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Journal of Scientific Computing

Erschienen in:

23.05.2017

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

verfasst von: Arbaz Khan, Pravir Dutt, Chandra Shekhar Upadhyay

Erschienen in: Journal of Scientific Computing | Ausgabe 2-3/2017

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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.

Vorheriger Artikel Offline-Enhanced Reduced Basis Method Through Adaptive Construction of the Surrogate Training Set

Nächster Artikel An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations

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

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

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

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

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

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

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

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)CrossRefMATHMathSciNet

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

10.

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

11.

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

12.

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

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)CrossRefMathSciNet

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)CrossRefMathSciNet

15.

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

16.

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

17.

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

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)CrossRefMATHMathSciNet

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)CrossRefMATHMathSciNet

20.

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

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)CrossRefMATHMathSciNet

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)CrossRefMATHMathSciNet

23.

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

24.

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

25.

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

26.

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

27.

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

28.

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

29.

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

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)CrossRefMATHMathSciNet

Titel: Spectral Element Method for Parabolic Initial Value Problem with Non-Smooth Data: Analysis and Application
verfasst von: Arbaz Khan
Pravir Dutt
Chandra Shekhar Upadhyay
Publikationsdatum: 23.05.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2-3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0457-0

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