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Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. (English) Zbl 1381.65082
The authors analyse two schemes for subdiffusion and diffusion-wave equations. The schemes use the Galerkin finite element method in space and the convolution quadrature, developed in [E. Cuesta et al., Math. Comput. 75, No. 254, 673–696 (2006; Zbl 1090.65147)], generated by the backward Euler method and second-order backward difference in time. Optimal error estimates for both smooth and nonsmooth initial data are derived. In particular, it is shown that the schemes achieve first-order and second-order convergence in time. Several numerical experiments are presented to support the theoretical results.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R11 Fractional partial differential equations
35L05 Wave equation
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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[1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resources Res., 28 (1992), pp. 3293–3307.
[2] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus, World Scientific, Hackensack, NJ, 2012. · Zbl 1248.26011
[3] E. Bazhlekova, B. Jin, R. Lazarov, and Z. Zhou, An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), pp. 1–31. · Zbl 1325.76113
[4] C.-M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), pp. 886–897. · Zbl 1165.65053
[5] F. Chen, Q. Xu, and J. S. Hesthaven, A multi-domain spectral method for time-fractional differential equations, J. Comput. Phys., 293 (2015), pp. 157–172. · Zbl 1349.65506
[6] S. Chen, J. Shen, and L.-L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comp., to appear. · Zbl 1335.65066
[7] E. Cuesta, C. Lubich, and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), pp. 673–696. · Zbl 1090.65147
[8] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Math. 2004, Springer, New York, 2010.
[9] N. J. Ford, J. Xiao, and Y. Yan, A finite element method for time fractional partial differential equations, Fract. Calc. Appl. Anal., 14 (2011), pp. 454–474. · Zbl 1273.65142
[10] H. Fujita and T. Suzuki, Evolution problems, in Handbook of Numerical Analysis, Handb. Numer. Anal. II, North–Holland, Amsterdam, 1991, pp. 789–928. · Zbl 0875.65084
[11] Y. Fujita, Integrodifferential equation which interpolates the heat and the wave equation, Osaka J. Math., 27 (1990), pp. 309–321. · Zbl 0790.45009
[12] G.-H. Gao, Z.-Z. Sun, and H.-W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), pp. 33–50. · Zbl 1349.65088
[13] R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dynam., 29 (2002), pp. 129–143. · Zbl 1009.82016
[14] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin, 1993. · Zbl 0789.65048
[15] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Res., 34 (1998), pp. 1027–1033.
[16] B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Galerkin FEM for fractional order parabolic equations with initial data in \(H^{-s}\), \(0 ≤ s ≤ 1\), in Proceedings of the 5th Conference on Numerical Analysis and Applications (2012), Lecture Notes in Comput. Sci. 8236, Springer, New York, 2013, pp. 24–37. · Zbl 1352.65351
[17] B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of a finite element method for the space-fractional parabolic equation, SIAM J. Numer. Anal., 52 (2014), pp. 2272–2294. · Zbl 1310.65126
[18] B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal., 35 (2015), pp. 561–582. · Zbl 1321.65142
[19] B. Jin, R. Lazarov, and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51 (2013), pp. 445–466. · Zbl 1268.65126
[20] B. Jin, R. Lazarov, and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., to appear. · Zbl 1336.65150
[21] B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010. · Zbl 1247.35203
[22] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003. · Zbl 1323.34027
[23] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. · Zbl 1092.45003
[24] T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), pp. 719–736. · Zbl 1072.65123
[25] C. Li and H. Ding, Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model., 38 (2014), pp. 3802–3821.
[26] W. Li and D. Xu, Finite central difference/finite element approximations for parabolic integro-differential equations, Computing, 90 (2010), pp. 89–111. · Zbl 1207.65161
[27] X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108–2131. · Zbl 1193.35243
[28] Y. Lin, X. Li, and C. Xu, Finite difference/spectral approximations for the fractional cable equation, Math. Comp., 80 (2011), pp. 1369–1396. · Zbl 1220.78107
[29] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533–1552. · Zbl 1126.65121
[30] Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), pp. 704–719. · Zbl 0624.65015
[31] C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math., 52 (1988), pp. 129–145. · Zbl 0637.65016
[32] C. Lubich, I. H. Sloan, and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp., 65 (1996), pp. 1–17. · Zbl 0852.65138
[33] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7 (1996), pp. 1461–1477. · Zbl 1080.26505
[34] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. · Zbl 1210.26004
[35] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), pp. 123–138. · Zbl 1228.35266
[36] W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithms, 52 (2009), pp. 69–88. · Zbl 1177.65194
[37] W. McLean and V. Thomée, Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation, IMA J. Numer. Anal., 30 (2010), pp. 208–230. · Zbl 1416.65381
[38] E. W. Montroll and G. H. Weiss, Random walks on lattices. II, J. Math. Phys., 6 (1965), pp. 167–181. · Zbl 1342.60067
[39] K. Mustapha, B. Abdallah, and K. M. Furati, A discontinuous Petrov–Galerkin method for time-fractional diffusion equations, SIAM J. Numer. Anal., 52 (2014), pp. 2512–2529. · Zbl 1323.65109
[40] K. Mustapha and W. McLean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal., 51 (2013), pp. 491–515. · Zbl 1267.26005
[41] K. Mustapha and D. Schötzau, Well-posedness of \(hp\)-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA J. Numer. Anal., 34 (2014), pp. 1426–1446. · Zbl 1310.65128
[42] R. R. Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), pp. 425–430.
[43] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. · Zbl 0292.26011
[44] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. · Zbl 0924.34008
[45] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), pp. 426–447. · Zbl 1219.35367
[46] J. M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25 (1988), pp. 319–327. · Zbl 0643.65098
[47] H. Seybold and R. Hilfer, Numerical algorithm for calculating the generalized Mittag-Leffler function, SIAM J. Numer. Anal., 47 (2008), pp. 69–88. · Zbl 1190.65033
[48] Z.-Z. Sun and X. Wu, A fully discrete scheme for a diffusion wave system, Appl. Numer. Math., 56 (2006), pp. 193–209. · Zbl 1094.65083
[49] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math. 25, Springer-Verlag, Berlin, 2006.
[50] S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006), pp. 264–274. · Zbl 1094.65085
[51] S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862–1874. · Zbl 1119.65379
[52] F. Zeng, C. Li, F. Liu, and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), pp. A2976–A3000. · Zbl 1292.65096
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