Strong optimal error estimates of discontinuous Galerkin method for multiplicative noise driving nonlinear SPDEs

This paper investigates the strong convergence of a fully discrete numerical method for the stochastic partial differential equations driven by multiplicative noise. The fully discrete space-time approximation consists of the symmetric interior penalty discontinuous Galerkin method for the spatial discretization and the implicit Euler method for the temporal discretization. Rather than the usual semi group analysis techniques, in this paper, we present an analysis framework in the variational formulation by introducing new weak variational approximation techniques. Some error estimates in a strong sense are established for the proposed fully discrete scheme. The optimal convergence rates are then obtained in both space and time. Numerical results for the nonlinear stochastic partial differential equations are finally presented to confirm the theoretical results.