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In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal. , 50 , 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme. [ABSTRACT FROM AUTHOR]

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this article, we study a multi-physics model for the closed-loop geothermal system consisting of some pipelines and a reservoir. The mathematical model for simulating this system mainly focuses on heat transfer between porous media in the reservoir and fluid flow in the pipelines. Besides, to describe the fluid and heat transfer in two regions, the governing equations of the mathematical model are denoted by the steady Navier-Stokes/Darcy equations coupled with the convection-diffusion equations. Next, we design a perturbation system to prove the well-posedness of the governing equations. Moreover, in variational formulation, a Nitsche's penalty term is added to overcome the difficulty of instability caused by the interface condition of heat flux. Furthermore, a decoupled iterative algorithm is proposed to solve the considered equations based on the finite element method, and the stability and convergence of the presented algorithm are shown. Finally, some numerical tests are performed to verify the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]