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Numerische Mathematik

, Volume 134, Issue 3, pp 637–666 | Cite as

A nearly optimal multigrid method for general unstructured grids

  • Lars Grasedyck
  • Lu WangEmail author
  • Jinchao Xu
Article
  • 456 Downloads

Abstract

In this paper, we develop a multigrid method on unstructured shape-regular grids. For a general shape-regular unstructured grid of \({\mathcal O}(N)\) elements, we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique. Such a construction is realized by a cluster tree which can be obtained in \({\mathcal O}(N\log N)\) operations for a grid of N elements. This tree structure in turn is used for the definition of the grid hierarchy from coarse to fine. For the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned CG for an elliptic PDE is \(1 - {\mathcal O}({1}/{\log N})\). Numerical experiments confirm the theoretical bounds and show that the total complexity is in \({\mathcal O}(N\log N)\).

Keywords

Clustering Multigrid Auxiliary space Finite elements 

Mathematics Subject Classification

65N22 65F10 65N30 65N55 

References

  1. 1.
    Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput. 36(153), 35–51 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bank, R.E., Xu, J.: A hierarchical basis multigrid method for unstructured grids. Notes Numer. Fluid Mech. 49, 1–1 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Braess, D.: Towards algebraic multigrid for elliptic problems of second order. Computing 55(4), 379–393 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including the v-cycle. SIAM J. Numer. Anal. 20(5), 967–975 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bramble, J.H., Pasciak, J.E.: New convergence estimates for multigrid algorithms. Math. Comput. 49(180), 311–329 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bramble, J.H., Pasciak, J.E., Wang, J.P., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57(195), 23–45 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bramble, J.H., Pasciak, J.E., Xu, J.: Parallel multilevel preconditioners. Math. Comput. 55(191), 1–22 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comput. 56(193), 1–34 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brandt, A., McCormick, S., Ruge, J.: Algebraic multigrid (AMG) for sparse matrix equations. In: Evans, D. (ed.) Sparsity and its Applications, pp. 257–284. Cambridge University Press, Cambridge (1984)Google Scholar
  10. 10.
    Brannick, J., Chen, Y., Hu, X., Zikatanov, L.: Parallel unsmoothed aggregation algebraic multigrid algorithms on gpus. In: Numerical Solution of Partial Differential Equations: Theory. Algorithms, and Their Applications, pp. 81–102. Springer, Berlin (2013)Google Scholar
  11. 11.
    Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive smoothed aggregation (SA) multigrid. SIAM Rev. 47(2), 317–346 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brezina, M., Vaněk, P., Vassilevski, P.S.: An improved convergence analysis of smoothed aggregation algebraic multigrid. Numer. Linear Algebra Appl. 19(3), 441–469 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bulgakov, V.: Multi-level iterative technique and aggregation concept with semi-analytical preconditioning for solving boundary-value problems. Commun. Numer. Methods Eng. 9(8), 649–657 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, L., Nochetto, R., Xu, J.: Optimal multilevel methods for graded bisection grids. Numer. Math. 120(1), 1–34 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, L., Zhang, C.: A coarsening algorithm on adaptive grids by newest vertex bisection and its applications. J. Comput. Math 28(6), 767–789 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ciarlet, P.: Basic error estimates for elliptic problems. In: Ciarlet, P., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 17–352. North Holland, Amsterdam (1991)Google Scholar
  17. 17.
    Falgout, R.D., Vassilevski, P.S.: On generalizing the algebraic multigrid framework. SIAM J. Numer. Anal. 42(4), 1669–1693 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Falgout, R.D., Vassilevski, P.S., Zikatanov, L.T.: On two-grid convergence estimates. Numer. Linear Algebra Appl. 12(5–6), 471–494 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feuchter, D., Heppner, I., Sauter, S., Wittum, G.: Bridging the gap between geometric and algebraic multigrid methods. Comput. Visual. Sci. 6, 1–13 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Feuchter, D., Heppner, I., Sauter, S., Wittum, G.: Bridging the gap between geometric and algebraic multi-grid methods. Comput. Visual. Sci. 6(1), 1–13 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Finkel, R., Bentley, J.: Quad trees a data structure for retrieval on composite keys. Acta Inf. 4(1), 1–9 (1974)CrossRefzbMATHGoogle Scholar
  22. 22.
    Grasedyck, L., Hackbusch, W., Le Borne, S.: Adaptive geometrically balanced clustering of \({\cal H}\)-matrices. Computing 73, 1–23 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Grasedyck, L., Hackbusch, W., LeBorne, S.: Adaptive refinement and clustering of \(\cal H \)-matrices. Tech. Rep. 106, Max Planck Institute of Mathematics in the Sciences (2001)Google Scholar
  24. 24.
    Grasedyck, L., Kriemann, R., LeBorne, S.: Domain-decomposition based \(\cal H \)-matrix preconditioners. In: Proceedings of DD16, LNSCE. Springer, Berlin (2005, to appear)Google Scholar
  25. 25.
    Hackbusch, W.: Multi-grid Methods and Applications, vol. 4. Springer, Berlin (1985)zbMATHGoogle Scholar
  26. 26.
    Henson, V., Yang, U.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hu, X., Xu, J., Zhang, C.: Application of auxiliary space preconditioning in field-scale reservoir simulation. In: Science China Mathematics, pp. 1–15 (2013)Google Scholar
  28. 28.
    Hu, X., Zhang, C.S., Wu, S., Zhang, S., Wu, X., Xu, J., Zikatanov, L.: Combined preconditioning with applications in reservoir simulation. Multisc. Model. Simul. 11(2), 507–521 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jones, J., Vassilevski, P.: AMGe based on element agglomeration. SIAM J. Sci. Comput. 23(1), 109–133 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lashuk, I., Vassilevski, P.: On some versions of the element agglomeration AMGe method. Numer. Linear Algebra Appl. 15(7), 595–620 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Liehr, F., Preusser, T., Rumpf, M., Sauter, S., Schwen, L.: Composite finite elements for 3D image based computing. Comput. Visual. Sci. 12, 171–188 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Nepomnyaschikh, S.: Decomposition and fictitious domains methods for elliptic boundary value problems. In: Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 62–72 (1992)Google Scholar
  33. 33.
    Ruge, R.W., Stüben, K.: Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Paddon, H.H.D.J. (ed.) Multigrid Methods for Integral and Differential Equations, pp. 169–212. Clarenden Press, Oxford (1985)Google Scholar
  34. 34.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Stüben, K.: Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput. 13(3–4), 419–451 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Stüben, K.: Algebraic multigrid (AMG): an introduction with applications. In: Trottenberg, U., Oosterlee, C.W., Schüller, A. (eds.) Multigrid. Academic Press, New York (2000). Also GMD Report 53, March 1999Google Scholar
  37. 37.
    Stüben, K.: A review of algebraic multigrid. J. Comput. Appl. Math. 128(1), 281–309 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Thum, P., Diersch, H.J., Gründler, R.: SAMG—the linear solver for groundwater simulation. In: MODELCARE 2011. Leipzig, Germany (2011)Google Scholar
  39. 39.
    Vaněk, P.: Acceleration of convergence of a two-level algorithm by smoothing transfer operators. Appl. Math. 37(4), 265–274 (1992)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Vaněk, P.: Fast multigrid solver. Appl. Math. 40(1), 1–20 (1995)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Vanek, P., Mandel, J., Brezina, M.: Algebraic multigrid on unstructured meshes, vol. 34. UCD/CCM Report (1994)Google Scholar
  42. 42.
    Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56(3), 179–196 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wang, L., Hu, X., Cohen, J., Xu, J.: A parallel auxiliary grid algebraic multigrid method for graphic processing units. SIAM J. Sci. Comput. 35(3), C263–C283 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Xu, J.: Theory of multilevel methods, vol. 8924558. Cornell Unversity, May (1989)Google Scholar
  45. 45.
    Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Xu, J.: An introduction to multigrid convergence theory. In: Chan, R., Chan, T., Golub, G. (eds.) Iterative Methods in Scientific Computing. Springer, Berlin (1997)Google Scholar
  48. 48.
    Xu, J., Chen, L., Nochetto, R.H.: Optimal multilevel methods for \(H(grad)\), \(H(curl)\), and \(H(div)\) systems on graded and unstructured grids. In: DeVore, R., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation, pp. 599–659. Springer, Berlin (2009)CrossRefGoogle Scholar
  49. 49.
    Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15(3), 573–597 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math. 49(4), 379–412 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer. 2, 285–326 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yserentant, H.: Coarse grid spaces for domains with a complicated boundary. Numer. Algorith. 21(1), 387–392 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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