Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

Jin, Bangti; Li, Buyang; Zhou, Zhi

doi:10.1090/mcom/3413

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution
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Math. Comp. 88 (2019), 2157-2186 Request permission

Abstract:

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.

References

Georgios Akrivis, Buyang Li, and Christian Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations, Math. Comp. 86 (2017), no. 306, 1527–1552. MR 3626527, DOI 10.1090/mcom/3228
Anatoly A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015), 424–438. MR 3273144, DOI 10.1016/j.jcp.2014.09.031
Emilia Grigorova Bajlekova, Fractional evolution equations in Banach spaces, Eindhoven University of Technology, Eindhoven, 2001. Dissertation, Technische Universiteit Eindhoven, Eindhoven, 2001. MR 1868564
Samuil D. Eidelman and Anatoly N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), no. 2, 211–255. MR 2047909, DOI 10.1016/j.jde.2003.12.002
Charles M. Elliott and Stig Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp. 58 (1992), no. 198, 603–630, S33–S36. MR 1122067, DOI 10.1090/S0025-5718-1992-1122067-1
Hiroshi Fujita and Takashi Suzuki, Evolution problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 789–928. MR 1115241
Guang-hua Gao, Zhi-zhong Sun, and Hong-wei Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys. 259 (2014), 33–50. MR 3148558, DOI 10.1016/j.jcp.2013.11.017
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
Bangti Jin, Raytcho Lazarov, Joseph Pasciak, and Zhi Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal. 35 (2015), no. 2, 561–582. MR 3335216, DOI 10.1093/imanum/dru018
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview, Comput. Methods Appl. Mech. Engrg. 346 (2019), 332–358. MR 3894161, DOI 10.1016/j.cma.2018.12.011
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), no. 1, 445–466. MR 3033018, DOI 10.1137/120873984
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput. 38 (2016), no. 1, A146–A170. MR 3449907, DOI 10.1137/140979563
Bangti Jin, Buyang Li, and Zhi Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal. 56 (2018), no. 1, 1–23. MR 3742688, DOI 10.1137/16M1089320
Samir Karaa, Semidiscrete finite element analysis of time fractional parabolic problems: a unified approach, SIAM J. Numer. Anal. 56 (2018), no. 3, 1673–1692. MR 3816184, DOI 10.1137/17M1134160
Samir Karaa, Kassem Mustapha, and Amiya K. Pani, Finite volume element method for two-dimensional fractional subdiffusion problems, IMA J. Numer. Anal. 37 (2017), no. 2, 945–964. MR 3649431, DOI 10.1093/imanum/drw010
Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
Adam Kubica and Masahiro Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal. 21 (2018), no. 2, 276–311. MR 3814402, DOI 10.1515/fca-2018-0018
Peer C. Kunstmann, Buyang Li, and Christian Lubich, Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity, Found. Comput. Math. 18 (2018), no. 5, 1109–1130. MR 3857906, DOI 10.1007/s10208-017-9364-x
Hyoseop Lee, Jinwoo Lee, and Dongwoo Sheen, Laplace transform method for parabolic problems with time-dependent coefficients, SIAM J. Numer. Anal. 51 (2013), no. 1, 112–125. MR 3033003, DOI 10.1137/110824000
Buyang Li and Weiwei Sun, Regularity of the diffusion-dispersion tensor and error analysis of Galerkin FEMs for a porous medium flow, SIAM J. Numer. Anal. 53 (2015), no. 3, 1418–1437. MR 3355773, DOI 10.1137/140958803
Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533–1552. MR 2349193, DOI 10.1016/j.jcp.2007.02.001
Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), no. 3, 704–719. MR 838249, DOI 10.1137/0517050
Yury Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl. 351 (2009), no. 1, 218–223. MR 2472935, DOI 10.1016/j.jmaa.2008.10.018
Mitchell Luskin and Rolf Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal. 19 (1982), no. 1, 93–113. MR 646596, DOI 10.1137/0719003
William McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J. 52 (2010), no. 2, 123–138. MR 2832607, DOI 10.1017/S1446181111000617
William McLean and Kassem Mustapha, Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithms 52 (2009), no. 1, 69–88. MR 2533995, DOI 10.1007/s11075-008-9258-8
William McLean and Kassem Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, J. Comput. Phys. 293 (2015), 201–217. MR 3342467, DOI 10.1016/j.jcp.2014.08.050
Kassem Mustapha, FEM for time-fractional diffusion equations, novel optimal error analyses, Math. Comp. 87 (2018), no. 313, 2259–2272. MR 3802434, DOI 10.1090/mcom/3304
Kenichi Sakamoto and Masahiro Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. MR 2805524, DOI 10.1016/j.jmaa.2011.04.058
Peter Sammon, Fully discrete approximation methods for parabolic problems with nonsmooth initial data, SIAM J. Numer. Anal. 20 (1983), no. 3, 437–470. MR 701091, DOI 10.1137/0720031
Giuseppe Savaré, $A(\Theta )$-stable approximations of abstract Cauchy problems, Numer. Math. 65 (1993), no. 3, 319–335. MR 1227025, DOI 10.1007/BF01385755
Martin Stynes, Eugene O’Riordan, and José Luis Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057–1079. MR 3639581, DOI 10.1137/16M1082329
Zhi-zhong Sun and Xiaonan Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193–209. MR 2200938, DOI 10.1016/j.apnum.2005.03.003
Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
Yubin Yan, Monzorul Khan, and Neville J. Ford, An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data, SIAM J. Numer. Anal. 56 (2018), no. 1, 210–227. MR 3744997, DOI 10.1137/16M1094257
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), no. 5, 1862–1874. MR 2139227, DOI 10.1137/030602666
Rico Zacher, A De Giorgi–Nash type theorem for time fractional diffusion equations, Math. Ann. 356 (2013), no. 1, 99–146. MR 3038123, DOI 10.1007/s00208-012-0834-9