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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

Litan Yan 1,2, and  Xiuwei Yin 1,,

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  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.

Keywords: Fractional stochastic partial differential equation, fractional Brownian motion, spectral Galerkin method, strong approximation, optimal convergence rates.
Mathematics Subject Classification: 60H15, 60H35, 65C30.
Citation: Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & 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.  Google Scholar

[2]

M. Caputo, Linear models of dissipation whose $ Q $ is almost frequency independent, Part Ⅲ, Geophys, J. R. Astron. Soc., 13 (1967), 529-539.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[6]

G. DaPrato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972016.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[22]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, 2008. doi: 10.1007/978-3-540-75873-0.  Google Scholar

[23]

I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, 2012. doi: 10.1007/978-88-470-2823-4.  Google Scholar

[24]

D. Nualart, Malliavin Calculus and Related Topics, Springer, 2006.  Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[28]

T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25.  doi: 10.1214/EJP.v19-3058.  Google Scholar

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006.  Google Scholar

[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.  Google Scholar

[31]

C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer, 2013. doi: 10.1007/978-3-319-00936-0.  Google Scholar

[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.  Google Scholar

[33]

F. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. Caputo, Linear models of dissipation whose $ Q $ is almost frequency independent, Part Ⅲ, Geophys, J. R. Astron. Soc., 13 (1967), 529-539.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. DaPrato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972016.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Springer, 2014. doi: 10.1007/978-3-319-02231-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, 2008. doi: 10.1007/978-3-540-75873-0.  Google Scholar

[23]

I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, 2012. doi: 10.1007/978-88-470-2823-4.  Google Scholar

[24]

D. Nualart, Malliavin Calculus and Related Topics, Springer, 2006.  Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[28]

T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25.  doi: 10.1214/EJP.v19-3058.  Google Scholar

[29]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006.  Google Scholar

[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.  Google Scholar

[31]

C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer, 2013. doi: 10.1007/978-3-319-00936-0.  Google Scholar

[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.  Google Scholar

[33]

F. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Convergence rates for the spatial discretization
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Figure 2.  Convergence rates for the differential $\alpha$
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