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Article
Convergence of the Spectral Galerkin Method for the Stochastic Reaction–Diffusion–Advection Equation
Journal of Mathematical Analysis and Applications
  • Li Yang
  • Yanzhi Zhang, Missouri University of Science and Technology
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 Hr-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.

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
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2017 Elsevier, All rights reserved.
Publication Date
2-15-2017
Citation Information
Li Yang and Yanzhi Zhang. "Convergence of the Spectral Galerkin Method for the Stochastic Reaction–Diffusion–Advection Equation" Journal of Mathematical Analysis and Applications Vol. 446 Iss. 2 (2017) p. 1230 - 1254 ISSN: 0022-247X
Available at: http://works.bepress.com/yanzhi-zhang/2/