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

## Summary

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.

## Keywords

second-order elliptic problems heterogeneous media smoothed aggregation two-level additive Schwarz algebraic multigrid## AMS Subject Classifications

65F10 65N22 65N55## Preview

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## References

- 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 - Bastian, P.: Aggregation-based algebraic multigrid code. IWR, University of Heidelberg (1996) (available at http://www.dune-project.org/index.html)Google Scholar
- 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 - 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
- Brenner S.C. and Scott L.R. (1994). The mathematical theory of finite element methods. Springer, Berlin MATHGoogle Scholar
- Brezina M. and Vanek P. (1999). A black-box iterative solver based on a two-level Schwarz method.
*Computing*63: 233–263 MATHCrossRefGoogle Scholar - 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 MATHCrossRefGoogle Scholar - 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 MATHCrossRefGoogle Scholar - Chan, T. F., Mathew, T.: Domain decomposition methods. Acta Numerica 1994. Cambridge University Press (1994)Google Scholar
- 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 MATHCrossRefGoogle Scholar - Davies, T. A.: UMFPACKv4.4 multifrontal software, CISE, University of Florida (2005) (available at http://www.cise.ufl.edu/research/sparse/umfpack)Google Scholar
- 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
- 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 MATHCrossRefGoogle Scholar - 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
- 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 MATHCrossRefGoogle Scholar - Graham I.G., Lechner P.O. and Scheichl R.O. (2007). Domain decomposition for multiscale PDEs.
*Numer Math*106: 589–626 MATHCrossRefGoogle Scholar - Graham I.G. and Scheichl R. (2007). Robust domain decomposition algorithms for multiscale PDEs.
*Numer Meth Partial Differ Equ*23: 859–878 MATHCrossRefGoogle Scholar - 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
- Griebel M., Scherer K. and Schweitzer M.A. (2007). Robust norm equivalences and optimal preconditioners for diffusion problems.
*Math Comput*76: 1141–1161 MATHGoogle Scholar - Henson V.E. and Yang U.M. (2002). BoomerAMG: a parallel algebraic multigrid solver and preconditioner.
*Appl Numer Math*41: 155–177 MATHCrossRefGoogle Scholar - 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 MATHCrossRefGoogle Scholar - Jones J.E. and Vassilevski P.S. (2001). AMGe-based on element agglomeration.
*SIAM J Sci Comput*23: 109–133 MATHCrossRefGoogle Scholar - Kozintsev, B., Kedem, B.: Gaussian package, University of Maryland (1999) (available at http://www.math.umd.edu/ bnk/bak/generate.cgi)Google Scholar
- 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
- 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
- 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
- Sala M. (2004). Analysis of two-level domain decomposition preconditioners based on aggregation.
*ESAIM M2AN*38: 765–780 MATHCrossRefGoogle Scholar - 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 - Toselli A. and Widlund O. (2005). Domain decomposition methods – algorithms and theory. Springer, Berlin MATHGoogle Scholar
- 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 http://www.dougdevel.org)Google Scholar
- Vanek P., Mandel J. and Brezina M. (1996). Algebraic multigrid by smoothed aggregation for 2nd and 4th order elliptic problems.
*Computing*56: 179–196 MATHCrossRefGoogle Scholar - Vanek P., Brezina M. and Mandel J. (2001). Convergence of algebraic multigrid based on smoothed aggregation.
*Numer Math*88: 559–579 MATHCrossRefGoogle Scholar - 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 MATHCrossRefGoogle Scholar - 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 MATHCrossRefGoogle Scholar