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)
This paper develops and analyzes a fully discrete finite element method for a classof semilinear stochastic partial differential equations (SPDEs) with multiplicative noise.The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitzand the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitzcondition. These assumptions are the same ones as used in [18] where numerical methodsfor general nonlinear stochastic ordinary differential equations (SODEs) under "minimumassumptions" were studied. As a result, the semilinear SPDEs considered in this paper are adirect generalization of the SODEs considered in [18]. There are several difficulties whichneed 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 aspatial discretization must be designed to guarantee certain properties for the numericalscheme and its stiffness matrix. In this paper we use a finite element interpolation techniqueto discretize the nonlinear drift term. Second, in order to prove the strong convergenceof the proposed fully discrete finite element method, stability estimates for higher ordermoments of the $H^1$-seminorm of the numerical solution must be established, which aredifficult and delicate. A judicious combination of the properties of the drift and diffusionterms and a nontrivial technique borrowed from [28] is used in this paper to achieve the goal.Finally, stability estimates for the second and higher order moments of the $L^2$-norm of thenumerical solution are also difficult to obtain due to the fact that the mass matrix may notbe diagonally dominant. This is done by utilizing the interpolation theory and the highermoment estimates for the $H^1$-seminorm of the numerical solution. After overcoming thesedifficulties, it is proved that the proposed fully discrete finite element method is convergentin strong norms with nearly optimal rates of convergence. Numerical experiment resultsare also presented to validate the theoretical results and to demonstrate the efficiency ofthe proposed numerical method.