A multilevel Monte Carlo ensemble and hybridizable discontinuous Galerkin method for a stochastic parabolic problem

Abstract

AbstractA second‐order, multilevel Monte Carlo ensemble, and hybridizable discontinuous Galerkin (MLMCE‐HDG) method is proposed to solve the stochastic parabolic partial differential equations (SPDEs). By introducing an ensemble average of the diffusion coefficient, the MLMCE‐HDG method results in a single discrete system with multiple right‐hand‐vectors, which can be solved more efficiently than a group of linear systems. A rigorous error estimate is obtained with a second‐order accuracy in time and optimal convergence rate in physical space. Comparing with the multilevel Monte Carlo and hybridizable discontinuous Galerkin (MLMC‐HDG) method, the MLMCE‐HDG method can reduce the computational cost. Finally, we provide several numerical experiments to illustrate the theoretical results.