Convergence of the spectral Galerkin method for the stochastic reaction-diffusion-advection equation

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• 作者：Li Yang^a ; ^{li.yang@inria.fr" class="auth_mail" title="E-mail the corresponding author} ; Yanzhi Zhang^b ; ^{zhangyanz@mst.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Stochastic reaction&ndash ; diffusion&ndash ; advection equation ; Galerkin approximation ; Convergence rate ; Allen&ndash ; Cahn equation ; Burgers' equation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1230-1254
• 全文大小：1525 K

文摘

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 xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305315&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305315&_rdoc=1&_issn=0022247X&md5=b229f50c5d287c8a233c3d3f60880326" title="Click to view the MathML source">H_r$H_r$-norm is lineImage" height="23" width="59" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305315-si18.gif">${(\frac{1}{2}-r)}^{-}$, for any lineImage" height="23" width="63" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305315-si17.gif">$r\in [0,\frac{1}{2})$ (lineImage" height="23" width="104" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305315-si23.gif">$r\in (\frac{1}{2}-\frac{1}{2d},\frac{1}{2})$). 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.