A stochastic local discontinuous Galerkin method for stochastic two-point boundary-value problems driven by additive noises

The local discontinuous Galerkin (LDG) method has been successfully applied to deterministic boundary-value problems (BVPs) arising from a wide range of applications. In this paper, we propose a stochastic analogue of the LDG method for stochastic two-point BVPs. We first approximate the white noise process by a piecewise constant random process to obtain an approximate BVP. We show that the solution of the new BVP converges to the solution of the original problem. The new problem is then discretized using the LDG method for deterministic problems. We prove that the solution to the new approximate BVP has better regularity which facilitates the convergence proof for the proposed LDG method. More precisely, we prove L² error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be two in the mean-square sense, when piecewise polynomials of degree at most p are used. Finally, several numerical examples are provided to illustrate the theoretical results.