Strong convergence rates of a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise

Abstract: We consider a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator $A$ and the covariance operator $Q$ of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for SPDEs pointed out by Jentzen, Kloeden and Winkel. The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs with additive noise, we establish optimal strong convergence rates in both space and time for our fully discrete scheme. We also present numerical experiments which are consistent with our theoretical results.