Journal of Scientific Computing

, Volume 45, Issue 1–3, pp 48–63 | Cite as

Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D

Article

Abstract

Conventional high-order finite element methods are rarely used for industrial problems because the Jacobian rapidly loses sparsity as the order is increased, leading to unaffordable solve times and memory requirements. This effect typically limits order to at most quadratic, despite the favorable accuracy and stability properties offered by quadratic and higher order discretizations. We present a method in which the action of the Jacobian is applied matrix-free exploiting a tensor product basis on hexahedral elements, while much sparser matrices based on Q 1 sub-elements on the nodes of the high-order basis are assembled for preconditioning. With this “dual-order” scheme, storage is independent of spectral order and a natural taping scheme is available to update a full-accuracy matrix-free Jacobian during residual evaluation. Matrix-free Jacobian application circumvents the memory bandwidth bottleneck typical of sparse matrix operations, providing several times greater floating point performance and better use of multiple cores with shared memory bus. Computational results for the p-Laplacian and Stokes problem, using block preconditioners and AMG, demonstrate mesh-independent convergence rates and weak (bounded) dependence on order, even for highly deformed meshes and nonlinear systems with several orders of magnitude dynamic range in coefficients. For spectral orders around 5, the dual-order scheme requires half the memory and similar time to assembled quadratic (Q 2) elements, making it very affordable for general use.

Keywords

High-order Finite element method Newton-Krylov Preconditioning 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ainsworth, M., Senior, B.: An adaptive refinement strategy for hp-finite element computations. Appl. Numer. Math. 26(1), 165–178 (1998) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Balay, S., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., Curfman McInnes, L., Smith, B.F., Zhang, H.: PETSc users manual. Technical Report ANL-95/11—Revision 3.0.0, Argonne National Laboratory (2008) Google Scholar
  3. 3.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blacker, T., Bohnhoff, W., Edwards, T., Hipp, J., Lober, R., Mitchell, S., Sjaardema, G., Tautges, T., Wilson, T., Oakes, W., et al.: CUBIT mesh generation environment. Technical report, Sandia National Labs., Albuquerque, NM. Cubit Development Team (1994) Google Scholar
  5. 5.
    Demkowicz, L., Oden, J.T., Rachowicz, W., Hardy, O.: Toward a universal hp adaptive finite element strategy. I: Constrained approximation and data structure. Comput. Methods Appl. Mech. Eng. 77, 79–112 (1989) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Demkowicz, L., Rachowicz, W., Devloo, P.: A fully automatic hp-adaptivity. J. Sci. Comput. 17(1), 117–142 (2002) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deville, M., Mund, E.: Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning. J. Comput. Phys. 60, 517 (1985) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deville, M.O., Mund, E.H.: Finite-element preconditioning for pseudospectral solutions of elliptic problems. SIAM J. Sci. Stat. Comput. 11, 311 (1990) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eisenstat, S.C., Walker, H.F.: Choosing the forcing terms in an inexact newton method. SIAM J. Sci. Comput. 17(1), 16–32 (1996) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Elman, H.C., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations. J. Comput. Phys. 227(1), 1790–1808 (2008) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Evans, L.C.: The 1-Laplacian, the ∞-Laplacian and differential games. Perspect. Nonlinear Partial Differ. Equ.: In Honor of Haim Brezis 446, 245 (2007) Google Scholar
  12. 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. 13.
    Gropp, W.D., Kaushik, D.K., Keyes, D.E., Smith, B.: Performance modeling and tuning of an unstructured mesh cfd application. In: Supercomputing ’00: Proceedings of the 2000 ACM/IEEE Conference on Supercomputing (CDROM), Washington, DC, USA, 2000, p. 34. IEEE Computer Society, New York (2000) Google Scholar
  14. 14.
    Henson, V.E., Yang, U.M.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Heys, J.J., Manteuffel, T.A., McCormick, S.F., Olson, L.N.: Algebraic multigrid for higher-order finite elements. J. Comput. Phys. 204(2), 520–532 (2005) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Interoperable technologies for advanced petascale simulations (ITAPS). http://www.itaps.org/
  17. 17.
    Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, Oxford (2005) MATHCrossRefGoogle Scholar
  18. 18.
    Kim, S.D.: Piecewise bilinear preconditioning of high-order finite element methods. Electron. Trans. Numer. Anal. 26, 228–242 (2007) MATHMathSciNetGoogle Scholar
  19. 19.
    Kirk, B., Peterson, J.W., Stogner, R.H., Carey, G.F.: libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22(3–4), 237–254 (2006). http://dx.doi.org/10.1007/s00366-006-0049-3 CrossRefGoogle Scholar
  20. 20.
    Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Knoll, D.A., McHugh, P.R.: Enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow. SIAM J. Sci. Comput. 19(1), 291–301 (1998) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lottes, J.W., Fischer, P.F.: Hybrid multigrid/Schwarz algorithms for the spectral element method. J. Sci. Comput. 24(1), 45–78 (2005) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    May, D.A., Moresi, L.: Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics. Phys. Earth Planet. Inter. 171(1–4), 33–47 (2008). Recent Advances in Computational Geodynamics: Theory, Numerics and Applications CrossRefGoogle Scholar
  24. 24.
    Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969–1972 (2000) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Nachtigal, N.M., Reddy, S.C., Trefethen, L.N.: How fast are nonsymmetric matrix iterations? SIAM J. Matrix Anal. Appl. 13, 778 (1992) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Oden, J.T., Demkowicz, L., Rachowicz, W., Westermann, T.A.: Toward a universal hp adaptive finite element strategy. II: A posteriori error estimation. Comput. Methods Appl. Mech. Eng. 77, 113–180 (1989) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92 (1980) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Rachowicz, W., Oden, J.T., Demkowicz, L.: Toward a universal hp adaptive finite element strategy. III: Design of hp meshes. Comput. Methods Appl. Mech. Eng. 77, 181–212 (1989) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Schötzau, D., Schwab, C., Stenberg, R.: Mixed hp-FEM on anisotropic meshes. Math. Models Methods Appl. Sci. 8, 787–820 (1998) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Schwab, C.: P- and Hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, Oxford (1998) MATHGoogle Scholar
  31. 31.
    Tautges, T.J.: CGM: a geometry interface for mesh generation, analysis and other applications. Eng. Comput. 17(3), 299–314 (2001) MATHCrossRefGoogle Scholar
  32. 32.
    Tautges, T.J., Meyers, R., Merkley, K., Stimpson, C., Ernst, C.: MOAB: a mesh-oriented database. Technical report, Sandia National Laboratories, April 2004 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie (VAW)ETH ZürichZürichSwitzerland

Personalised recommendations