Skip to main content
Log in

Spectral Element Method for Parabolic Initial Value Problem with Non-Smooth Data: Analysis and Application

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

Abstract

In this paper, a least-squares spectral element method for parabolic initial value problem for two space dimension on parallel computers is presented. The theory is also valid for three dimension. This method gives exponential accuracy in both space and time. The method is based on minimization of residuals in terms of the partial differential equation and initial condition, in different Sobolev norms, and a term which measures the jump in the function and its derivatives across inter-element boundaries in appropriate fractional Sobolev norms. Rigorous error estimates for this method are given. Some specific numerical examples are solved to show the efficiency of this method.

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

Access this article

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
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  2. Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer, Berlin (2009)

    MATH  Google Scholar 

  4. Boyle, P.P.: A lattice framework for option pricing with two state variables. J. Financial Quant. Anal. 23, 1–12 (1988)

    Article  Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  6. Dutt, P.K., Singh, A.K.: Spetral methods for periodic intial value problems with non smooth data. Math. Comp. 61, 645–658 (1993)

    Article  MathSciNet  Google Scholar 

  7. Dutt, P., Biswas, P., Ghorai, S.: Spectral element methods for parabolic problems. J. Comput. Appl. Math. 203, 461–486 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Dutt, P., Bedekar, S.: Spetral methods for hyperbolic intial boundary value problem on parallel computers. J. Comput. Appl. Math. 134, 164–190 (2001)

    Article  Google Scholar 

  10. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Application. SIAM, Philadelphia (1977)

    Book  MATH  Google Scholar 

  11. Hilber, N., Reichmann, O., Schwab, C., Winter, C.: Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  12. Khan, A., Dutt, P., Upadhyay, C.S.: Nonconforming least-squares spectral element method for European options. Comput. Math. Appl. 70, 47–65 (2015)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. Leentvaar, C.C.W., Oosterlee, C.W.: Multi-asset option pricing using a parallel Fourier-based technique. J. Comput. Finance 12, 1–26 (2008)

    Article  MathSciNet  Google Scholar 

  15. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  16. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications II. Springer, Berlin (1973)

    Book  MATH  Google Scholar 

  17. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications III. Springer, Berlin (1973)

    Book  MATH  Google Scholar 

  18. Meng, Q.J., Ding, D.: An efficient pricing method for rainbow options based on two-dimensional modified sinesine series expansions. Int. J. Comput. Math. 90, 1096–1113 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pontaza, J.P., Reddy, J.N.: Space-time coupled spectral/\(hp\) least-squares finite element formulation for the incompressible NavierStokes equations. J. Comput. Phys. 197, 418–459 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Proot, M.M.J., Gerrtisma, M.I.: Least-squares spectral elements applied to the Stokes problem. J. Comput. Phys. 181, 454–477 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ruijter, M.J., Oosterlee, C.W.: Two-dimensional Fourier cosine series expansion method for pricing financial options. SIAM J. Sci. Comput. 34, B642–B671 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schötzau, D., Schwab, C.: Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38, 837–875 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  25. Strain, J.: Fast adaptive methods for the free-space heat equation. SIAM J. Sci. Comput. 15, 185–206 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. Stulz, R.: Options on the minimum or the maximum of two risky assets: analysis and applications. J. Financial Econ. 10, 161–185 (1982)

    Article  Google Scholar 

  27. Tadmor, E.: Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numerica 16, 305–378 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tanner, J.: Optimal filter and mollifier for piecewise smooth spectral data. Math. Comp. 75, 767–790 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Tomar, S.K.: h-p spectral element method for elliptic problems on non-smooth domains using parallel computers. Computing 78, 117–143 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhu, W., Kopriva, D.A., Spectral, A.: Element method to price European options. II. The Black–Scholes model with two underlying assets. J. Sci. Comput. 39, 323–339 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arbaz Khan.

Additional information

Research supported by CSIR (Council of Scientific and Industrial Research, India) Grant 09/092(0712)/2009-EMR-I.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, A., Dutt, P. & Upadhyay, C.S. Spectral Element Method for Parabolic Initial Value Problem with Non-Smooth Data: Analysis and Application. J Sci Comput 73, 876–905 (2017). https://doi.org/10.1007/s10915-017-0457-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0457-0

Keywords

Navigation