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February  2019, 24(2): 615-635. doi: 10.3934/dcdsb.2018199

## Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion

 1 College of Information Science and Technology, Donghua University, 2999 North Renmin Rd., Shanghai 201620, China 2 Department of Mathematics, College of Science, Donghua University, 2999 North Renmin Rd., Shanghai 201620, China

§ Corresponding author: Xiuwei Yin

Received  May 2017 Revised  December 2017 Published  February 2019 Early access  June 2018

Fund Project: The Project-sponsored by NSFC (11571071), Innovation Program of Shanghai Municipal Education Commission (12ZZ063)) and The Fundamental Research Funds for the Central Universities (17D310403).

In this paper, we consider the numerical approximation for a class of fractional stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with hurst index $H∈ (\frac{1}{2}, 1)$. By using spectral Galerkin method, we analyze the spatial discretization, and we give the temporal discretization by using the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. Under some suitable assumptions, we prove the sharp regularity properties and the optimal strong convergence error estimates for both semi-discrete and fully discrete schemes.

Citation: Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
##### References:
 [1] F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, 2008. doi: 10.1007/978-1-84628-797-8. [2] M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent, Part Ⅲ, Geophys, J. R. Astron. Soc., 13 (1967), 529-539. [3] L. Chen, Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: Moments, Hölder regularity and intermittency, Trans. Amer. Math. Soc., 369 (2017), 8497-8535.  doi: 10.1090/tran/6951. [4] J. Cui and L. Yan, Controllability of neutral stochastic evolution equation driving by fractional Brownian motion, Acta. Math. Sci., 37 (2017), 108-118.  doi: 10.1016/S0252-9602(16)30119-9. [5] M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804.  doi: 10.1016/j.jcp.2009.07.021. [6] G. DaPrato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014. doi: 10.1017/CBO9781107295513. [7] T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Semilinear stochastic equations in a Hilbert space with a fractional brownian motion, SIAM J. Math. Anal., 40 (2009), 2286-2315.  doi: 10.1137/08071764X. [8] T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250.  doi: 10.1142/S0219493702000340. [9] Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc., 175 (2005), ⅷ+127 pp. doi: 10.1090/memo/0825. [10] A. Jentzen and P. E. Kloeden, The numerical approximation of stochastic partial differential equations, Milan J. Math., 77 (2009), 205-244.  doi: 10.1007/s00032-009-0100-0. [11] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972016. [12] 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.  doi: 10.1137/120873984. [13] M. Kamrani and N. Jamshidi, Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion, Commun. Nonlinear Sci. Numer., 44 (2017), 1-10.  doi: 10.1016/j.cnsns.2016.07.023. [14] 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.  doi: 10.1090/S0025-5718-2014-02803-2. [15] R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217-251.  doi: 10.1093/imanum/drs055. [16] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4. [17] X. Li and X. Yang, Error estimates of finite element methods for stochastic fractional differential equations, J. Comput. Math., 35 (2017), 346-362.  doi: 10.4208/jcm.1607-m2015-0329. [18] B. Maslowski and D. Nualart, Evolution equations driven by a fractional brownian motion, J. Funct. Anal., 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4. [19] W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a fractional diffusion equation, Numer. Algor., 52 (2009), 69-88.  doi: 10.1007/s11075-008-9258-8. [20] W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with nonsmooth initial data, J. Comput. Phys., 293 (2015), 201-217.  doi: 10.1016/j.jcp.2014.08.050. [21] A. L. Mehaute, T. Machado, J. C. Trigeassou and J. Sabatier, Fractional differential and its applications, FDA'04, Proceedings of the first IFAC workshop, vol. 2004-1, International Federation of Autromatic Control, ENSEIRB, Bordeaux, France, July 19-21, 2004. [22] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, 2008. doi: 10.1007/978-3-540-75873-0. [23] I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, 2012. doi: 10.1007/978-88-470-2823-4. [24] D. Nualart, Malliavin Calculus and Related Topics, Springer, 2006. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [26] J. Quintana-Murillo and S. B. Yuste, A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations, Eur. Phys. J. Spec. Top., 222 (2013), 1987-1998. [27] 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), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [28] T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25.  doi: 10.1214/EJP.v19-3058. [29] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006. [30] S. Tindel, C. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Relat. Field., 127 (2003), 186-204.  doi: 10.1007/s00440-003-0282-2. [31] C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer, 2013. doi: 10.1007/978-3-319-00936-0. [32] S. Umarov, On fractional Duhamels principle and its applications, J. Diff. Equa., 252 (2012), 5217-5234.  doi: 10.1016/j.jde.2012.01.029. [33] F. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5. [34] X. Wang, R. Qi and F. Jiang, Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations, BIT Numer. Math., 57 (2017), 557-585.  doi: 10.1007/s10543-016-0639-4. [35] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384.  doi: 10.1137/040605278. [36] Y. Zhang, Z. Sun and H. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys., 265 (2014), 195-210.  doi: 10.1016/j.jcp.2014.02.008. [37] T. Zhang, Lattice approximations of reflected stochastic partial differential equations driven by space-time white noise, Ann. Appl. Probab., 26 (2016), 3602-3629.  doi: 10.1214/16-AAP1186.

show all references

##### References:
 [1] F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, 2008. doi: 10.1007/978-1-84628-797-8. [2] M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent, Part Ⅲ, Geophys, J. R. Astron. Soc., 13 (1967), 529-539. [3] L. Chen, Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: Moments, Hölder regularity and intermittency, Trans. Amer. Math. Soc., 369 (2017), 8497-8535.  doi: 10.1090/tran/6951. [4] J. Cui and L. Yan, Controllability of neutral stochastic evolution equation driving by fractional Brownian motion, Acta. Math. Sci., 37 (2017), 108-118.  doi: 10.1016/S0252-9602(16)30119-9. [5] M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804.  doi: 10.1016/j.jcp.2009.07.021. [6] G. DaPrato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014. doi: 10.1017/CBO9781107295513. [7] T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Semilinear stochastic equations in a Hilbert space with a fractional brownian motion, SIAM J. Math. Anal., 40 (2009), 2286-2315.  doi: 10.1137/08071764X. [8] T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250.  doi: 10.1142/S0219493702000340. [9] Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc., 175 (2005), ⅷ+127 pp. doi: 10.1090/memo/0825. [10] A. Jentzen and P. E. Kloeden, The numerical approximation of stochastic partial differential equations, Milan J. Math., 77 (2009), 205-244.  doi: 10.1007/s00032-009-0100-0. [11] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972016. [12] 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.  doi: 10.1137/120873984. [13] M. Kamrani and N. Jamshidi, Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion, Commun. Nonlinear Sci. Numer., 44 (2017), 1-10.  doi: 10.1016/j.cnsns.2016.07.023. [14] 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.  doi: 10.1090/S0025-5718-2014-02803-2. [15] R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217-251.  doi: 10.1093/imanum/drs055. [16] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4. [17] X. Li and X. Yang, Error estimates of finite element methods for stochastic fractional differential equations, J. Comput. Math., 35 (2017), 346-362.  doi: 10.4208/jcm.1607-m2015-0329. [18] B. Maslowski and D. Nualart, Evolution equations driven by a fractional brownian motion, J. Funct. Anal., 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4. [19] W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a fractional diffusion equation, Numer. Algor., 52 (2009), 69-88.  doi: 10.1007/s11075-008-9258-8. [20] W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with nonsmooth initial data, J. Comput. Phys., 293 (2015), 201-217.  doi: 10.1016/j.jcp.2014.08.050. [21] A. L. Mehaute, T. Machado, J. C. Trigeassou and J. Sabatier, Fractional differential and its applications, FDA'04, Proceedings of the first IFAC workshop, vol. 2004-1, International Federation of Autromatic Control, ENSEIRB, Bordeaux, France, July 19-21, 2004. [22] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, 2008. doi: 10.1007/978-3-540-75873-0. [23] I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, 2012. doi: 10.1007/978-88-470-2823-4. [24] D. Nualart, Malliavin Calculus and Related Topics, Springer, 2006. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [26] J. Quintana-Murillo and S. B. Yuste, A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations, Eur. Phys. J. Spec. Top., 222 (2013), 1987-1998. [27] 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), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [28] T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25.  doi: 10.1214/EJP.v19-3058. [29] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006. [30] S. Tindel, C. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Relat. Field., 127 (2003), 186-204.  doi: 10.1007/s00440-003-0282-2. [31] C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer, 2013. doi: 10.1007/978-3-319-00936-0. [32] S. Umarov, On fractional Duhamels principle and its applications, J. Diff. Equa., 252 (2012), 5217-5234.  doi: 10.1016/j.jde.2012.01.029. [33] F. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5. [34] X. Wang, R. Qi and F. Jiang, Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations, BIT Numer. Math., 57 (2017), 557-585.  doi: 10.1007/s10543-016-0639-4. [35] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384.  doi: 10.1137/040605278. [36] Y. Zhang, Z. Sun and H. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys., 265 (2014), 195-210.  doi: 10.1016/j.jcp.2014.02.008. [37] T. Zhang, Lattice approximations of reflected stochastic partial differential equations driven by space-time white noise, Ann. Appl. Probab., 26 (2016), 3602-3629.  doi: 10.1214/16-AAP1186.
Convergence rates for the spatial discretization
Convergence rates for the differential $\alpha$
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