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

Abstract

In this article, a decoupled modified characteristic finite element method is proposed for the time-dependent Navier-Stokes/Biot problem. In the numerical scheme, the implicit backward Euler scheme is used for the time discretization, whereas the coupling terms are treated explicitly. At each time step, we only need to solve two decoupled problems, one is the Navier-Stokes equations solved by the modified characteristic finite element method, and the other is Biot equations. The stability and the error estimates are established for the proposed fully discrete scheme. Numerical experiments are provided to illustrate the theory.

A decoupled stabilized finite element method for the time–dependent Navier–Stokes/Biot problem

“…With the aid of (38), the error estimate of the pressure p 𝑓 can be obtained in a similar way as Guo and Chen. 30 However, the use of the inverse inequality will lead to the loss of half order for the pressure in Ω 𝑓 , similar results can also be found in Shan andd Zheng. 12 Remark 4.…”

“…The estimates ( 86) and ( 87) can be found in Guo and Chen. 30 Now, we are in a position to prove the error estimate for  m+1 e , 0 ≤ m ≤ N − 1.…”

“…By the Cauchy-Schwarz inequality, Poincaré's inequality (30), inverse inequality (33), (39), and (43), and Young's inequality (34), for any 𝛿 2 , 𝛿 4 > 0, we have…”

“…Compared with the Stokes problem, the nonlinear convection term in Navier–Stokes equation makes it difficult for the stability analysis and the error estimates in the numerical analysis. We refer the readers to previous studies 27–30 …”

“…We refer the readers to previous studies. [27][28][29][30] In order to ensure the compatibility between the approximations of velocity and pressure, we should choose suitable finite elements to satisfy the so-called inf-sup stability condition. In practice, the lowest equal-order mixed finite elements, such as P 1 − P 1 and Q 1 − Q 1 , have some nice features, such as convenient computation in parallel processing and lowest computational costs.…”