A posteriori error analysis for approximations of time-fractional subdiffusion problems

Banjai, Lehel; Makridakis, Charalambos

doi:10.1090/mcom/3723

A posteriori error analysis for approximations of time-fractional subdiffusion problems
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Math. Comp. 91 (2022), 1711-1737 Request permission

Abstract:

In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori error estimates. Our approach is based on appropriate pointwise representations of the numerical schemes as perturbed evolution equations and on stability estimates for the evolution equation. A posteriori error estimates in $L^2(H)$ and $L^\infty (H)$ norms of optimal order are derived. Extensive numerical experiments indicate the reliability and the optimality of the estimators for the schemes considered, as well as their efficiency as error indicators driving adaptive mesh selection locating singularities of the problem.

References

Georgios Akrivis, Charalambos Makridakis, and Ricardo H. Nochetto, Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods, Numer. Math. 114 (2009), no. 1, 133–160. MR 2557872, DOI 10.1007/s00211-009-0254-2
K. Baker and L. Banjai, Numerical analysis of a wave equation for lossy media obeying a frequency power law, IMA J. of Numer. Anal.5 (2021), drab028.
L. Banjai and M. López-Fernández, Efficient high order algorithms for fractional integrals and fractional differential equations, Numer. Math. 141 (2019), no. 2, 289–317. MR 3905428, DOI 10.1007/s00211-018-1004-0
L. Banjai and F.-J. Sayas, Integral Equation Methods for Evolutionary PDE: A Convolution Quadrature Approach, Springer Series in Computational Mathematics, in preparation.
E. Bänsch, F. Karakatsani, and C. G. Makridakis, A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem, Calcolo 55 (2018), no. 2, Paper No. 19, 32. MR 3797748, DOI 10.1007/s10092-018-0259-2
Emilia Bazhlekova, The abstract Cauchy problem for the fractional evolution equation, Fract. Calc. Appl. Anal. 1 (1998), no. 3, 255–270. MR 1657572
Hermann Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004. MR 2128285, DOI 10.1017/CBO9780511543234
Andrea Cangiani, Emmanuil H. Georgoulis, Irene Kyza, and Stephen Metcalfe, Adaptivity and blow-up detection for nonlinear evolution problems, SIAM J. Sci. Comput. 38 (2016), no. 6, A3833–A3856. MR 3584583, DOI 10.1137/16M106073X
Carl de Boor, Divided differences, Surv. Approx. Theory 1 (2005), 46–69. MR 2221566
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Emmanuil H. Georgoulis, Omar Lakkis, Charalambos G. Makridakis, and Juha M. Virtanen, A posteriori error estimates for leap-frog and cosine methods for second order evolution problems, SIAM J. Numer. Anal. 54 (2016), no. 1, 120–136. MR 3448347, DOI 10.1137/140996318
Rudolf Gorenflo, Francesco Mainardi, Daniele Moretti, and Paolo Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dynam. 29 (2002), no. 1-4, 129–143. Fractional order calculus and its applications. MR 1926470, DOI 10.1023/A:1016547232119
Wolfgang Hackbusch, Integral equations, International Series of Numerical Mathematics, vol. 120, Birkhäuser Verlag, Basel, 1995. Theory and numerical treatment; Translated and revised by the author from the 1989 German original. MR 1350296, DOI 10.1007/978-3-0348-9215-5
Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Res. Res. 34 (1998), no. 5, 1027-1033
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal. 36 (2016), no. 1, 197–221. MR 3463438, DOI 10.1093/imanum/dru063
Bangti Jin, Buyang Li, and Zhi Zhou, Correction of high-order BDF convolution quadrature for fractional evolution equations, SIAM J. Sci. Comput. 39 (2017), no. 6, A3129–A3152. MR 3738850, DOI 10.1137/17M1118816
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
Theodoros Katsaounis and Irene Kyza, A posteriori error analysis for evolution nonlinear Schrödinger equations up to the critical exponent, SIAM J. Numer. Anal. 56 (2018), no. 3, 1405–1434. MR 3803285, DOI 10.1137/16M1108029
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
Omar Lakkis and Charalambos Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75 (2006), no. 256, 1627–1658. MR 2240628, DOI 10.1090/S0025-5718-06-01858-8
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
María López-Fernández, Christian Lubich, and Achim Schädle, Adaptive, fast, and oblivious convolution in evolution equations with memory, SIAM J. Sci. Comput. 30 (2008), no. 2, 1015–1037. MR 2385897, DOI 10.1137/060674168
M. Lopez-Fernandez and S. Sauter, Generalized convolution quadrature based on Runge-Kutta methods, Numer. Math. 133 (2016), no. 4, 743–779. MR 3520008, DOI 10.1007/s00211-015-0761-2
Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), no. 3, 704–719. MR 838249, DOI 10.1137/0517050
C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. MR 923707, DOI 10.1007/BF01398686
Christian Lubich, Convolution quadrature revisited, BIT 44 (2004), no. 3, 503–514. MR 2106013, DOI 10.1023/B:BITN.0000046813.23911.2d
Christian Lubich and Charalambos Makridakis, Interior a posteriori error estimates for time discrete approximations of parabolic problems, Numer. Math. 124 (2013), no. 3, 541–557. MR 3066039, DOI 10.1007/s00211-013-0520-1
Charalambos Makridakis, Space and time reconstructions in a posteriori analysis of evolution problems, ESAIM Proceedings. Vol. 21 (2007) [Journées d’Analyse Fonctionnelle et Numérique en l’honneur de Michel Crouzeix], ESAIM Proc., vol. 21, EDP Sci., Les Ulis, 2007, pp. 31–44. MR 2404052, DOI 10.1051/proc:072104
Charalambos Makridakis and Ricardo H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585–1594. MR 2034895, DOI 10.1137/S0036142902406314
William McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput. 34 (2012), no. 6, A3039–A3056. MR 3029841, DOI 10.1137/120870505
Ralf Metzler and Joseph Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (2000), no. 1-2, 107–125. MR 1763650, DOI 10.1016/S0378-4371(99)00503-8
R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, physica status solidi (b) 133 (1986), no. 1, 425-430.
Ricardo H. Nochetto, Giuseppe Savaré, and Claudio Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525–589. MR 1737503, DOI 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M
Ljubica Oparnica and Endre Süli, Well-posedness of the fractional Zener wave equation for heterogeneous viscoelastic materials, Fract. Calc. Appl. Anal. 23 (2020), no. 1, 126–166. MR 4069924, DOI 10.1515/fca-2020-0005
F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.1 of 2021-03-15.
H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A 27 (1994), no. 10, 3407–3410. MR 1282181, DOI 10.1088/0305-4470/27/10/017
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
Dominik Schötzau and Thomas P. Wihler, A posteriori error estimation for $hp$-version time-stepping methods for parabolic partial differential equations, Numer. Math. 115 (2010), no. 3, 475–509. MR 2640055, DOI 10.1007/s00211-009-0285-8
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
K. Šiškova, Inverse source problems in evolutionary PDEs, PhD thesis, Ghent University, 2018.
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