Unconditional convergence and superconvergence analysis for the transient Stokes equations with damping

Highlights

• The linearized backward Euler scheme is presented for the transient Stokes equations with damping.

• The lowest-order Bernadi-Raugel rectangular element pair is used to approximate the velocity and pressure, and unconditional optimal error estimates are derived on structured rectangular meshes.

• Unconditional superclose and superconvergent results are obtained for the velocity in the norm L^∞(H¹) and pressure in the norm L^∞(L²) on structured rectangular meshes.

Abstract

In this paper, the linearized backward Euler scheme for the transient Stokes equations with damping is presented, in which the velocity and pressure are approximated by the lowest-order Bernadi-Raugel rectangular element pair. Unconditional optimal error estimates of the velocity in the norms L^∞(L²) and L^∞(H¹), and the pressure in the norm L^∞(L²) are derived through the Stokes operator and the $H^{-1}$-norm estimate. Moreover, the superclose properties and global superconvergent results are obtained by the interpolation post-processing technique. Finally, some numerical results are provided to confirm the theoretical analysis.

Introduction

We consider the following transient Stokes equations with damping:

$$\{\begin{array}{c}{u}_t-\nu \Delta u+\alpha {\left|u\right|}^{r-2}u+\nabla p=f,\text{in}(0,\mathrm{T}]\times \Omega ,\hfill \\ \text{div}u=0,\text{in}(0,\mathrm{T}]\times \Omega ,\hfill \\ u=0,\text{on}\partial \Omega ,\hfill \\ u(0,·)=u_0,\text{in}\Omega ,\hfill \end{array}$$

where $\Omega \subset {\mathcal{R}}^2$ is a bounded domain with the boundary ∂Ω, $u=(u_1,u_2)$ and p are the fluid velocity and pressure, respectively, f is a given external force and ν is the viscosity coefficient. The nonlinear damping term $\alpha {\left|u\right|}^{r-2}u$ comes from the resistance to the motion of the flow, where 2 ≤ r < ∞ and α are two damping parameters, and | · | denotes the Euclidean norm.

The systems (1.1) describes various physical phenomena such as porous media flow, drag or friction effects, and some dissipative mechanisms [1], [2]. The existence, uniqueness and regularity of the solutions for the Navier-Stokes equations with damping were investigated in [3], [4]. At the same time, some works have been contributed to the numerical methods of the stationary incompressible Stokes or Navier-Stokes equations with damping. In [5], the existence and uniqueness of weak solutions were proved, and the conforming finite element method was developed. In [6], the superclose and superconvergence phenomenon of some stable conforming elements were investigated. In [7], [8], the local projection stabilized finite element methods with the $P_1-P_1$ element pair were proposed for the Stokes or Navier-Stokes equations with damping. In [9], [10], the two-level and multi-level algorithms were applied to the problem to save computation cost. In [11], the MAC finite difference scheme was established for the Stokes equations with damping on non-uniform grids. However, there were few numerical methods reported for the transient Stokes or Navier-Stokes equations with damping.

In this paper, we propose the linearized backward Euler scheme for the problem (1.1). The velocity and pressure are approximated by the lowest-order Bernadi-Raugel rectangular element pair. We aim to obtain unconditional optimal error estimates and some superconvergent results. It requires the boundedness of the velocity in the norm L^∞(H¹) in the error analysis, which is hard to be arrived in prior. The traditional techniques always employ the inverse inequality, but this results in some time-step restrictions. Recently, by introducing a time discrete system, the error splitting technique was presented to derive unconditional error estimates. This technique was already applied to various nonlinear problems [12], [13], [14], [15], [16], [17], [18]. In addition, some unconditional superclose and superconvergent results were also obtained, see [19], [20], [21], [22], [23], [24]. Observing that we only need the bounds of the velocity in the norm L^∞(H¹), we try a more simple method. By use of mathematical induction, optimal error estimate of the velocity in the norm L^∞(H¹) and the boundedness of the velocity in the norm L^∞(H¹) are obtained. Subsequently, through the $H^{-1}$-norm estimate and the Stokes operator, we deduce optimal error estimate for the pressure in the norm L^∞(L²), while in many previous works [17], [25], [26], it is only optimal in L²(L²). We also obtain optimal error estimate of the velocity in the norm L^∞(L²) by employing the Stokes projection. Moreover, we derive superclose properties and global superconvergent results. Finally, numerical results are provided to confirm the theoretical analysis.

Throughout this paper, we use the classical Sobolev spaces L^l(Ω), H^m(Ω) and $H_0^m\left(\Omega \right),$ where 1 ≤ l ≤ ∞ and m ≥ 0. Let ‖ · ‖_0,l and ‖ · ‖_m be the norms on L^l(Ω) and H^m(Ω), respectively, and ( · ,  · ) be the natural inner product in L²(Ω). We use $H^{-1}\left(\Omega \right)$ to denote the dual space of $H_0^1\left(\Omega \right),$ and  <  · ,  ·  >  the duality product. Additionally, we define the space L^l(Y) with the norm ${\parallel u\parallel }_{L^l\left(Y\right)}={\left({\int }_0^T{\parallel u(·,t)\parallel }_Y^{l}dt\right)}^{\frac{1}{l}},$ and we denote by C a generic positive constant, which may be different in different places, but independent of the mesh size h, the time step τ and the time level n.