| Abstract: |
| In this talk, we consider a new approach for the time discretization of 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 while the nonlinearity in the drift term is only considered to satisfy a one-sided Lipschitz condition that has a broader application in reality. Our new strategy for the time discretization is based on the classical Milstein method from stochastic differential equations. In addition, for the spatial discretization, we also use a finite element interpolation technique to discretize the nonlinear drift term. We use the energy method to show a strong convergence order of at most $1$ for the time discrete solution. The proof is based on new H\older continuity estimates of the PDE solution and higher moment estimates for the $H^1$-norm of the numerical solution. |
|