Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

Abstract

In this paper, we present a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for solving a linear stochastic parabolic partial differential equation driven by additive noises. When the diffusion coefficient in the stochastic parabolic partial differential equation varies in multiple scales, it is very challenging to resolve all scales for the model using traditional finite element methods. To overcome the difficulty, we use the CEM-GMsFEM to solve the stochastic multiscale parabolic differential equation and construct a coarse computational model. For numerical computation, the infinite dimensional additive noise is approximated by a finite dimensional noise. Convergence analysis is carried out for semi-discretization and full discretization. The convergence rate is characterized by the coarse grid size, eigenvalue decay of local spectral problems and the stochastic noise approximation. A few numerical results for the stochastic parabolic equation driven by different noises are presented to confirm the theoretic analysis and show the computational performance of the approach.