Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Abstract

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.

Decoupled modified characteristic finite element method for the time‐dependent Navier–Stokes/Biot problem

“…which together with the triangle inequality, (118), ( 142), (140), and ( 33) implies (127).▪ Remark 3 The use of the inverse inequality in ( 136)-(138) leads to the loss of half order for the pressure in Ω f , similar results can also been found in References [33,42]. It is very interesting that whether the half order can be improved theoretically.…”

“…Si et al [44] initially proposed a MCFEM to NSD problem with Beavers–Joseph–Saffman condition. Jia et al [33] presented a new decoupled characteristic stabilized finite element method to solve the nonstationary NSD model. Recently, the MCFEM is also applied to NSD problem with Beavers–Joseph condition [12].…”

A hybridizable discontinuous Galerkin method for the coupled Navier–Stokes and Darcy problem

“…Existence and uniqueness of a weak solution to the physically more relevant model, without inertia effects on the interface, was proven in [9], while convergence of a discontinuous Galerkin method for this model was proven in [12]. Conforming methods for the transient problem have been studied in [25,43].…”

Characteristic stabilized finite element method for non-stationary conduction-convection problems

“…Among them, pressure projection stabilized method is a popular method (Bochev et al , 2006; Li and He, 2008; Li et al , 2007; Shan and Hou, 2009; Shang, 2010; Jia et al , 2010; Zhang et al , 2010), which has some excellent features: parameter-free, avoiding higher-order derivatives and stabilization being completely local at the element level. The above technique has also been extended to the Stokes/Darcy problems (Li et al , 2016), the Navier–Stokes/Darcy problems (Jia et al , 2019), and the conduction–convection problems (Huang et al , 2012; Zhang and Liang, 2017).…”