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

Litan Yan; Xiuwei Yin

doi:10.3934/dcdsb.2018199

Article Contents

2019, Volume 24, Issue 2: 615-635. Doi: 10.3934/dcdsb.2018199

This issue Previous Article Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation Next Article Advection-diffusion equation on a half-line with boundary Lévy noise

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
§ Corresponding author: Xiuwei Yin

Received: April 30, 2017

Revised: November 30, 2017

Early access: June 2018

Published: January 2019

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

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Abstract

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:

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Figure 1. Convergence rates for the spatial discretization

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Figure 2. Convergence rates for the differential $\alpha$

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