Optimal Convergence Analysis of a Fully Discrete Scheme for the Stochastic Stokes-Darcy Equations

In this paper, we consider numerical analysis for a fully discrete scheme for the stochastic Stokes-Darcy equations with multiplicative noise. Implicit Euler scheme is used for the time discretization, and interior penalty discontinuous Galerkin (IPDG) scheme based on the BDM1-P0 finite element space is used for the space discretization. Physical interface conditions are imposed to couple the fluid equations in free fluid and porous media regions. It is proved that the implicit Euler scheme for the stochastic Stokes-Darcy equations is unconditionally stable. Under usual assumptions for the multiplicative noise and regularity of the velocity, we present the optimal convergence analysis in both time and space discretizations. Moreover, our stability result and error estimates for the velocity are independent of pressure. Numerical results are given to verify the theoretical analysis.