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Convergence rates for the spatial discretization
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Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation
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 |
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 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. |
Figure 2.
Convergence rates for the differential $\alpha$
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