Skip to main content
Springer Nature Link
Log in

Fast Tensor-Product Solvers: Partially Deformed Three-dimensional Domains

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

Abstract

We discuss the numerical solution of partial differential equations in a particular class of three-dimensional geometries; the two-dimensional cross section (in the xy-plane) can have a general shape, but is assumed to be invariant with respect to the third direction. Earlier work has exploited such geometries by approximating the solution as a truncated Fourier series in the z-direction. In this paper we propose a new solution algorithm which also exploits the tensor-product feature between the xy-plane and the z-direction. However, the new algorithm is not limited to periodic boundary conditions, but works for general Dirichlet and Neumann type of boundary conditions. The proposed algorithm also works for problems with variable coefficients as long as these can be expressed as a separable function with respect to the variation in the xy-plane and the variation in the z-direction. For problems where the new method is applicable, the computational cost is very competitive with the best iterative solvers. The new algorithm is easy to implement, and useful, both in a serial and parallel context. Numerical results demonstrating the superiority of the method are presented for three-dimensional Poisson and Helmholtz problems using both low order finite elements and high order spectral element discretizations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernardi, B., Maday, Y.: Approximations Spectrales de Problèmes aux Limites Elliptiques. Springer, Berlin (1992)

    MATH  Google Scholar 

  2. Bjørstad, P.E., Tjøstheim, B.P.: Efficient algorithms for solving a fourth-order equation with the spectral Galerkin method. SIAM J. Sci. Comput. 18(2), 621–632 (1997)

    Article  MathSciNet  Google Scholar 

  3. Carvalho, M.S., Scriven, L.E.: Three-dimensional stability analysis of free surface flows: Application to forward deformable roll coating. J. Comput. Phys. 151(2), 534–562 (1999)

    Article  MATH  Google Scholar 

  4. Chu, D., Henderson, R., Karniadakis, G.E.: Parallel spectral-element-Fourier simulation of turbulent flow over riblet-mounted surfaces. Theor. Comput. Fluid Dyn. 3, 219–229 (1992)

    Article  MATH  Google Scholar 

  5. Couzy, W., Deville, M.O.: A fast Schur complement method for the spectral element discretization of the incompressible Navier-Stokes equations. J. Comput. Phys. 116, 135–142 (1995)

    Article  MATH  Google Scholar 

  6. Dryja, M., Widlund, O.B.: Towards a unified theory of domain decomposition algorithms for elliptic problems. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O.B. (eds.) Third Int. Symp. on Domain Decomposition Methods for PDE’s, pp. 3–21. SIAM, Philadelphia (1990)

    Google Scholar 

  7. Fischer, P.F.: An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. J. Comput. Phys. 133, 84–101 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Golub, G., Van Loan, C.F.: Matrix Computations. John Hopkins Press, Baltimore (1983)

    MATH  Google Scholar 

  9. Henderson, R.: Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 353, 65–112 (1997)

    Article  Google Scholar 

  10. Kwan, Y.-Y., Shen, J.: An efficient direct parallel spectral-element solver for separable elliptic problems. J. Comput. Phys. 225, 1721–1735 (2007). DOI: 10.1016/j.jcp2007.02.013

    Article  MATH  MathSciNet  Google Scholar 

  11. Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial differential equations by tensor product methods. Numer. Math. 6, 185–199 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  12. Maday, Y., Patera, A.T.: Spectral element methods for the Navier-Stokes equations. In: Noor, A.K. (ed.) State of the Art Surveys in Computational Mechanics, pp. 71–143. ASME, New York (1989)

    Google Scholar 

  13. Patera, A.T.: Fast direct Poisson solvers for high-order finite element discretizations in rectangularly decomposable domains. J. Comput. Phys. 65, 474–480 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shen, J.: Efficient spectral-Galerkin method I. Direct solvers for the second and fourth order equations using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Toselli, A., Widlund, O.B.: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2004)

    Google Scholar 

  16. Tufo, H.M., Fischer, P.F.: Terascale Spectral Element Algorithms and Implementations. In: Proc. of the ACM/IEEE SC99 Conf. on High Performance Networking and Computing. IEEE Comput. Soc., Los Alamitos (1999), CDROM, Gordon Bell Prize paper

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Tormod Bjøntegaard & Einar M. Rønquist

  2. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France

    Yvon Maday

Authors
  1. Tormod Bjøntegaard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yvon Maday
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Einar M. Rønquist
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Einar M. Rønquist.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bjøntegaard, T., Maday, Y. & Rønquist, E.M. Fast Tensor-Product Solvers: Partially Deformed Three-dimensional Domains. J Sci Comput 39, 28–48 (2009). https://doi.org/10.1007/s10915-008-9246-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9246-0

Keywords