Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise

In this paper, a new time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise is developed and studied. This model considers heat transfer between the free flow in the pipe region and the porous media flow in the porous media region. Darcy’s law and stochastic Navier-Stokes equations are used to control the flows in the pipe and porous media regions, respectively. The heat equation is coupled with the flow equation to describe the heat transfer in these both regions. In order to avoid sub-optimal convergence, a new mixed finite element method is proposed by using the Helmholtz decomposition that drives the multiplicative noise. Then, the stability of the proposed method is proved, and we obtain the optimal convergence order \(o(\Delta t^{\frac{1}{2}}+h)\) $o(\mathrm{\Delta }{t}^{\frac{1}{2}}+h)$ of global error estimation. Finally, numerical results indicate the efficiency of the proposed model and the accuracy of the numerical method.