STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE

作     者：Xiaobing Feng Yukun Li Yi Zhang Xiaobing Feng;Yukun Li;Yi Zhang

作者机构：Department of MathematicsThe University of TennesseeKnoxvilleTN 37996USA Department of MathematicsUniversity of Central FloridaOrlandoFL 32816USA Department of Mathematics and StatisticsThe University of North Carolina at GreensboroGreensboroNC 27402USA 

出 版 物：《Journal of Computational Mathematics》 (计算数学（英文）)

年 卷 期：2021年第39卷第4期

页      面：574-598页

核心收录：

学科分类：07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070102[理学-计算数学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：work of the first author was partially supported by the NSF grant DMS-1318486 The work of the second author was partially supported by the startup grant from University of Central Florida 

主　　题：Stochastic partial differential equations One-sided Lipschitz Strong convergence 

摘      要：This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations(SPDEs)with multiplicative *** nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz *** assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations(SODEs)under“minimum assumptionswere *** a result,the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear *** are several difficulties which need to be overcome for this ***,obviously the spatial discretization,which does not appear in the SODE case,adds an extra layer of *** turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness *** this paper we use a finite element interpolation technique to discretize the nonlinear drift ***,in order to prove the strong convergence of the proposed fully discrete finite element method,stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established,which are difficult and delicate.A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the ***,stability estimates for the second and higher order moments of the L^(2)-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally *** is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical *** overcoming these difficulties,it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of ***