, Volume 80, Issue 4, pp 319–343 | Cite as

Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients

  • R. ScheichlEmail author
  • E. Vainikko


We develop a new coefficient-explicit theory for two-level overlapping domain decomposition preconditioners with non-standard coarse spaces in iterative solvers for finite element discretisations of second-order elliptic problems. We apply the theory to the case of smoothed aggregation coarse spaces introduced by Vanek, Mandel and Brezina in the context of algebraic multigrid (AMG) and are particularly interested in the situation where the diffusion coefficient (or the permeability) α is highly variable throughout the domain. Our motivating example is Monte Carlo simulation for flow in rock with permeability modelled by log–normal random fields. By using the concept of strong connections (suitably adapted from the AMG context) we design a two-level additive Schwarz preconditioner that is robust to strong variations in α as well as to mesh refinement. We give upper bounds on the condition number of the preconditioned system which do not depend on the size of the subdomains and make explicit the interplay between the coefficient function and the coarse space basis functions. In particular, we are able to show that the condition number can be bounded independent of the ratio of the two values of α in a binary medium even when the discontinuities in the coefficient function are not resolved by the coarse mesh. Our numerical results show that the bounds with respect to the mesh parameters are sharp and that the method is indeed robust to strong variations in α. We compare the method to other preconditioners and to a sparse direct solver.


second-order elliptic problems heterogeneous media smoothed aggregation two-level additive Schwarz algebraic multigrid 

AMS Subject Classifications

65F10 65N22 65N55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Almond D.P., Bowen C.R. and Rees D.A.S. (2006). Composite dielectrics and conductors: simulation, characterization and design. J Phys D Appl Phys 39: 1295–1304 CrossRefGoogle Scholar
  2. Bastian, P.: Aggregation-based algebraic multigrid code. IWR, University of Heidelberg (1996) (available at Scholar
  3. Bastian P. and Helmig R. (1999). Efficient fully-coupled solution techniques for two-phase flow in porous media: parallel multigrid solution and large scale computations. Adv Water Res 23: 199–216 CrossRefGoogle Scholar
  4. Blatt, M., Bastian, P.: The iterative solver template library. In: Proc. Workshop on State-of-the-Art in Scientific and Parallel Computing, PARA ’06, Umea, June 18–21, 2006. Lecture Notes in Scientific Computing. Springer, Berlin (forthcoming)Google Scholar
  5. Brenner S.C. and Scott L.R. (1994). The mathematical theory of finite element methods. Springer, Berlin zbMATHGoogle Scholar
  6. Brezina M. and Vanek P. (1999). A black-box iterative solver based on a two-level Schwarz method. Computing 63: 233–263 zbMATHCrossRefGoogle Scholar
  7. Cai X., Nielsen B.F. and Tveito A. (1999). An analysis of a preconditioner for the discretized pressure equation arising in reservoir simulation. IMA J Numer Anal 19: 291–316 zbMATHCrossRefGoogle Scholar
  8. Carvalho L.M., Giraud L. and Le Tallec P. (2001). Algebraic two-level preconditioners for the Schur complement method. SIAM J Sci Comput 22: 1987–2005 zbMATHCrossRefGoogle Scholar
  9. Chan, T. F., Mathew, T.: Domain decomposition methods. Acta Numerica 1994. Cambridge University Press (1994)Google Scholar
  10. Cliffe K.A., Graham I.G., Scheichl R. and Stals L. (2000). Parallel computation of flow in heterogeneous media modelled by mixed finite elements. J Comput Phys 164: 258–282 zbMATHCrossRefGoogle Scholar
  11. Davies, T. A.: UMFPACKv4.4 multifrontal software, CISE, University of Florida (2005) (available at Scholar
  12. Gee, M. W., Siefert, C. M., Hu, J. J., Tuminaro, R. S., Sala, M. G.: ML 5.0 Smoothed Aggregation User’s Guide, Technical Report SAND2006-2649, Sandia National Laboratories (2006)Google Scholar
  13. Giraud L., Guevara Vasquez F. and Tuminaro R.S. (2003). Grid transfer operators for highly-variable coefficient problems in two-level non-overlapping domain decomposition methods. Numer Linear Algebra Appl 10: 467–484 zbMATHCrossRefGoogle Scholar
  14. Graham I.G. and Hagger M.J. (1998). Additive Schwarz, CG and discontinuous coefficients. In: Bjorstad, P., Espedal, M. and Keyes, D.E. (eds) Proc. 9th Int. Conf. on Domain Decomposition Methods, Bergen, 1996, pp. Domain Decomposition Press, Bergen Google Scholar
  15. Graham I.G. and Hagger M.J. (1999). Unstructured additive Schwarz–CG method for elliptic problems with highly discontinuous coefficients. SIAM J Sci Comput 20: 2041–2066 zbMATHCrossRefGoogle Scholar
  16. Graham I.G., Lechner P.O. and Scheichl R.O. (2007). Domain decomposition for multiscale PDEs. Numer Math 106: 589–626 zbMATHCrossRefGoogle Scholar
  17. Graham I.G. and Scheichl R. (2007). Robust domain decomposition algorithms for multiscale PDEs. Numer Meth Partial Differ Equ 23: 859–878 zbMATHCrossRefGoogle Scholar
  18. Graham, I. G., Scheichl, R.: Coefficient-explicit condition number bounds for overlapping additive Schwarz. In: Proc. 17th Int. Conf. on Domain Decomposition Methods, Strobl, Austria, 3–7 July 2006. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2007) (forthcoming)Google Scholar
  19. Griebel M., Scherer K. and Schweitzer M.A. (2007). Robust norm equivalences and optimal preconditioners for diffusion problems. Math Comput 76: 1141–1161 zbMATHGoogle Scholar
  20. Henson V.E. and Yang U.M. (2002). BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl Numer Math 41: 155–177 zbMATHCrossRefGoogle Scholar
  21. Jenkins E.W., Kelley C.T., Miller C.T. and Kees C.E. (2001). An aggregation-based domain decomposition preconditioner for groundwater flow. SIAM J Sci Comput 23: 430–441 zbMATHCrossRefGoogle Scholar
  22. Jones J.E. and Vassilevski P.S. (2001). AMGe-based on element agglomeration. SIAM J Sci Comput 23: 109–133 zbMATHCrossRefGoogle Scholar
  23. Kozintsev, B., Kedem, B.: Gaussian package, University of Maryland (1999) (available at bnk/bak/generate.cgi)Google Scholar
  24. Lasser C. and Toselli A. (2002). Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces. In: Pavarino, L. and Toselli, A. (eds) Recent developments in domain decomposition methods. Lecture Notes in Computational Science and Engineering, vol. 23, pp 95–117. Springer, Berlin Google Scholar
  25. Raw, M.: A coupled algebraic multigrid method for the 3D Navier–Stokes equations. In: Proc. 34th Aerospace Sciences Meeting and Exhibit, Reno, AIAA-1996-297. AIAA Press, Washington (1996)Google Scholar
  26. Ruge J. and Stüben K. (1985). Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Paddon, D.J. and Holstein, H. (eds) Multigrid methods for integral and differential equations. IMA Conference Series, pp 169–212. Oxford, Clarendon Google Scholar
  27. Sala M. (2004). Analysis of two-level domain decomposition preconditioners based on aggregation. ESAIM M2AN 38: 765–780 zbMATHCrossRefGoogle Scholar
  28. Sala M., Shadid J.N. and Tuminaro R.S. (2006). An improved convergence bound for aggregation-based domain decomposition preconditioners. SIAM J Matrix Anal 27: 744–756 CrossRefGoogle Scholar
  29. Toselli A. and Widlund O. (2005). Domain decomposition methods – algorithms and theory. Springer, Berlin zbMATHGoogle Scholar
  30. Vainikko, E., Tehver, M., Batrashev, O., Niitsoo, M., Skaburskas, K., Scheichl, R.: ADOUG – aggregation-based domain decomposition on unstructured grids. Universities of Tartu and Bath (2006) (available at Scholar
  31. Vanek P., Mandel J. and Brezina M. (1996). Algebraic multigrid by smoothed aggregation for 2nd and 4th order elliptic problems. Computing 56: 179–196 zbMATHCrossRefGoogle Scholar
  32. Vanek P., Brezina M. and Mandel J. (2001). Convergence of algebraic multigrid based on smoothed aggregation. Numer Math 88: 559–579 zbMATHCrossRefGoogle Scholar
  33. Vuik C., Segal A. and Meijerink J.A. (1999). An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients. J Comput Phys 152: 385–403 zbMATHCrossRefGoogle Scholar
  34. Wan W.L., Chan T.F. and Smith B. (2000). An energy-minimizing interpolation for robust multigrid methods. SIAM J Sci Comput 21: 1632–1649 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Bath Institute for Complex Systems, Department of Mathematical SciencesUniversity of BathClaverton DownUK
  2. 2.Institute of Computer ScienceUniversity of TartuTartuEstonia

Personalised recommendations