A generalized finite element θ-scheme for backward stochastic partial differential equations and its error estimates

Abstract

In this paper, we study numerical methods for solving a class of nonlinear backward stochastic partial differential equations. By utilizing finite element methods in space and $\theta$-scheme in time, the proposed scheme forms a generalized spatio-temporal full discrete scheme, which can be solved in parallel. We rigorously prove the boundedness and error estimates, and obtain the optimal convergence rates in both time (first order/second order) and space ($k+1$, $k$ in $L^2$ and $H^1$, respectively). Numerical results are finally provided to demonstrate the effectiveness of the proposed scheme and validate the theoretical analyses.The analytical techniques we presented can be directly applied to other more complicated BSPDEs.