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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


IMPACT FACTOR 2017: 0.658

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1609-9389
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Volume 18, Issue 1

Issues

Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator

Petr N. Vabishchevich
  • Corresponding author
  • Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya,115191 Moscow; and Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow,Russia
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Published Online: 2017-08-17 | DOI: https://doi.org/10.1515/cmam-2017-0028

Abstract

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Padé-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

Keywords: Elliptic Operator; Fractional Power of an Operator; Finite Element Approximation; Two-Level Schemes; Stability Of Difference Schemes

MSC 2010: 26A33; 35R11; 65F60; 65M06

References

  • [1]

    L. Aceto and P. Novati, Rational approximation to the fractional Laplacian operator in reaction-diffusion problems, SIAM J. Sci. Comput. 39 (2017), no. 1, 214–228. CrossrefWeb of ScienceGoogle Scholar

  • [2]

    G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal. 55 (2017), no. 2, 472–495. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Softw. 3 (2015), no. 100. Google Scholar

  • [4]

    A. Bonito and J. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp. 84 (2015), no. 295, 2083–2110. CrossrefGoogle Scholar

  • [5]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2008. Google Scholar

  • [6]

    A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT 54 (2014), no. 4, 937–954. Web of ScienceCrossrefGoogle Scholar

  • [7]

    K. Burrage, N. Hale and D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput. 34 (2012), no. 4, A2145–A2172. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    C. M. Carracedo, M. S. Alix and M. Sanz, The Theory of Fractional Powers of Operators, Elsevier, Amsterdam, 2001. Google Scholar

  • [9]

    H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon Press, New York, 1986. Google Scholar

  • [10]

    A. Frommer, S. Güttel and M. Schweitzer, Efficient and stable Arnoldi restarts for matrix functions based on quadrature, SIAM J. Matrix Anal. Appl. 35 (2014), no. 2, 661–683. Web of ScienceCrossrefGoogle Scholar

  • [11]

    W. Gautschi, Quadrature formulae on half-infinite intervals, BIT 31 (1991), no. 3, 437–446. CrossrefGoogle Scholar

  • [12]

    W. Gautschi, Algorithm 726: ORTHPOL – A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software 20 (1994), no. 1, 21–62. CrossrefGoogle Scholar

  • [13]

    W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004. Google Scholar

  • [14]

    I. Gavrilyuk, W. Hackbusch and B. Khoromskij, Data-sparse approximation to the operator-valued functions of elliptic operator, Math. Comp. 73 (2004), no. 247, 1297–1324. Google Scholar

  • [15]

    I. Gavrilyuk, W. Hackbusch and B. Khoromskij, Data-sparse approximation to a class of operator-valued functions, Math. Comp. 74 (2005), no. 250, 681–708. Google Scholar

  • [16]

    V. Hernandez, J. E. Roman and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software (TOMS) 31 (2005), no. 3, 351–362. CrossrefGoogle Scholar

  • [17]

    N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008. Google Scholar

  • [18]

    Q. Huang, G. Huang and H. Zhan, A finite element solution for the fractional advection–dispersion equation, Adv. Water Res. 31 (2008), no. 12, 1578–1589. CrossrefGoogle Scholar

  • [19]

    M. Ilic, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation. II: With nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal. 9 (2006), no. 4, 333–349. Google Scholar

  • [20]

    M. Ilić, I. W. Turner and V. Anh, A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional Poisson equation, Int. J. Stoch. Anal. 2008 (2008), Article ID 104525. Google Scholar

  • [21]

    B. Jin, R. Lazarov, J. Pasciak and Z. Zhou, Error analysis of finite element methods for space-fractional parabolic equations, SIAM J. Numer. Anal. 52 (2014), no. 5, 2272–2294. CrossrefGoogle Scholar

  • [22]

    P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer, New York, 2003. Google Scholar

  • [23]

    M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik and P. E. Sobolevskij, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976. Google Scholar

  • [24]

    A. Logg, K.-A. Mardal, G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer, Berlin, 2012. Google Scholar

  • [25]

    M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65–77. CrossrefGoogle Scholar

  • [26]

    I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, 1998. Google Scholar

  • [27]

    A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, 2002. Google Scholar

  • [28]

    A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1994. Google Scholar

  • [29]

    A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, Dover Publications, Mineola, 2001. Google Scholar

  • [30]

    Y. Saad, Numerical Methods for Large Eigenvalue Problems, SIAM, Philadelphia, 2011. Google Scholar

  • [31]

    A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, 2001. Google Scholar

  • [32]

    A. A. Samarskii, P. P. Matus and P. N. Vabishchevich, Difference Schemes with Operator Factors, Kluwer Academic, Dordrecht, 2002. Google Scholar

  • [33]

    E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation, Comput. Math. Appl. 64 (2012), no. 10, 3141–3152. Web of ScienceCrossrefGoogle Scholar

  • [34]

    G. W. Stewart, A Krylov–Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl. 23 (2001), no. 3, 601–614. Google Scholar

  • [35]

    B. J. Szekeres and F. Izsák, Finite element approximation of fractional order elliptic boundary value problems, J. Comput. Appl. Math. 292 (2016), 553–561. Web of ScienceCrossrefGoogle Scholar

  • [36]

    C. Tadjeran, M. M. Meerschaert and H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), no. 1, 205–213. CrossrefGoogle Scholar

  • [37]

    V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 2006. Google Scholar

  • [38]

    V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Applications, Higher Education Press, Beijing, 2013. Google Scholar

  • [39]

    V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Background and Theory, Higher Education Press, Beijing, 2013. Google Scholar

  • [40]

    P. N. Vabishchevich, Additive Operator-Difference Schemes: Splitting Schemes, De Gruyter, Berlin, 2014. Google Scholar

  • [41]

    P. N. Vabishchevich, Numerically solving an equation for fractional powers of elliptic operators, J. Comput. Phys. 282 (2015), no. 1, 289–302. CrossrefWeb of ScienceGoogle Scholar

  • [42]

    P. N. Vabishchevich, Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Model. 13 (2016), no. 2, 296–309. Google Scholar

  • [43]

    P. N. Vabishchevich, Numerical solution of nonstationary problems for a space-fractional diffusion equation, Fract. Calc. Appl. Anal. 19 (2016), no. 1, 116–139. Web of ScienceGoogle Scholar

  • [44]

    P. N. Vabishchevich, Numerical solving unsteady space-fractional problems with the square root of an elliptic operator, Math. Model. Anal. 21 (2016), no. 2, 220–238. Web of ScienceCrossrefGoogle Scholar

  • [45]

    A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer, Berlin, 2009. Google Scholar

About the article

Received: 2017-04-17

Revised: 2017-07-14

Accepted: 2017-07-18

Published Online: 2017-08-17

Published in Print: 2018-01-01


The publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement # 02.a03.21.0008).


Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 1, Pages 111–128, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0028.

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