Convergence of the Spectral Galerkin Method for the Stochastic Reaction–Diffusion–Advection Equation

Author

Li Yang
Yanzhi Zhang, Missouri University of Science and TechnologyFollow

Abstract

We study the convergence of the spectral Galerkin method in solving the stochastic reaction-diffusion-advection equation under different Lipschitz conditions of the reaction function f. When f is globally (locally) Lipschitz continuous, we prove that the spectral Galerkin approximation strongly (weakly) converges to the mild solution of the stochastic reaction–diffusion–advection equation, and the rate of convergence in H_r-norm is (1/2−r)^-, for any r ∈ [0, 1/2) (r ∈ (1/2 – 1/2d ,1/2)). The convergence analysis in the local Lipschitz case is challenging, especially in the presence of an advection term. We propose a new approach based on the truncation techniques, which can be easily applied to study other stochastic partial differential equations. Numerical simulations are also provided to study the convergence of Galerkin approximations.

Recommended Citation

L. Yang and Y. Zhang, "Convergence of the Spectral Galerkin Method for the Stochastic Reaction–Diffusion–Advection Equation," Journal of Mathematical Analysis and Applications, vol. 446, no. 2, pp. 1230 - 1254, Elsevier, Feb 2017.

The definitive version is available at https://doi.org/10.1016/j.jmaa.2016.09.028

Department(s)

Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research

Keywords and Phrases

Stochastic reaction–diffusion–advection equation; Galerkin approximation; Convergence rate; Allen–Cahn equation; Burgers' equation

International Standard Serial Number (ISSN)

0022-247X

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2017 Elsevier, All rights reserved.

Publication Date

15 Feb 2017