Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines, in SuperComputing 2000 Proceedings, J. Donnelley, Ed. (2000).

and ,[22] Two-grid method for linear elasticity on unstructured meshes. SIAM J. Sci. Comput. 21 (1999) 900-923. | Zbl 0952.65099

, and ,[23] Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88 (2001) 559-579. | Zbl 0992.65139

, and ,