STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE
Journal of Computational Mathematics
2021, 39 ( 4):
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The 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 condition. These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under “minimum assumptions” were studied. As a result, the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the
1-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 goal. Finally, stability estimates for the second and higher order moments of the
2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the
1-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence. Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.
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