Skip to main content
SpringerLink
Log in

Multigrid with Nonstandard Coarse-Level Operators and Coarsening Factors

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We consider the numerical solution of Poisson’s equation on structured grids using geometric multigrid with nonstandard coarse grids and coarse-level operators. We are motivated by the problem of developing high-order accurate numerical solvers for elliptic boundary value problems on complex geometry using overset grids. For flexibility in grid generation, we would like to consider lower-order accurate coarse-level approximations, and coarsening factors other than two. We show that second-order accurate coarse-level approximations are very effective for fourth- or sixth-order accurate fine-level finite-difference discretizations. We study the use of different Galerkin and non-Galerkin coarse-level operators. Using local Fourier analysis (LFA) we choose the smoothing parameter \(\omega \) and the coarse-level operators to optimize the overall multigrid convergence rate. We show that the results based on LFA for periodic problems also hold for more general boundary conditions provided these are discretized using compatibility conditions. Numerical results for Poisson’s equation on a sample overset grid show that our multigrid solver is many times faster, and uses less memory, than selected Krylov solvers and an algebraic multigrid solver. We also study grid coarsening by a general factor and show that good convergence rates are retained for a range of coarsening factors around two. We ask the question of which coarsening factor leads to the most efficient multigrid algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability

Not applicable.

Notes

  1. The LFA results presented in Sect. 3, in particular, are obtained using the software from Wienands [27], which we extended for sixth-order accuracy as well as for additional choices of coarse-level operators. The LFA results in Sect. 5 are produced using Matlab codes we developed.

  2. We acknowledge that FLOPS do not tell the whole story on modern computer architectures and so the results presented here are more of a rough guideline.

  3. Here “Galerkin” is in quotations since the coarse-level operator does not come from the actual operator on the fine level \(L_h= L_h^{(4)}\).

  4. For multi-level cycles the same \(\omega \) for smoothing is used on every level.

  5. We estimate the work-units for a multigrid cycle based on using as many levels as possible[7]. Non-Galerkin coarse-level operators have some advantage in terms of WU over Galerkin operators because of their sparser stencils, especially for the simple example of the standard discrete Laplacians.

  6. Available at: http://www.overtureframework.org

  7. The results were computed using a Intel Xeon 3.0 GHz processor.

  8. The floor function \(\lfloor a \rfloor \) denotes the largest integer less than or equal to \(a \in \mathbb {R}\).

References

  1. Henshaw, W.D.: On multigrid for overlapping grids. SIAM J. Sci. Comput. 26(5), 1547–157228 (2005). https://doi.org/10.1137/040603735

    Article  MathSciNet  MATH  Google Scholar 

  2. Henshaw, W.D.: A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids. J. Comput. Phys. 113(1), 13–25240 (1994). https://doi.org/10.1006/jcph.1994.1114

    Article  MathSciNet  MATH  Google Scholar 

  3. Henshaw, W.D., Schwendeman, D.W.: An adaptive numerical scheme for high-speed reactive flow on overlapping grids. J. Comput. Phys. 191, 420–44796 (2003). https://doi.org/10.1016/S0021-9991(03)00323-1

    Article  MathSciNet  MATH  Google Scholar 

  4. Henshaw, W.D., Chand, K.K.: A composite grid solver for conjugate heat transfer in fluid-structure systems. J. Comput. Phys. 228, 3708–374175 (2009). https://doi.org/10.1016/j.jcp.2009.02.007

    Article  MathSciNet  MATH  Google Scholar 

  5. Henshaw, W.D.: A high-order accurate parallel solver for Maxwell’s equations on overlapping grids. SIAM J. Sci. Comput. 28(5), 1730–176555 (2006). https://doi.org/10.1137/050644379

    Article  MathSciNet  MATH  Google Scholar 

  6. Henshaw, W.D., Chesshire, G.S.: Multigrid on composite meshes. SIAM J. Sci. Stat. Comput. 8(6), 914–92347 (1987). https://doi.org/10.1137/0908074

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu, K., Henshaw, W.D.: Multigrid with nonstandard coarsening. Preprint at arXiv:2008.03885 [math.NA] (2020)

  8. Stüben, K., Trottenberg, U.: Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid Methods, pp. 1–176. Springer, Berlin (1982)

    MATH  Google Scholar 

  9. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  10. Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, London (2001)

    MATH  Google Scholar 

  11. Wesseling, P.: An Introduction To Multigrid Methods. John Wiley & Sons, New York (1991)

    MATH  Google Scholar 

  12. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977). https://doi.org/10.1090/S0025-5718-1977-0431719-X

    Article  MathSciNet  MATH  Google Scholar 

  13. Stüben, K.: A review of algebraic multigrid. J. Comput. Appl. Math. 128, 281–309 (2001). https://doi.org/10.1016/S0377-0427(00)00516-1

    Article  MathSciNet  MATH  Google Scholar 

  14. Chang, Q., Wong, Y.S., Fu, H.: On the algebraic multigrid method. J. Comput. Phys. 125, 279–292 (1996). https://doi.org/10.1006/jcph.1996.0094

    Article  MathSciNet  MATH  Google Scholar 

  15. Grauschopf, T., Griebel, M., Regler, H.: Additive multilevel preconditioners based on bilinear interpolation, matrix-dependent geometric coarsening and algebraic multigrid coarsening for second-order elliptic PDEs. Appl. Numer. Math. 23, 63–95 (1997). https://doi.org/10.1016/S0168-9274(96)00062-1

    Article  MathSciNet  MATH  Google Scholar 

  16. Krechel, A., Stüben, K.: Operator dependent interpolation in algebraic multigrid. In: Hackbusch, W., Wittum, G. (eds.) Multigrid Methods V. Lecture Notes in Computational Science and Engineering, vol. 3, pp. 189–211. Springer, Berlin (1998)

    MATH  Google Scholar 

  17. Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive algebraic multigrid. SIAM J. Sci. Comput. 27, 1261–1286 (2006). https://doi.org/10.1137/040614402

    Article  MathSciNet  MATH  Google Scholar 

  18. Hinatsu, M., Ferziger, J.H.: Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique. Int. J. Numer. Methods Fluids 13, 971–997 (1991). https://doi.org/10.1002/fld.1650130804

    Article  MATH  Google Scholar 

  19. Johnson, R.A., Belk, D.M.: Multigrid approach to overset grid communication. AIAA J. 33(12), 2305–2308 (1995). https://doi.org/10.2514/3.12984

    Article  MATH  Google Scholar 

  20. Tu, J.Y., Fuchs, L.: Calculation of flows using three-dimensional overlapping grids and multigrid methods. Int. J. Numer. Methods Eng. 38, 259–282 (1995). https://doi.org/10.1002/nme.1620380207

    Article  MATH  Google Scholar 

  21. Zang, Y., Street, R.L.: A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains. Int. J. Numer. Methods Fluids 20, 341–361 (1995). https://doi.org/10.1002/fld.1650200502

    Article  MATH  Google Scholar 

  22. Yavneh, I.: On red-black SOR smoothing in multigrid. SIAM J. Sci. Comput. 17(1), 180–192 (1996). https://doi.org/10.1137/0917013

    Article  MathSciNet  MATH  Google Scholar 

  23. Rodrigo, C., Gaspar, F.J., Zikatanov, L.T.: On the validity of the local fourier analysis. Preprint at arXiv:1710.00408 [math.NA] (2017)

  24. Hackbusch, W.: On multi-grid iterations with defect correction. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid Methods, pp. 461–473. Springer, Berlin (1982)

    Chapter  Google Scholar 

  25. Bernert, K.: \(\tau \)-extrapolation–theoretical foundation, numerical experiment, and application to Navier-Stokes equations. SIAM J. Sci. Comput. 18(2), 460–478 (1997). https://doi.org/10.1137/S1064827594276266

    Article  MathSciNet  MATH  Google Scholar 

  26. Hemker, P.W.: On the order of prolongations and restrictions in multigrid procedures. J. Comput. Appl. Math. 32, 423–429 (1990). https://doi.org/10.1016/0377-0427(90)90047-4

    Article  MathSciNet  MATH  Google Scholar 

  27. Wienands, R.: Lfa00_2d_scalar. GMD - Institute for Algorithms and Scientific Computing. Technical Report. http://www.mgnet.org/mgnet-codes-wienands.html, (SCAI) (2000)

  28. Stüben, K., Trottenberg, U.: Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Hackbusch, W., Trottenberg, U. (eds.) Multigrid Methods, pp. 1–176. Springer, Berlin, Heidelberg (1982)

    MATH  Google Scholar 

  29. Hassanieh, N.A., Banks, J.W., Henshaw, W.D., Schwendeman, D.W.: Local compatibility boundary conditions for high-order accurate finite-difference approximations of PDEs. Preprint at arXiv:2111.02915 (2021)

  30. Henshaw, W.D., Kreiss, H.-O., Reyna, L.G.M.: A fourth-order accurate difference approximation for the incompressible Navier-Stokes equations. Comput. Fluids 23(4), 575–59352 (1994). https://doi.org/10.1016/0045-7930(94)90053-1

    Article  MathSciNet  MATH  Google Scholar 

  31. Chesshire, G.S., Henshaw, W.D.: Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90(1), 1–64527 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

Research supported by the National Science Foundation under grants DMS-1519934 and DMS-1818926.

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Utah, 201 Presidents Circle, Salt Lake City, UT, 84112, USA

    Chang Liu

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY, 12180, USA

    William Henshaw

Authors
  1. Chang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William Henshaw
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang Liu.

Ethics declarations

Conflict of interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, C., Henshaw, W. Multigrid with Nonstandard Coarse-Level Operators and Coarsening Factors. J Sci Comput 94, 58 (2023). https://doi.org/10.1007/s10915-023-02103-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02103-x

Keywords

Search

Navigation