nach oben

Journal of Scientific Computing

Erschienen in:

01.03.2021

Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval

verfasst von: Xiangcheng Zheng, V. J. Ervin, Hong Wang

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2021

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov–Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.

Vorheriger Artikel A Local Radial Basis Function Method for the Laplace–Beltrami Operator

Nächster Artikel Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington (1964)

Acosta, G., Borthagaray, J.P., Bruno, O., Maas, M.: Regularity theory and high order numerical methods for the (1-d)-fractional Laplacian. Math. Comput. 87, 1821–1857 (2018)MathSciNetCrossRef

Babuška, I., Guo, B.: Direct and inverse approximation theorems for the \(p\)-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces. SIAM J. Numer. Anal. 39(5):1512–1538 (2001/02)

Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36(6), 1413–1424 (2000)CrossRef

Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev World. Preprint IRMAR 07-14, Université de Rennes 1 (2007)

Bernardi, C., Dauge, M., Maday, Y.: Polynomials in weighted Sobolev spaces: basics and trace liftings. Internal Report 92039, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris (1992)

Buades, A., Coll, B., Morel, J.M.:Image denoising methods. A new nonlocal principle. SIAM Rev. 52(1):113–147 (2010). Reprint of “A review of image denoising algorithms, with a new one” [MR2162865]

Chen, H., Wang, H.: Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation. J. Comput. Appl. Math. 296, 480–498 (2016)MathSciNetCrossRef

Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)MathSciNetCrossRef

10.

Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228(20), 7792–7804 (2009)MathSciNetCrossRef

11.

Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRef

12.

Ervin, V.J.: Regularity of the solution to fractional diffusion, advection, reaction equations. https://arxiv.org/abs/1911.03261 (2019)

13.

Ervin, V.J., Heuer, N., Roop, J.P.: Regularity of the solution to 1-D fractional order diffusion equations. Math. Comput. 87, 2273–2294 (2018)MathSciNetCrossRef

14.

Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)MathSciNetCrossRef

15.

Gatto, P., Hesthaven, J.S.: Numerical approximation of the fractional Laplacian via \(hp\)-finite elements, with an application to image denoising. J. Sci. Comput. 65(1), 249–270 (2015)MathSciNetCrossRef

16.

Ginting, V., Li, Y.: On the fractional diffusion–advection–reaction equation in \({\mathbb{R}}\). Fract. Calc. Appl. Anal. 22(4), 1039–1062 (2019)MathSciNetCrossRef

17.

Gui, W., Babuška, I.: The \(h,\;p\) and \(h\)-\(p\) versions of the finite element method in \(1\) dimension. II. The error analysis of the \(h\)- and \(h\)-\(p\) versions. Numer. Math. 49(6), 613–657 (1986)MathSciNetCrossRef

18.

Guo, B.-Y., Wang, L.-L.: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 128(1), 1–41 (2004)MathSciNetCrossRef

19.

Hao, Z., Lin, G., Zhang, Z.: Error estimates of a spectral Petrov–Galerkin method for two-sided fractional reaction–diffusion equations. Appl. Math. Comput. 374, 125045 (2020)MathSciNetMATH

20.

Hao, Z., Zhang, Z.: Optimal regularity and error estimates of a spectral galerkin method for fractional advection–diffusion–reaction equations. SIAM J. Numer. Anal. 58(1), 211–233 (2020)MathSciNetCrossRef

21.

Jia, L., Chen, H., Ervin, V.J.: Existence and regularity of solutions to 1-D fractional order diffusion equations. Electron. J. Differ. Equ. 93, 1–21 (2019)MathSciNetMATH

22.

Jin, B., Lazarov, R., Lu, X., Zhou, Z.: A simple finite element method for boundary value problems with a Riemann–Liouville derivative. J. Comput. Appl. Math. 293, 94–111 (2016)MathSciNetCrossRef

23.

Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84(296), 2665–2700 (2015)MathSciNetCrossRef

24.

Jin, B., Lazarov, R., Zhou, Z.: A Petrov–Galerkin finite element method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 54(1), 481–503 (2016)MathSciNetCrossRef

25.

Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15(3), 383–406 (2012)MathSciNetCrossRef

26.

Li, Y., Chen, H., Wang, H.: A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations. Math. Methods Appl. Sci. 40(14), 5018–5034 (2017)MathSciNetCrossRef

27.

Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. In: Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods (BAIL 2002), vol. 166, pp. 209–219 (2004)

28.

Liu, Q., Liu, F., Turner, I., Anh, V.: Finite element approximation for a modified anomalous subdiffusion equation. Appl. Math. Model. 35(8), 4103–4116 (2011)MathSciNetCrossRef

29.

Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine. 1996), Volume 378 of CISM Courses and Lectures, pp. 291–348. Springer, Vienna (1997)

30.

Mao, Z., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)MathSciNetCrossRef

31.

Mao, Z., Em-Karniadakis, G.: A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56(1), 24–49 (2018)MathSciNetCrossRef

32.

Mao, Z., Shen, J.: Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016)MathSciNetCrossRef

33.

Mao, Z., Shen, J.: Spectral element method with geometric mesh for two-sided fractional differential equations. Adv. Comput. Math. 44(3), 745–771 (2018)MathSciNetCrossRef

34.

Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)MathSciNetCrossRef

35.

Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987)MathSciNetCrossRef

36.

Szegő, G.: Orthogonal polynomials. American Mathematical Society, Providence, R.I., 4th edition, 1975. American Mathematical Society, Colloquium Publications, vol. XXIII

37.

Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220(2), 813–823 (2007)MathSciNetCrossRef

38.

Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34(5), A2444–A2458 (2012)MathSciNetCrossRef

39.

Wang, H., Yang, D.: Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51(2), 1088–1107 (2013)MathSciNetCrossRef

40.

Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014)MathSciNetCrossRef

41.

Zaslavsky, G.M., Stevens, D., Weitzner, H.: Self-similar transport in incomplete chaos. Phys. Rev. E (3) 48(3), 1683–1694 (1993)MathSciNetCrossRef

42.

Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)MathSciNetCrossRef

43.

Zheng, X., Ervin, V.J., Wang, H.: Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension. Appl. Math. Comput. 361, 98–111 (2019)MathSciNetMATH

44.

Zheng, X., Ervin, V.J., Wang, H.: An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates. Commun. Appl. Math. Comput. 2(1), 147–162 (2020)MathSciNetCrossRef

Titel: Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval
verfasst von: Xiangcheng Zheng
V. J. Ervin
Hong Wang
Publikationsdatum: 01.03.2021
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2021
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01366-y

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2021

Developing and Analyzing New Unconditionally Stable Finite Element Schemes for Maxwell’s Equations in Complex Media

Weak Convergence Rates for an Explicit Full-Discretization of Stochastic Allen–Cahn Equation with Additive Noise

Volumetric Density-Equalizing Reference Map with Applications

Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

A Posteriori Subcell Finite Volume Limiter for General Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

On a New Spatial Discretization for a Regularized 3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System of Equations with Boundary Conditions

Premium Partner