TY - JOUR
T1 - Strong Convergence of a Fully Discrete Finite Element Method for a Class of Semilinear Stochastic Partial Differential Equations with Multiplicative Noise
AU - Feng , Xiaobing
AU - Li , Yukun
AU - Zhang , Yi
JO - Journal of Computational Mathematics
VL - 4
SP - 574
EP - 598
PY - 2021
DA - 2021/05
SN - 39
DO - http://doi.org/10.4208/jcm.2003-m2019-0250
UR - https://global-sci.org/intro/article_detail/jcm/19156.html
KW - Stochastic partial differential equations, One-sided Lipschitz, Strong convergence.
AB - <p style="text-align: justify;">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 used in [18] where numerical methods
for general nonlinear stochastic ordinary differential equations (SODEs) under &quot;minimum
assumptions&quot; were studied. As a result, the semilinear SPDEs considered in this paper are a
direct generalization of the SODEs considered in [18]. 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 $H^1$-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 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 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 $H^1$-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.</p>