Local Discontinuous Galerkin Method for Nonlinear BSPDEs of Neumann Boundary Conditions with Deep Backward Dynamic Programming Time-Marching (Communications in Computational Physics, 2026)
This paper aims to present a local discontinuous Galerkin (LDG) method for solving nonlinear backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.
Keywords: Local discontinuous Galerkin method; backward stochastic partial differential equations; Neumann boundary problems; stability analysis; error estimates; deep learning algorithm