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Numerical approximation of stochastic time-fractional diffusion. (English) Zbl 1447.60126

Summary: We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order \(\alpha \in\) (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order \(\gamma \in\) [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the \(L^2\)-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] E.J. Allen, S.J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Stoch. Rep. 64 (1998) 117-142. · Zbl 0907.65147 · doi:10.1080/17442509808834159
[2] A. Andersson, M. Kovács and S. Larsson, Weak error analysis for semilinear stochastic Volterra equations with additive noise. J. Math. Anal. Appl. 437 (2016) 1283-1304. · Zbl 1333.60154
[3] A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximation of SPDE. Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016) 113-149. · Zbl 1357.60063
[4] A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation. Math. Comp. 85 (2016) 1335-1358. · Zbl 1332.65143 · doi:10.1090/mcom/3016
[5] V.V. Anh, N.N. Leonenko and M. Ruiz-Medina, Space-time fractional stochastic equations on regular bounded open domains. Fract. Calc. Appl. Anal. 19 (2016) 1161-1199. · Zbl 1354.60065
[6] W. Arendt, C.J. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. 2nd edition. Birkhäuser, Basel (2011) · Zbl 1226.34002 · doi:10.1007/978-3-0348-0087-7
[7] D. Baffet and J.S. Hesthaven, A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal. 55 (2017) 496-520. · Zbl 1359.65106
[8] 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) 1-31. · Zbl 1325.76113
[9] C.-E. Bréhier, M. Hairer and A.M. Stuart, Weak error estimates for trajectories of SPDEs under spectral Galerkin discretization. J. Comput. Math. 36 (2018) 159-182. · Zbl 1413.65008
[10] L. Chen, Nonlinear stochastic time-fractional diffusion equations on R: moments, Hölder regularity and intermittency. Trans. Amer. Math. Soc. 369 (2017) 8497-8535. · Zbl 1406.60093 · doi:10.1090/tran/6951
[11] L. Chen, Y. Hu and D. Nualart, Nonlinear stochastic time-fractional slow and fast diffusion equations on Formula . Preprint (2015).
[12] Z.-Q. Chen, K.-H. Kim and P. Kim, Fractional time stochastic partial differential equations. Stochastic Process. Appl. 125 (2015) 1470-1499. · Zbl 1322.60106 · doi:10.1016/j.spa.2014.11.005
[13] E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75 (2006) 673-696. · Zbl 1090.65147 · doi:10.1090/S0025-5718-06-01788-1
[14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd edition. Cambridge University Press, Cambridge (2014). · Zbl 1317.60077 · doi:10.1017/CBO9781107295513
[15] A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation. Math. Comp. 78 (2009) 845-863. · Zbl 1215.60043 · doi:10.1090/S0025-5718-08-02184-4
[16] Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal. 40 (2002) 1421-1445. · Zbl 1030.65002
[17] M. Foondun, Remarks on a fractional-time stochastic equation. Preprint (2018). · Zbl 1467.60047
[18] M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30 (2002) 1397-1465. · Zbl 1017.60076
[19] H. Fujita and T. Suzuki, Evolution problems. In: Handbook of Numerical Analysis. Vol. II. NorthHolland, Amsterdam (1991) 789-928. · Zbl 0875.65084
[20] M.B. Giles, Multilevel Monte Carlo methods. Acta Numer. 24 (2015) 259-328. · Zbl 1316.65010 · doi:10.1017/S096249291500001X
[21] M. Gunzburger, B. Li and J. Wang, Convergence of finite element solution of stochastic partial integral-differential equations driven by white noise. Preprint (2017).
[22] A. Jentzen and P.E. Kloeden, The numerical approximation of stochastic partial differential equations. Milan J. Math. 77 (2009) 205-244. · Zbl 1205.60130 · doi:10.1007/s00032-009-0100-0
[23] 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) 445-466. · Zbl 1268.65126
[24] B. Jin, R. Lazarov and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38 (2016) A146-A170. · Zbl 1381.65082
[25] B. Jin, R. Lazarov and Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview. Comput. Methods Appl. Mech. Eng. 346 (2019) 332-358. · Zbl 1440.65138
[26] B. Jin, B. Li and Z. Zhou, Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39 (2017) A3129-A3152. · Zbl 1379.65078
[27] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006). · Zbl 1092.45003
[28] M. Kovács and J. Printems, Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation. Math. Comp. 83 (2014) 2325-2346. · Zbl 1312.60082 · doi:10.1090/S0025-5718-2014-02803-2
[29] M. Kovács and J. Printems, Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term. J. Math. Anal. Appl. 413 (2014) 939-952. · Zbl 1325.60116
[30] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, In Vol. 2093, Lecture Notes in Mathematics. Springer, Heidelberg (2014). · Zbl 1285.60002 · doi:10.1007/978-3-319-02231-4
[31] X. Li and X. Yang, Error estimates of finite element methods for stochastic fractional differential equations. J. Comput. Math. 35 (2017) 346-362. · Zbl 1399.65212
[32] W. Liu, M. Röckner and J.L. da Silva, Quasi-linear (stochastic) partial differential equations with time-fractional derivatives. SIAM J. Math. Anal. 50 (2018) 2588-2607. · Zbl 1407.60086 · doi:10.1137/17M1144593
[33] S.V. Lototsky and B.L. Rozovsky, Classical and generalized solutions of fractional stochastic differential equations. Preprint. (2018) . · Zbl 1434.60004
[34] C. Lubich, Discretized fractional calculus. SIAM J. Math. Anal. 17 (1986) 704-719. · Zbl 0624.65015 · doi:10.1137/0517050
[35] 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) 1-17. · Zbl 0852.65138 · doi:10.1090/S0025-5718-96-00677-1
[36] W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data. J. Comput. Phys. 293 (2015) 201-217. · Zbl 1349.65469
[37] R. Metzler, J.H. Jeon, A.G. Cherstvy and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16 (2014) 24128-24164. · doi:10.1039/C4CP03465A
[38] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. 2nd edition. Springer-Verlag, Berlin (2006). · Zbl 1105.65102
[39] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43 (2005) 1363-1384. · Zbl 1112.60049
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