Skip to main content
SpringerLink
Log in

Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

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

Abstract

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.

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
Fig. 12

Similar content being viewed by others

References

  1. Altas, I., Dym, J., Gupta, M.M., Manohar, R.: Mutigrid solution of automatically generated high-order discretizations for the biharmonic equation. SIAM J. Sci. Comput. 19, 1575–1585 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babuška, I., Guo, B.: Approximation properties of the \(h-p\) version of the finite element method. Comput. Methods Appl. Mech. Eng. 133, 319–346 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuška, I., Guo, B.: Regularity of the solutions of elliptic problems with piecewise analytic data, part I: boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19(1), 172–203 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Guo, B.: The \(h-p\) version of the finite element method on domains with curved boundaries. SIAM J. Numer. Anal. 25, 837–861 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babuška, I., Guo, B.: The \(h\)-\(p\) version of the finite element method, part I: the basic approximation results. Comput. Mech. 1, 21–41 (1986)

    Article  MATH  Google Scholar 

  6. Babuška, I., Guo, B.: The \(h-p\) version of the finite element method, part II: general results and applications. Comput. Mech. 1, 203–220 (1986)

    Article  MATH  Google Scholar 

  7. Bernardi, C., Coppoletta, G., Maday, Y.: Some spectral approximations of two-dimensional fourth-order problems. Math. Comput. 59(199), 63–76 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bernardi, C., Maday, Y.: Spectral methods. Handb. Numer. Anal. 5, 209–485 (1997)

    Google Scholar 

  9. Ben-Artzi, M., Croisille, J.P., Fishelov, D.: A fast direct solver for the biharmonic problem in a rectangular grid. SIAM J. Sci. Comput. 31(1), 303–333 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bialecki, B., Karageorghis, A.: A Legendre spectral Galerkin method for the biharmonic Dirichlet problem. SIAM J. Sci. Comput. 22(5), 1549–1569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bialecki, B.: A Legendre spectral collocation method for the biharmonic Dirichlet problem. ESAIM: Math. Model. Numer. Anal. 34(3), 637–662 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bialecki, B.: A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles. J. Comput. Phys. 191, 601–621 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bjørstad, P.: Fast numerical solution of the biharmonic Dirichlet problem on rectangles. SIAM J. Numer. Anal. 20(1), 59–71 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buzzbee, B.L., Dorr, F.W.: The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions. SIAM J. Numer. Anal. 11(4), 753–763 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  16. Ciarlet, P.G.: The finite element method for elliptic problems, classics in applied mathematics. SIAM. 40 (2002). doi:10.1137/1.9780898719208

  17. Dutt, P., Biswas, P., Raju, G.N.: Preconditioners for spectral element methods for elliptic and parabolic problems. J. Comput. Appl. Math. 215(1), 152–166 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dutt, P.K., Kumar, N.K., Upadhyay, C.S.: Non-conforming \(h-p\) spectral element methods for elliptic problems. Proc. Indian Acad. Sci. (Math. Sci.) 117(1), 109–145 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dutt, P., Tomar, S.: Stability estimates for \(h-p\) spectral element methods for general elliptic problems on curvilinear domains. Proc. Indian Acad. Sci. (Math. Sci.) 113(4), 395–429 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dutt, P., Tomar, S., Kumar, B.V.R.: Stability estimates for \(h-p\) spectral element methods for elliptic problems. Proc. Indian Acad. Sci. (Math Sci.) 112(4), 601–639 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gerritsma, M., Maerschalck, D.B.: Least-squares spectral element methods in computational fluid dynamics. Advanced Computational Methods in Science and Engineering Lecture Notes in Computational Science and Engineering, vol. 71, pp. 179–227 (2010)

  22. Grisvard, P.: Elliptic problems in nonsmooth domains, classics in applied mathematics. SIAM, 69 (2011). doi:10.1137/1.9781611972030

  23. Husain, A., Khan, A.: Least-squares spectral element preconditioners for fourth order elliptic problems. arXiv:1608.08416

  24. Karniadakis, G., Sherwin, J.: Spectral/h-p Element Methods for CFD. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  25. Khan, A., Upadhyay, C.S.: Exponentially accurate nonconforming least-squares spectral element method for elliptic problems on unbounded domain. Comput. Methods Appl. Mech. Eng. 305, 607–633 (2016)

    Article  MathSciNet  Google Scholar 

  26. Kumar, N.K., Nagaraju, G.: Least-squares \(hp\)/spectral element method for elliptic problems. Appl. Numer. Math. 60(1–2), 38–54 (2010)

    Article  MathSciNet  Google Scholar 

  27. Kumar, N.K., Dutt, P.K., Upadhyay, C.S.: Nonconforming spectral/\(h-p\) element methods for elliptic systems. J. Numer. Math. 17(2), 119–142 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mandel, J.: Iterative methods for \(p-\)version finite elements: preconditioning thin solids. Comput. Methods Appl. Mech. Eng. 133, 247–257 (1996)

    Article  MATH  Google Scholar 

  29. Meleshko, V.V.: Selected topics in the history of the two-dimensional biharmonic problem. Appl. Mech. Rev. 56(1), 3385 (2003)

    Article  Google Scholar 

  30. Pathria, D., Karniadakis, G.E.: Spectral element methods for elliptic problems in non-smooth domains. J. Comput. Phys. 122, 83–95 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pontaza, J.P., Reddy, J.N.: Spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equation. J. Comput. Phys. 190(2), 523–549 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Proot, M.M.J., Gerritsma, M.I.: A least-squares spectral element formulation for Stokes problem. J. Comput. Phys. 17(1–4), 285–296 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Schwab, C.: \(p\) and \(h-p\) Finite Element Methods. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  34. 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  MathSciNet  MATH  Google Scholar 

  35. Shen, J., Tang, T.: Spectral and High-order Methods with Applications. Science Press, Beijing (2006)

  36. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  37. Tomar, S.K.: \(h-p\) Spectral element methods for elliptic problems over non-smooth domains using parallel computers. Computing 78, 117–143 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tomar, S.K., Dutt, P., Kumar, B.V.R.: An efficient and exponentially accurate parallel \(h-p\) spectral element method for elliptic problems on polygonal domains—the Dirichilet case. Lecture Notes in Computer Science 2552, High Performance Computing. Springer, Berlin (2002)

  39. Yu, X.H., Guo, B.Y.: Spectral element method for mixed inhomogeneous boundary value problems of fourth order. J. Sci. Comput. 61(3), 673–701 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yu, X.H., Guo, B.Y.: Spectral method for fourth-order problems on quadrilaterals. J. Sci. Comput. 66(2), 1–27 (2015)

    MathSciNet  Google Scholar 

  41. Zhang, S., Xu, J.: Optimal solvers for fourth-order PDEs discretized on unstructured grids. SIAM J. Numer. Anal. 52(1), 282–307 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhao, J.: Multigrid methods for fourth order problems. Ph.D. Thesis, University of South Carolina. www.math.sc.edu/~fem/zhaodiss.pdf (2004)

  43. Zhu, Z.: The least-squares spectral element method solution of the gas-liquid multi-fluid model coupled with the population balance equation. Ph.D. Thesis, NTNU Norway. www.diva-portal.org/smash/get/diva2:297012/FULLTEXT03.pdf (2009)

  44. Žitňan, P.: Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary. Appl. Math. Comput. 217(22), 8973–8982 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Ruprecht-Karls-Universität Heidelberg, 69120, Heidelberg, Germany

    Arbaz Khan

  2. School of Engineering and Technology, BML Munjal University, Gurgaon, Haryana, 122413, India

    Akhlaq Husain

Authors
  1. Arbaz Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akhlaq Husain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arbaz Khan.

Additional information

The first author is thankful to the LNM Institute of Information Technology (LNMIIT), Jaipur for supporting his visits to carry out this work and also thankful to IIT Kanpur for providing the parallel computers facilities to compute the numerical results. Research is also supported by Mathematics Center Heidelberg (MATCH), Ruprecht-Karls-Universität Heidelberg, 69120 Heidelberg, Germany.

This work was carried out during the second author’s stay at the LNM Institute of Information Technology (LNMIIT), Jaipur as assistant professor.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, A., Husain, A. Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems. J Sci Comput 71, 303–328 (2017). https://doi.org/10.1007/s10915-016-0300-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0300-z

Keywords

Search

Navigation