1 Introduction

We consider the incompressible Navier–Stokes equations on a bounded domain \(\varOmega \subset \mathbb {R}^d\) (d=2 or 3):

$$\begin{aligned} -\nu \varDelta u+(u\cdot \nabla )u+\nabla p=f&\quad \text{ in } \varOmega , \end{aligned}$$
(1)
$$\begin{aligned} \nabla \cdot u=0&\quad \text{ in } \varOmega , \end{aligned}$$
(2)

subject to homogeneous Dirichlet boundary conditions \(u=0\) on \(\partial \varOmega \). In the above equations, \(u =(u_1(x),\ldots , u_d(x))^T\) denotes the velocity field, \(p =p(x) \) denotes the pressure, \(f=(f_1(x), \ldots , f_d(x))^T\) a given body force field, and \(\nu >0\) is the viscosity of fluid. Let U be a characteristic velocity and L a characteristic length, then the Reynolds number is defined as \(Re=UL/\nu \).

In this paper, we are interested in the simulation of the above Navier–Stokes equations at high Reynolds numbers or with small viscosities. It is well understood that when the Reynolds number increases, the Navier–Stokes equations become convection dominated, causing troubles for the classical Galerkin finite element methods which may lead to spurious oscillations. Therefore, stabilization methods are essential in simulation of the Navier–Stokes equations at high Reynolds numbers. Among various stabilization methods in the literature, we are here interested in a finite element variational multiscale method based on two local Gauss integrations presented by Zheng et al. in [1]. The idea of two local Gauss integrations was first considered in [2, 3] to stabilize the equal-order \(P_1-P_1\) elements approximations of Stokes and Navier–Stokes equations, which can be viewed as an variant of the general stabilization method of Bochev and co-workers [4]. As we will see later, this finite element variational multiscale method based on two local Gauss integrations is equivalent to a LPS (local projection stabilization, cf. [5, 6])-like method for these mixed finite element approximations where the interpolation of velocity is of second order. Its advantage over the commonly used projection-based variational multiscale methods (see, e.g., [7] and references therein) is that it is easy to implement, free of extra storage, and hence has attracted considerable attentions from researchers (cf. [8,9,10,11,12,13,14]).

In this paper, by combining the above-mentioned finite element variational multiscale method based on two local Gauss integrations with two-grid discretization strategy, we present a three-step Oseen correction method for the Navier–Stokes equations at high Reynolds numbers. The method proceeds as follows:

  • Step 1: Begin by solving a finite element variational multiscale-stabilized nonlinear Navier–Stokes system for \((u_H, p_H)\) upon a coarse mesh of width H.

  • Step 2: Pass the coarse mesh solution \(u_H\) to the fine mesh of width h and solve a discrete finite element variational multiscale-stabilized Oseen problem for \((u_{h}^*, p_{h}^*)\).

  • Step 3: Solve a discrete Oseen problem (which has the same stiffness matrix as Step 2, with only different right-hand side involving \(u_H\) and \(u_{h}^*\)) for the update \((u^{h},p^{h})\) upon the same fine mesh.

Our theoretical analysis shows that, if we choose \(h\sim H^{{(s+1)}/{s}}\) for a certain integer s, the approximate solution \((u^{h}, p^{h})\) produced by the three-step Oseen correction method is asymptotically as accurate as the approximation produced by solving directly the finite element variational multiscale-stabilized Navier–Stokes system on the fine mesh. We numerically demonstrate that each step of the method significantly improves the accuracy of the approximate solution of the previous step.The novelty of the presented three-step Oseen correction method is that it combines the best algorithmic features of the two-grid discretization strategy and the finite element variational multiscale method, as well as the Oseen linearization. Specifically, on the one hand, it allows high Reynolds number flows that are challenging to simulate for the standard two-grid methods. On the other hand, it can save a large amount of computational time compared with the one-level finite element variational multiscale method used directly on the fine mesh. An additional decoration to the proposed method consists in using an Oseen-type partial linearization for the fine mesh corrections, making the resulting linear systems be more tractable than the ones from the commonly used full linearizations based on Newton iterations, such as those in [15,16,17]. Moreover, with the extra third step, the presented method can yield a much better approximate solution with a greater accuracy compared with the standard stabilized two-grid method based on Oseen correction.

Two-grid discretization strategy is a well-established technique for steady nonlinear elliptic PDEs, see [18,19,20]. The basic idea is to capture the low modes or global solution envelope by solving an initial nonlinear approximation problem on a coarse mesh, and then capture the fine structures by computing a linearized system on a fine mesh. This two-grid discretization strategy has been applied to numerically solving the steady Navier–Stokes equations by Layton [15], Dai and Cheng [21], He et al. [22,23,24,25], Liu and Hou [26]. It was also combined with the subgrid stabilization method [17, 27, 28] and the domain decomposition method [29,30,31]. Two-grid methods for the unsteady Navier–Stokes equations were also studied in [32,33,34,35,36,37,38,39]. In most of the above works, a full linearization of the nonlinear Navier–Stokes system based on Newton’s method is used to define the fine mesh problem, and consequently, the resulting linear systems of equations are generally both highly nonsymmetric and indefinite. As mentioned before, by means of an Oseen-type partial linearization studied herein, the linear systems originating from the fine mesh problem become more tractable. The Oseen-type linearization had also been used extensively in, e.g., [40,41,42,43,44] to resolve the nonlinearity in the Navier–Stokes system. It is worth mentioning that there are other types of two-grid methods (see, e.g., [45,46,47,48]) which are preconditioning methods for a given discretization scheme, not for designing a discretization scheme as studied herein.

The rest of the paper proceeds as follows. In Sect. 2, we introduce some notations and preliminaries. In Sect. 3, we present and analyze the three-step Oseen correction method. Section 4 shows some numerical results followed by conclusions in Sect. 5.

2 Preliminaries and notations

2.1 Functional setting of the Navier–Stokes equations

In this paper, for an integer \(k\ge 1\), \(H^k(\varOmega )^d\) denotes the usual Sobolev spaces of functions with kth distributional derivatives in \(L^2(\varOmega )^d\) and equipped with norms \(\Vert \cdot \Vert _{k}\). \(H_0^1(\varOmega )^d\) is the subspace of \(H^1(\varOmega )^d\) of functions trace of which is zero on \(\partial \varOmega \) and \(L^2_0(\varOmega )\) is the space of \(L^2(\varOmega )\) functions mean value of which is zero. \((\cdot ,\cdot )\) denotes the the standard inner-product of \(L^2(\varOmega )\) or \(L^2(\varOmega )^d\) (cf. [49]). We will denote by \(\Vert \cdot \Vert _{-1}\) the norm of the dual space of \(H^1_0(\varOmega )^d\). The letter c or C (with or without subscript) stands for a generic positive constant which is independent of the viscosity \(\nu \), the stabilization, and mesh parameters, and may stand for different values at its different occurrences.

Let

$$\begin{aligned} X=H^1_0(\varOmega )^d,~~Y=L^2(\varOmega )^d,~~M=L^2_0(\varOmega ). \end{aligned}$$

We define

$$\begin{aligned} b(u,v,w)&=((u\cdot \nabla )v,w)+\frac{1}{2}((\nabla \cdot u)v,w)\\&=\frac{1}{2}((u\cdot \nabla )v,w)-\frac{1}{2}((u\cdot \nabla )w,v),\quad \forall ~u,~v,~w\in X, \end{aligned}$$

which has the following properties (cf. [50,51,52,53,54]):

$$\begin{aligned} b(u,v,w)&=-\,b(u,w,v),\quad \forall ~u, v, w\in X, \end{aligned}$$
(3)
$$\begin{aligned} |b(u,v,w)|&\le N\Vert \nabla u\Vert _0\Vert \nabla v\Vert _0\Vert \nabla w\Vert _0,\quad \forall ~ u, v, w\in X, \end{aligned}$$
(4)
$$\begin{aligned} |b(u,v,w)|&\le c\Vert u\Vert _0\Vert \nabla v\Vert _0\Vert w\Vert _2,\quad \forall ~ u, v\in X, w\in H^2(\varOmega )^d, \end{aligned}$$
(5)

where N and c are positive constants depending only on \(\varOmega \). The weak form of (1)–(2) reads: find a pair \((u,p)\in X\times M\) such that

$$\begin{aligned} \nu (\nabla u,\nabla v)+b(u,u,v)-(\nabla \cdot v,p)+(\nabla \cdot u,q)=(f,v), \quad \forall (v,q)\in X\times M. \end{aligned}$$
(6)

A nonsingular solution u of the Navier–Stokes equations is defined as [55]

$$\begin{aligned} \inf _{v\in V}\sup _{w\in V}\frac{\nu (\nabla v,\nabla w)+b(u,v,w)+b(v,u,w)}{\Vert \nabla v\Vert _0~\Vert \nabla w\Vert _0}\ge \sigma >0, \end{aligned}$$
(7)

and satisfies (cf. [56])

$$\begin{aligned} \Vert \nabla u\Vert _0\le \nu ^{-1}\Vert f\Vert _{-1},\quad \Vert f\Vert _{-1}=\sup _{v\in X, v\ne 0}\frac{|(f,v)|}{\Vert \nabla v\Vert _{0}}, \end{aligned}$$
(8)

where \(\sigma =\sigma (\nu ,u)\) (see [51]) is a constant and \(V=\{v\in X: (\nabla \cdot v,q)=0,~\forall q\in M\}\).

2.2 Mixed finite element spaces

Let \(\mu >0\) be a discretization parameter. For each \(\mu \), let \(T^\mu (\varOmega )=\{K\}\) be a regular family of triangulations of \({\overline{\varOmega }}\) (see, e.g., [51]), consisting of triangles (when \(d=2\)) or tetrahedrons (when \(d=3\)). For each triangle or tetrahedron K, we denote by \(\mu _K\) the diameter of K and by \(\mu \) the maximum of \(\mu _K\). Let \((X_\mu , M_\mu )\) be a pair of compatible finite element spaces for discretizing the velocity and the pressure. We assume that the following Ladyzhenskaya–Babuska–Brezzi (LBB) condition holds:

$$\begin{aligned} \beta \Vert q\Vert _{0}\le \sup _{v\in X_\mu ,v\ne 0} \frac{(\nabla \cdot v,q)}{\Vert \nabla v\Vert _{0}},\quad \forall q\in M_\mu , \end{aligned}$$
(9)

and for each \((u,p)\in H^{k+1}(\varOmega )^d\times H^k(\varOmega )\) (\(1 \le k\)), there is an approximation pair \((\pi _\mu u,\rho _\mu p)\in X_\mu \times M_\mu \) such that

$$\begin{aligned} \Vert u-\pi _\mu u\Vert _{1} \le c\mu ^{s}\Vert u\Vert _{1+s}, \quad \Vert p-\rho _\mu p\Vert _{0}\le c\mu ^{s}\Vert p\Vert _{s},\quad 1\le s\le k, \end{aligned}$$
(10)

where \(\beta >0\) is a constant.

We consider those LBB stable, mixed finite element approximations where the finite element interpolant for the velocity is of second order. Examples of such compatible, mixed finite element pairs are the Taylor–Hood (\(P_2-P_1\)) elements [57], the \(P_2-P_0\) elements [58], the augmented \(P_2-P_1\) elements [59, 60], and the Scott–Vogelius (\(P_2- P^\mathrm{{disc}}_1\)) elements under the restriction that the mesh is a barycenter refinement of a regular mesh when \(d=2\) or a Powell–Sabin tetrahedralization when \(d=3\) (see [61,62,63] for details).

We define

$$\begin{aligned} V_\mu =\{v_\mu \in X_\mu : (\nabla \cdot v_\mu ,q_\mu )=0,~\forall q_\mu \in M_\mu \}, \end{aligned}$$

and have [51, 56]

$$\begin{aligned} \inf _{v_\mu \in V_\mu }\Vert \nabla (v-v_\mu )\Vert _0\le c\left( 1+\frac{1}{\beta }\right) \inf _{v_\mu \in X_\mu }\Vert \nabla (v-v_\mu )\Vert _0,\quad \forall v\in V. \end{aligned}$$
(11)

2.3 The stabilization term

We define the stabilization term as follows [1]:

$$\begin{aligned} G_\mu (u_\mu ,v_\mu )=\,\alpha _\mu \sum _{K\in T^\mu (\varOmega )}\left( \int _{K,m}\nabla u_\mu \cdot \nabla v_\mu \mathrm{{d}}x-\int _{K,1}\nabla u_\mu \cdot \nabla v_\mu \mathrm{{d}}x\right) , \quad \forall u_\mu ,v_h\in X_\mu , \end{aligned}$$
(12)

where \(\int _{K,m}(\cdot ) \mathrm{{d}}x\) (\(m\ge 2\)) denotes the mth-order Gauss integral over the element K and \(\int _{K,1}(\cdot ) \mathrm{{d}}x\) the first-order Gauss integral over the element K. \(\alpha _\mu >0\) is the stabilization coefficient.

Let \(P_0\) be the piecewise constant space on the element K, so that we set

$$\begin{aligned} R_0=\Bigg \{v\in L^2(\varOmega ): v|_{K}\in P_{0}, \quad \forall K\in T^\mu (\varOmega )\Bigg \},\quad L_\mu =R_0^{d\times d},\quad L=L^2(\varOmega )^{d\times d}, \end{aligned}$$

and define \(L^2\)-orthogonal projection \(\varPi _\mu : L\rightarrow L_\mu \), which projects onto the piecewise constant space, can be computed locally and satisfies

$$\begin{aligned}&(\varPi _\mu \nabla u,v)=(\nabla u, v),\quad \forall u\in X, v\in L_\mu , \end{aligned}$$
(13)
$$\begin{aligned}&\Vert \varPi _\mu \nabla v\Vert _0\le \Vert \nabla v\Vert _0,\quad \forall v\in X, \end{aligned}$$
(14)
$$\begin{aligned}&\Vert (I-\varPi _\mu ) \nabla v\Vert _0\le \Vert \nabla v\Vert _0,\quad \forall v\in X. \end{aligned}$$
(15)

Then the stabilization term (12) can be rewritten as [1]

$$\begin{aligned} G_\mu (u_\mu ,v_\mu )&=\alpha _\mu ((I-\varPi _\mu )\nabla u_h,(I-\varPi _\mu )\nabla v_\mu ) = \alpha _\mu (\nabla u_\mu ,\nabla v_\mu )-\alpha _\mu (\varPi _\mu \nabla u_\mu ,\nabla v_\mu ). \end{aligned}$$
(16)

Remark 2.1

The equivalence of the stabilizations (12) and (16) is only valid for those mixed finite element pairs where the interpolant of the velocity is of second order (see [1] for details). However, our following error estimates based on the stabilization (16) are also valid for higher-order LBB stable elements \(P_k-P_{k-1} (k\ge 2)\) and non-LBB stable elements such as equal-order \(P_k-P_k (k\ge 2) \) elements (in such a case, appropriate pressure stabilizations should be added to the formulation).

3 Three-step Oseen correction method

In our method, we need two levels meshes: a coarse mesh \(T^H(\varOmega )\) of width H and a fine mesh \(T^h(\varOmega )\) of width h \((0<h<H<1)\). We assume they are nested since it will simplify our analysis substantially. Let \((X_H, M_H)\) be a pair of compatible finite element spaces associated with the coarse mesh \(T^H(\varOmega )\), and \((X_h, M_h)\) the one associated with the fine mesh \(T^h(\varOmega )\). Then our three-step Oseen correction method is stated as follows.

Algorithm I: Three-step Oseen correction algorithm.

Step 1: Find a coarse mesh solution \((u_H,p_H)\in X_H\times M_H\) satisfying

$$\begin{aligned}&\nu (\nabla u_H,\nabla v_H)+b(u_H,u_H,v_H)-(\nabla \cdot v_H,p_H)+(\nabla \cdot u_H,q_H)+G_H(u_H,v_H) =(f,v_H),\nonumber \\&\forall (v_H,q_H)\in X_H\times M_H. \end{aligned}$$
(17)

Step 2: Find a fine mesh solution \((u_{h}^*,p_{h}^*) \in X_h\times M_h\) such that for all \((v_h,q_h)\in X_h\times M_h\)

$$\begin{aligned} \begin{array}{rr} \nu (\nabla u_{h}^*,\nabla v_h)+b(u_H,u_{h}^*,v_h)-(\nabla \cdot v_h,p_{h}^*)+(\nabla \cdot u_{h}^*,q)+G_h(u_{h}^*,v_h) =(f,v_h)\end{array} \end{aligned}$$
(18)

Step 3: Find \((u^{h},p^{h}) \in X_h\times M_h\) such that

$$\begin{aligned} \begin{array}{ll} \nu (\nabla u^{h},\nabla v_h)+b(u_H,u^{h},v_h)-(\nabla \cdot v_h,p^{h})+(\nabla \cdot u^{h},q_h)+G_h(u^{h},v_h)\\ \qquad =(f,v_h)+b(u_H-u_{h}^*,u_{h}^*,v_h) , \quad \forall (v_h,q_h)\in X_h\times M_h. \end{array} \end{aligned}$$
(19)

Remark 3.1

It is worth mentioning that on the left side of (19) in the above algorithm, we linearize the nonlinear convection term by \(b(u_H,u^{h},v_h)\) rather than \(b(u_h^*,u^{h},v_h)\) as used usually in the standard Newton or Oseen iterations. There are two reasons for this. First, by using \(u_H\) instead of the newly computed solution \(u_h^*\) to linearize the nonlinearity, we keep the same coefficient matrices as the previous step. Second, since there is a term \(b(u_H-u_{h}^*,u_{h}^*,v_h)\) in the right-hand side of (19), Step 3 differs from the standard Newton or Oseen iterations which usually use the solution newly computed at the previous iteration step to linearize the nonlinearity in the current iteration step. In fact, our numerical results show that replacing \(b(u_H,u^{h},v_h)\) by \(b(u_h^*,u^{h},v_h)\) in the left-hand side of (19) will degrade the accuracy of the approximate solutions.

We now give basic error analysis for Algorithm I in the natural energy norm for the Navier–Stokes equations. At the beginning, we introduce some notations and definitions:

$$\begin{aligned}&\kappa _1=\nu +\alpha _H+2\nu ^{-1}N\Vert f\Vert _{-1}, \quad \kappa _2=\nu +\alpha _h+\nu ^{-1}N\Vert f\Vert _{-1},\\&\delta =\nu ^{-1}(\nu +\alpha _h)^{-1}\Vert f\Vert _{-1} . \end{aligned}$$

By a standard argument (cf. [13]), we have the following results for the approximate solution of Step 1:

Theorem 3.1

Assume \((u,p)\in (X\cap H^{k+1}(\varOmega )^d)\times (M\cap H^k(\varOmega ))\) be the exact solution of the Navier–Stokes equations, and \(\alpha _H\) tends to zero as H tends to zero. Then, there is \(H_0 > 0\) such that for all \(H\in (0, H_0]\), the approximate solution \((u_H,p_H)\) of Step 1 in Algorithm I satisfies

$$\begin{aligned}&\Vert \nabla u_H\Vert _0\le \nu ^{-1} \Vert f\Vert _{-1}, \end{aligned}$$
(20)
$$\begin{aligned}&\Vert \nabla (u-u_H)\Vert _{0}\le C(\beta )\left( 1+\frac{2\kappa _1}{\sigma }\right) \inf _{w_H\in X_H}\Vert \nabla (u-w_H)\Vert _{0}+\frac{2}{\sigma }\inf _{\lambda _H\in M_H}\Vert p-\lambda _h\Vert _{0} +\frac{4\alpha _H}{\nu \sigma }\Vert f\Vert _{-1}, \end{aligned}$$
(21)
$$\begin{aligned}&\Vert p-p_H\Vert _{0}\le \left( 1+\frac{1}{\beta }\right) \inf _{\lambda _h\in M_H}\Vert p-\lambda _h\Vert _{0} +\frac{\kappa _1}{\beta }\Vert \nabla (u-u_H)\Vert _{0}+ \frac{2\alpha _H}{\nu \beta }\Vert f\Vert _{-1}.\end{aligned}$$
(22)
$$\begin{aligned} \Vert u-u_H\Vert _{0}+\Vert p-p_H\Vert _{-1}&\le c_1H\left( \Vert \nabla (u-u_H)\Vert _{0}+\Vert p-p_H\Vert _{0}\right) +c_2\alpha _H\Vert \nabla (u-u_H)\Vert _{0}\nonumber \\&\quad +c_3\Vert \nabla (u-u_H)\Vert _{0}^2+c_4\alpha _H H\Vert u\Vert _{2}. \end{aligned}$$
(23)

By applying (10), we have with \(1\le s\le k\) that

$$\begin{aligned}&\Vert \nabla (u-u_H)\Vert _0 \le C_1\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) H^s+\frac{C_2\alpha _H}{\nu \sigma }, \end{aligned}$$
(24)
$$\begin{aligned}&\Vert p-p_H\Vert _0 \le C_3\left( 1+\kappa _1+\frac{2\kappa _1}{\sigma }(1+\kappa _1)\right) H^s+\frac{C_4(\kappa _1+\sigma )\alpha _H}{\nu \sigma }, \end{aligned}$$
(25)
$$\begin{aligned} \Vert u-u_H\Vert _0+\Vert p-p_H\Vert _{-1}&\le C_5\left( 1+\kappa _1+\frac{2}{\sigma }(1+\kappa _1)^2\right) H^{s+1}+C_6\left( 1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\right) \alpha _HH\nonumber \\&\quad +C_7\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) \alpha _HH^s+C_8\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) ^2H^{2s}+C_9\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\alpha _H^2. \end{aligned}$$
(26)

Remark 3.2

The results (24)–(26) show that for the solution \((u_H,p_H)\) to attain the same asymptotic convergence rate \(\mathcal {O} (H^s)\) as the classical finite element Galerkin solution, the stabilization parameter \(\alpha _H\) should be chosen as \(\alpha _H=\mathcal {O} (H^s)\).

Remark 3.3

In general, the choice of the stabilization parameters \(\alpha _H\) and \(\alpha _h\) not only depends on the mesh parameters H and h, respectively, but also on the viscosity \(\nu \) of the physical fluid being simulated. However, it is a nontrivial task to derive the explicit dependence of stabilization parameters on the viscosity. In our analysis, we only give the dependence of the stabilization parameters on the mesh parameters.

We now give error estimates for the approximate solution of Step 2.

Theorem 3.2

Under the conditions of Theorem 3.1, the solution \((u_{h}^*,p_{h}^*)\in X_h\times M_h\) of Step 2 in Algorithm I satisfies for \(1\le s\le k\)

$$\begin{aligned} \Vert \nabla u_h^*\Vert _0&\le \nu ^{-1}\Vert f\Vert _{-1}, \end{aligned}$$
(27)
$$\begin{aligned} \Vert \nabla (u-u_{h}^*)\Vert _0&\le c_1\left( 2+\delta N+(\nu +\alpha _h)^{-1}\right) h^s\nonumber \\&\quad +2\delta \alpha _h+(\nu +\alpha _h)^{-1}\Bigg \{ c_2\left( 1+\kappa _1+\frac{2}{\sigma }(1+\kappa _1)^2\right) H^{s+1}\nonumber \\&\quad +c_3\left( 1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\right) \alpha _HH+c_4\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) \alpha _HH^s\nonumber \\&\quad +c_5\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) ^2H^{2s}+c_6\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\alpha _H^2\Bigg \}, \end{aligned}$$
(28)
$$\begin{aligned} \Vert p-p_h^*\Vert _0&\le c_7\left( 1+\kappa _2(2+\delta N+(\nu +\alpha _h)^{-1})\right) h^s+c_8(2\delta \kappa _2+\nu ^{-1})\alpha _h\nonumber \\&\quad +\left( 1+\kappa _2(\nu +\alpha _h)^{-1}\right) \Bigg \{ c_9\left( 1+\kappa _1+\frac{2}{\sigma }(1+\kappa _1)^2\right) H^{s+1}\nonumber \\&\quad +c_{10}\left( 1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\right) \alpha _HH+c_{11}\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) \alpha _HH^s\nonumber \\&\quad +c_{12}\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) ^2H^{2s}+c_{13}\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\alpha _H^2\Bigg \}. \end{aligned}$$
(29)

Proof

Taking \((v_h,q_h)=(u_h^*, p_h^*)\) in (18), using (3), (16) and the Schwarz inequality, we have

$$\begin{aligned} \nu \Vert \nabla u_h^*\Vert _0^2+\alpha _h\Vert (I-\varPi _h)\nabla u_h^*\Vert _0^2\le \Vert f\Vert _{-1}\Vert \nabla u_h^*\Vert _0, \end{aligned}$$

which leads to the required result (27).

Subtracting (18) from (6), we obtain

$$\begin{aligned}&(\nu +\alpha _h) (\nabla e_h^*,\nabla v_h)+b(u_H,e_h^*,v_h) -(\nabla \cdot v_h,\theta _h^*)+(\nabla \cdot e_h^*,q_h)\nonumber \\&\quad =\alpha _h(\nabla u,\nabla v_h) -\alpha _h(\varPi _h \nabla u_h^*,\nabla v_h)+b(u_H-u,u,v_h),\quad \forall (v_h,q_h)\in X_h\times M_h, \end{aligned}$$
(30)

where \((e_h^*,\theta _h^*)=(u-u_{h}^*,p-p_{h}^*)\). We assume \((w_h,\lambda _h)\in V_h\times M_h\) be an approximation of (u, p) and set \(e_h^*=\chi -\varphi _h\) with \(\chi =u-w_h,~\varphi _h=u_{h}^*-w_h\). Since \(V_h\subset X_h\), limiting \(v_h\in V_h\) and setting \(q_h=0\) in (30), we reach

$$\begin{aligned} (\nu +\alpha _h)&(\nabla \varphi _h,\nabla v_h)+b(u_H,\varphi _h,v_h)\nonumber \\ =&\, (\nu +\alpha _h)(\nabla \chi ,\nabla v_h)+b(u_H,\chi ,v_h)-(\nabla \cdot v_h,p-\lambda _h)\nonumber \\&-\alpha _h(\nabla u,\nabla v_h)+\alpha _h(\varPi _h \nabla u_h^*,\nabla v_h)+b(u-u_H,u,v_h),\quad \forall v_h\in V_h. \end{aligned}$$
(31)

Taking \(v_h=\varphi _h\) in (31), using (3)–(5), (8), (14) (20) and (27) we infer

$$\begin{aligned} (\nu +\alpha _h) \Vert \nabla \varphi _h\Vert _0\le \,&(\nu +\alpha _h+N\Vert \nabla u_H\Vert _0) \Vert \nabla \chi \Vert _0+\Vert p-\lambda _h\Vert _0\nonumber \\&+\alpha _h(\Vert \nabla u\Vert _0+\Vert \varPi _h\nabla u_h^*\Vert _0)+c\Vert u\Vert _2\Vert u-u_H\Vert _0\nonumber \\ \le \,&(\nu +\alpha _h+\nu ^{-1}N\Vert f\Vert _{-1}) \Vert \nabla \chi \Vert _0+\Vert p-\lambda _h\Vert _0\nonumber \\&+2\nu ^{-1}\alpha _h\Vert f\Vert _{-1}+c\Vert u\Vert _2\Vert u-u_H\Vert _0 \end{aligned}$$
(32)

Taking the infimum over \(w_h\in V_h\) and \(\lambda _h\in M_h\), applying (8), (11) and the triangle inequality, we obtain

$$\begin{aligned} \Vert \nabla (u-u_{h}^*)\Vert _0\le \, \,&C(\beta )(2+\delta N)\inf _{w_h\in X_h}\Vert \nabla (u-w_h)\Vert _0 +(\nu +\alpha _h)^{-1}\inf _{\lambda _h\in M_h}\Vert p-\lambda _h\Vert _0\nonumber \\&+2\delta \alpha _h+c_2(\nu +\alpha _h)^{-1} \Vert u-u_H\Vert _0. \end{aligned}$$
(33)

Inserting (10) (with \(\mu =h\)) and (26) into the right-hand side of estimator (33), we get

$$\begin{aligned} \Vert \nabla (u-u_{h}^*)\Vert _0\le \, \,&c_1\left( 2+\delta N+(\nu +\alpha _h)^{-1}\right) h^s +2\delta \alpha _h+(\nu +\alpha _h)^{-1}\Bigg \{ c_2\left( 1+\kappa _1+\frac{2}{\sigma }(1+\kappa _1)^2\right) H^{s+1}\\&+c_3\left( 1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\right) \alpha _HH+c_4\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) \alpha _HH^s\\&+c_5\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) ^2H^{2s}+c_6\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\alpha _H^2\Bigg \}. \end{aligned}$$

The required result (28) follows.

We now give error estimate for the pressure. Taking \(q_h=0\) in (30), applying (4), (8), (14), (20) and the Schwarz inequality, we have

$$\begin{aligned} (\nabla \cdot v_h,p_h^*-\lambda _h)&=(\nabla \cdot v_h,p-\lambda _h)-(\nu +\alpha _h) (\nabla e_h^*,\nabla v_h)-b(u_H,e_h^*,v_h)\\&\quad \,\,+b(u-u_H,u,v_h)+\alpha _h(\nabla u,\nabla v_h)-\alpha _h(\varPi _h\nabla u_h^*,\nabla v_h)\\&\le \left( \Vert p-\lambda _h\Vert _0+(\nu +\alpha _h+\nu ^{-1}N\Vert f\Vert _{-1})\Vert \nabla e_h^*\Vert _0\right) \Vert \nabla v_h\Vert _0\\&\quad \,\,+\left( c\Vert u||_2\Vert u-u_H\Vert _0+2\alpha _h\nu ^{-1}\Vert f\Vert _{-1}\right) \Vert \nabla v_h\Vert _0. \end{aligned}$$

The applications of LBB condition and the triangle inequality lead to

$$\begin{aligned} \Vert p-p_h^*\Vert _0&\le \left( 1+\frac{1}{\beta }\right) \Vert p-\lambda _h\Vert _0+\frac{1}{\beta }\left( \nu +\alpha _h+\nu ^{-1}N\Vert f\Vert _{-1}\right) \Vert \nabla e_h^*\Vert _0 +\frac{c}{\beta }\Vert u\Vert _2\Vert u-u_H\Vert _0+\frac{2\alpha _h}{\nu \beta }\Vert f\Vert _{-1}. \end{aligned}$$

Taking the infimum over \(\lambda _h\in M_h\), applying (10), (26) and (28), we obtain

$$\begin{aligned} \Vert p-p_h^*\Vert _0&\le \left( 1+\frac{1}{\beta }\right) \inf _{\lambda _h\in M_h}\Vert p-\lambda _h\Vert _0+\frac{\kappa _2}{\beta }\Vert \nabla (u-u_h^*)\Vert _0 +\frac{c}{\beta }\Vert u\Vert _2\Vert u-u_H\Vert _0+\frac{2\alpha _h}{\nu \beta }\Vert f\Vert _{-1}\\&\le c_7\left( 1+\kappa _2(2+\delta N+(\nu +\alpha _h)^{-1})\right) h^s+c_8(2\delta \kappa _2+\nu ^{-1})\alpha _h\\&\quad +(1+\kappa _2(\nu +\alpha _h)^{-1})\Bigg \{ c_9\left( 1+\kappa _1+\frac{2}{\sigma }(1+\kappa _1)^2\right) H^{s+1}\\&\quad +c_{10}\left( 1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\right) \alpha _HH+c_{11}\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) \alpha _HH^s\\&\quad +c_{12}\left( 1+\frac{2}{\sigma }(1+\kappa _1)\right) ^2H^{2s}+c_{13}\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\alpha _H^2\Bigg \}, \end{aligned}$$

which leads to (29). \(\square \)

Remark 3.4

From Theorem 3.2, we see that when \(H=\mathcal {O} (h^{s/(s+1)}), \alpha _h=\mathcal {O} (h^s)\) and \(\alpha _H=\mathcal {O} (H^{s})\), the desired asymptotic convergence rate \(\mathcal {O} (h^s)\) for the solution \((u_h^*,p_h^*)\) is expectedly obtained.

Remark 3.5

Due to technical difficulty, there are terms \((\nu +\alpha _h)^{-1}h^s\) and \((\nu +\alpha _h)^{-1}H^{s+1}\) in the estimates (28) and (29), making the error estimates not being optimal with respect to the viscosity \(\nu \). However, compared with the standard two-level Oseen correction method without stabilizations which has a term \(\nu ^{-1}h^s\) in the error estimates (see, e.g., [24, 64, 65]), the stabilizations in our method improve the error factor from \(\nu ^{-1}\) to \((\nu +\alpha _h)^{-1}\). It is worth mentioning that the full Navier–Stokes equations with the established stabilization methods have also usually a term \(C(\nu ^{-1})h^s\) in their error estimates; for example, there is a term \(\nu ^{-2}h^s\) in the the error estimator for the streamline diffusion method (see Theorem 4.3 and Remark 4.2 in [66]).

Theorem 3.3

Under the conditions of Theorem 3.1, the solution \((u^h,p^h)\in X_h\times M_h\) obtained from Algorithm I satisfies for \(1\le s\le k\)

$$\begin{aligned} \Vert \nabla u^h\Vert _0&\le \nu ^{-1}\Vert f\Vert _{-1}+2\nu ^{-3}N\Vert f\Vert _{-1}^2, \end{aligned}$$
(34)
$$\begin{aligned} \Vert \nabla (u-u^h)\Vert _0&\le c_1\Bigg \{1+\kappa _2 (\nu +\alpha _h)^{-1}+\delta N \left( 2+\delta N+(\nu +\alpha _h)^{-1}\right) \Bigg \}h^s\nonumber \\&\quad +c_2(\nu +\alpha _h)^{-1}\Bigg \{(1+\kappa _1)(1+\delta N)+\frac{2}{\sigma }(1+\kappa _1)[1+\delta N(1+\kappa _1)]\Bigg \}H^{s+1}\nonumber \\&\quad +c_3(\nu +\alpha )^{-1}\big (1+\delta N\big )\bigg (1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\bigg )\alpha _HH\nonumber \\&\quad +2\delta \big (1+\nu ^{-1}\Vert f\Vert _{-1}+\delta N\big )\alpha _h +c_4\big (1+\delta N\big )(\nu +\alpha )^{-1}\bigg (\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\bigg )\alpha _H^2, \end{aligned}$$
(35)
$$\begin{aligned} \Vert p-p^h\Vert _0&\le c_1\kappa _2\Bigg \{1+\kappa _2 (\nu +\alpha _h)^{-1}+\delta N \left( 2+\delta N+(\nu +\alpha _h)^{-1}\right) \Bigg \}h^s\nonumber \\&\quad +c_2\kappa _2(\nu +\alpha _h)^{-1}\Bigg \{(1+\kappa _1)(1+\delta N)+\frac{2}{\sigma }(1+\kappa _1)[1+\delta N(1+\kappa _1)]\Bigg \}H^{s+1}\nonumber \\&\quad +c_3\kappa _2(\nu +\alpha )^{-1}(1+\delta N)\bigg (1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\bigg )\alpha _HH\nonumber \\&\quad +\big \{2\kappa _2\delta (1+\nu ^{-1}\Vert f\Vert _{-1}+\delta N)+2\nu ^{-1}\Vert f\Vert _{_1}(1+\nu ^{-2}\Vert f\Vert _{-1})\big \}\alpha _h\nonumber \\&\quad +c_4\kappa _2\big (1+\delta N\big )(\nu +\alpha )^{-1}\bigg (\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\bigg )\alpha _H^2. \end{aligned}$$
(36)

Proof

The proof is similar to that for Theorem 3.2. Taking \((v_h,q_h)=(u^h, p^h)\) in (19), using (3), (4), (16), (20), (27), and the Schwarz inequality, we have

$$\begin{aligned} \nu \Vert \nabla u^h\Vert _0^2+\alpha _h\Vert (I-\varPi _h)\nabla u^h\Vert _0^2 \le \Vert f\Vert _{-1}\Vert \nabla u^h\Vert _0+N\Vert \nabla (u_H-u^*_h)\Vert _0\Vert |\nabla u^*_h\Vert _0\Vert \nabla u^h\Vert _0. \end{aligned}$$

The applications of (20), (27) and the triangle inequality yield the required result (34).

By setting \((e^h,\theta ^h)=(u-u^h,p-p^h)\) and subtracting (19) from (6), we arrive at

$$\begin{aligned}&(\nu +\alpha _h) (\nabla e^h,\nabla v_h)+b(u_H,e^h,v_h) -(\nabla \cdot v_h,\theta ^h)+(\nabla \cdot e^h,q_h)\nonumber \\&\qquad =\alpha _h(\nabla u,\nabla v_h)-\alpha _h(\varPi _h\nabla u^h,\nabla v_h)+b(u_H-u,u,v_h)\nonumber \\&\qquad \quad +b(u_h^*-u_H,u_h^*,v_h), \quad \forall (v_h,q_h)\in X_h\times M_h. \end{aligned}$$
(37)

Letting \((w^h,\lambda ^h)\in V_h\times M_h\) be an approximation of (u, p), splitting \(e^h=\xi -\varphi ^h\) with \(\xi =u-w^h,~\varphi ^h=u^h-w^h\) and setting \(q_h=0\) in (37), we obtain for all \(v_h\in V_h\) that

$$\begin{aligned} (\nu +&\alpha _h) (\nabla \varphi ^h,\nabla v_h)+b(u_H,\varphi ^h,v_h)\nonumber \\&= (\nu +\alpha _h)(\nabla \xi ,\nabla v_h)+b(u_H,\xi ,v_h) -(\nabla \cdot v_h,p-\lambda ^h)-\alpha _h\left( \nabla u,\nabla v_h\right) \nonumber \\&\quad +\alpha _h\big (\varPi _h\nabla u^h,\nabla v_h\big )-b(u_H-u,u,v_h)-b(u_h^*-u_H,u_h^*,v_h). \end{aligned}$$
(38)

Noting that

$$\begin{aligned} b(u_h^*-u_H, u_h^*,v_h)&=b(u_h^*-u, u^*_h,v_h)+b(u-u_H, u_h^*,v_h)\\&=b(u_h^*-u, u^*_h-u,v_h)+b(u_h^*-u, u,v_h)\\&~~~+b(u-u_H, u_h^*-u,v_h)+b(u-u_H, u,v_h), \end{aligned}$$

taking \(v_h=\varphi ^h\) in (38), and applying (3)–(5) and (14), we infer

$$\begin{aligned} (\nu +\alpha _h) \Vert \nabla \varphi _h\Vert _0\le \,\,&(\nu +\alpha _h+N\Vert \nabla u_H\Vert _0) \Vert \nabla \xi \Vert _0+\Vert p-\lambda ^h\Vert _0\nonumber \\&+\alpha _h(\Vert \nabla u\Vert _0+\Vert \nabla u^h\Vert _0)+c\Vert u\Vert _2\Vert u-u_H\Vert _0\nonumber \\&+N\Vert \nabla (u-u_h^*)\Vert _0^2 +N\Vert \nabla u\Vert _0\Vert \nabla (u-u_h^*)\Vert _0\nonumber \\&+N\Vert \nabla (u-u_h^*)\Vert _0\Vert \nabla (u-u_H)\Vert _0. \end{aligned}$$
(39)

Taking the infimum over \(w^h\in V_h\) and \(\lambda ^h\in M_h\), applying (8), (10), (11), Theorems 3.1 and 3.2, the triangle inequality, noting that \(0<\alpha _H, \alpha _h, H, h<1\), and omitting the higher-order terms, we obtain

$$\begin{aligned} \Vert \nabla (u-u^h)\Vert _0 \, \le&\left( 1+\kappa _2(\nu +\alpha _h)^{-1})\right) h^s\\&+2\delta (1+\nu ^{-2}\Vert f\Vert _1)\alpha _h+c(\nu +\alpha _h)^{-1}\Vert u-u_H\Vert _0\\&+N(\nu +\alpha _h)^{-1}\Vert \nabla (u-u_h^*)\Vert _0^2 +\delta N\Vert \nabla (u-u_h^*)\Vert _0\\&+N(\nu +\alpha _h)^{-1}\Vert \nabla (u-u_h^*)\Vert _0\Vert \nabla (u-u_H)\Vert _0,\\ \le \,\,&c_1\Bigg \{1+\kappa _2 (\nu +\alpha _h)^{-1}+\delta N \left( 2+\delta N+(\nu +\alpha _h)^{-1}\right) \Bigg \}h^s\\&+c_2(\nu +\alpha _h)^{-1}\Bigg \{(1+\kappa _1)(1+\delta N)+\frac{2}{\sigma }(1+\kappa _1)[1+\delta N(1+\kappa _1)]\Bigg \}H^{s+1}\\&+c_3(\nu +\alpha )^{-1}(1+\delta N)\bigg (1+\frac{1+\kappa _1+\sigma }{\nu \sigma }\bigg )\alpha _HH\\&+2\delta \big (1+\nu ^{-1}\Vert f\Vert _{-1}+\delta N\big )\alpha _h +c_4(1+\delta N)\big (\nu +\alpha \big )^{-1}\bigg (\frac{1+\nu \sigma }{\nu ^2\sigma ^2}\bigg )\alpha _H^2. \end{aligned}$$

The reminder of the proof follows exactly the same way as for Theorem 3.2. \(\square \)

Remark 3.6

From Theorem 3.3, we see that if we choose

$$\begin{aligned} H=\mathcal {O} (h^{\frac{s}{s+1}}), \quad \alpha _h=\mathcal {O} (h^s), \quad \alpha _H=\mathcal {O} (H^{s}), \end{aligned}$$
(40)

the desired asymptotic convergence rate \(\mathcal {O}(h^s)\) is expectedly obtained.

Remark 3.7

Comparing Remark 3.6 to Remark 3.4, we see that there is no improvement in the scaling between the coarse and fine meshes for the solution \((u^h,p^h)\) of Step 3 compared with the solution \((u^*_h,p^*_h)\) of Step 2. A similar result was also obtained by Layton and Tobiska in [40], where a coarse mesh correction step is employed based on a full linearization for the nonlinearity in the Navier–Stokes equations. However, although the scaling between the coarse and fine meshes is not improved, we will see in the followed section that there is a significant improvement in accuracy of the solution \((u^h,p^h)\) of Step 3 compared with that of \((u_h^*,p_h^*)\) of Step 2.

4 Numerical results

We present numerical results for two examples in incompressible flow problems. The main goals of this section are to illustrate the numerical performance of the proposed method and to numerically verify its effectiveness. In all the computations reported herein, uniform meshes and Taylor–Hood elements are used for the finite element discretizations, and the public domain software FreeFem++ [67] is used. In addition, the Oseen iterative method (cf. [68]) is applied to solve the nonlinear coarse mesh system. The stopping criteria for the nonlinear iterations is set as \(10^{-6}\), and the initial guess is obtained by solving a Stokes problem.

4.1 Assessment by the problem with an analytic solution

In this part, we first investigate the asymptotic errors provided by the proposed algorithm, and then compare our method with other related methods to illustrate its effectiveness. For convenience, we present that the exact solution describing the steady flow of an incompressible viscous Newtonian fluid in a bounded domain \(\varOmega =[0,1]\times [0,1]\subset \mathbb {R}^2\) is given by

$$\begin{aligned} \begin{array}{ll} &{}p=3x^2+3y^2-2, \quad u=(u_1,u_2)^T, \\ &{}u_{1}(x,y)=0.5\varphi (x)\varphi '(y), \quad u_{2}(x,y)=-0.5\varphi (y)\varphi '(x), \end{array} \end{aligned}$$

with polynomial \(\varphi (t)=t^{2}(t-1)^2\).

It is worth noting that using Taylor–Hood elements, an optimal convergence rate of \(\mathcal {O}(h^2)\) in the natural energy norm can be obtained. According to Remark 3.6, we choose the stabilization parameters as \(\alpha _H=0.1H^2\) and \(\alpha _h=0.1h^2\). We set \(h=n^{-3}~(n=2,3,4,5,6)\) with H satisfying \(H=h^{2/3}\) and compute the finite element solutions applying Algorithm I. Figure 1 describes the errors of the computed solutions with different values of \(\nu =10^{-t} (t=0, 1,2,3,4\)), from which we see that the numerical results are in good agreement with the theoretical predictions, where optimal convergence rates are obtained. Table 1 compares the computed solutions by our present method, the standard stabilized two-grid Oseen correction method (denoted by Std-Oseen) and the one-level finite element variational multiscale method based on two local Gauss integrations [1] (denoted by One-VMS) on the fine mesh with \(\nu =0.0001\), where it is the nonlinear iteration’s count satisfying the stopping criterion for the nonlinear coarse mesh system and the CPU time is the wall time of the program, which includes the mesh generation time, the computing time, and the error computing time. From Table 1 we see that our present method yields better approximate solutions than the standard stabilized two-grid Oseen correction method. Noting that the standard stabilized two-grid Oseen correction method (cf. [24]) consists of the first two steps of our present method, the extra third step in our method increases the accuracy of the solution by nearly two times when \(h=1/125\) and 1 / 256, compared with the standard stabilized two-grid Oseen correction method. On the other hand, our method yields approximate solutions with an accuracy comparable to that from the one-level variational multiscale method applied directly on the fine mesh, with a substantial reduce in computational time.

Fig. 1
figure 1

Errors in \(H^1\)-semimorm for the computed velocities (left) and \(L^2\)-norm for the pressures (right)

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Table 1 A comparison of the computed solutions with \(\nu =0.0001\)
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Table 2 A comparison of error \(\Vert \nabla (u-u^{h})\Vert _0\) for the computed solutions \(u^{h}\) by various methods
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Table 3 A comparison of CPU time in second by various methods
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Fig. 2
figure 2

Lid-driven cavity flow problem

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To further investigate the performance of our present method, we fix the mesh sizes as \(H=1/36, h=1/256\) and then compare our method to other related methods with various viscosities \(\nu =10^{-n} (n=1,2,3,4,5,6)\). The compared methods include the one (denoted by T-New) obtained by replacing the linearized term \(b(u_H,u^h,v_h)\) in the left-hand side of (19) with \(b(u_h^*,u^h,v_h)\) as mentioned in Remark 3.1, the present three-step method without stabilizations (i.e., \(\alpha _H=\alpha _h=0\), denoted by W-Stab), a four-step Oseen correction method (denoted by F-Oseen) obtained by adding an additional fine mesh stabilized Oseen correction step to the present Algorithm I, and a three-step Newton iteration method (denoted by T-Newton) which consists of one stabilized coarse mesh nonlinear system and two fine mesh steps of stabilized Newton iterations. Tables 2 and 3 report the computed velocities and the CPU time taken by the methods.

From Table 2 we see that when the viscosity \(\nu \) is big (not less than \(10^{-3}\)), there is no obvious difference among the computed solutions by the methods being compared. However, when the viscosity is small (less or equal to \(10^{-4}\)), our present method yields solutions with a much higher accuracy than the ones obtained by replacing \(b(u_H,u^h,v_h)\) with \(b(u_h^*,u^h,v_h)\) in the left-hand side of (19). The method without stabilizations yields better solutions than our present method when \(\nu =10^{-4}\) and \(10^{-5}\); however, it dosen’t work when \(\nu =10^{-6}\) due to the divergence of the iterative method for the nonlinear coarse mesh system, while our present method works well. The numerical results of the four-step Oseen correction method show that further correction doesn’t significantly improve the accuracy of the solutions. The accuracy of the computed solutions between our present method and the Newton corrections method is also comparable, except the case \(\nu =10^{-6}\) in which the Newton correction method performs better; however, it takes more CPU time than our present method (see Table 3 for details). We see that from Table 3, when the viscosity is small (i.e., \(\nu =10^{-t}, t=3,4,5,6\)), our present method takes the least CPU time among the methods being compared.

4.2 Assessment by the lid-driven cavity flow

We here consider the 2D lid-driven cavity flow. The flow domain is a unit square \([0,1]\times [0,1]\) with no-slip boundary conditions, only in the top boundary with \((u_1,u_2)^T=(1,0)^T\); see Fig. 2 for detailed information.

Fig. 3
figure 3

Comparison of \(u_1\)-velocity profiles along the vertical centreline (left) \(u_2\)-velocity profiles along the horizontal centreline (right) for the lid-driven cavity flow at \(Re=1000\)

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Fig. 4
figure 4

Comparison of \(u_1\)-velocity profiles along the vertical centreline (left) \(u_2\)-velocity profiles along the horizontal centreline for the lid-driven cavity flow at \(Re=2500\)

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Fig. 5
figure 5

Comparison of \(u_1\)-velocity profiles along the vertical centreline (left) \(u_2\)-velocity profiles along the horizontal centreline for the lid-driven cavity flow at \(Re=5000\)

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In this test case, we set the Reynolds number as \(Re=1000, 2500\) and 5000, where uniform meshes of sizes \(H=1/40\) and \(h=1/160\) are used and the stabilization parameters are set as \(\alpha _H=0.025 H\) and \(\alpha _h=0.025h\). We compute the solutions by the first step, the second step and the entire algorithm of present method. For comparison, as done in Sect. 4.1, we also compute the solutions by applying the three-step Newton iteration method (denoted by T-Newton) consisted of one stabilized coarse mesh nonlinear system and two steps of stabilized Newton iterations on the fine mesh and the four-step Oseen correction method (denoted by F-Oseen) obtained by adding an additional fine mesh third Oseen correction step to the present Algorithm I. Figures 3, 4, 5 compare the numerical results with the existed data of Erturk et al. [69] who used a very fine mesh grid of \(601\times 601\). From Figs. 3, 4, 5 we see that each step of our present method significantly improves the accuracy of the approximate solutions of the previous step. Again, noting that the standard stabilized two-grid Oseen correction method consists of the first two steps of our present method, this test case shows that our method can yield an approximate solution with a higher accuracy than the standard stabilized two-grid Oseen correction method. The accuracy of the computed solutions is very comparable among our present method, the three-step Newton iteration method, the four-step Oseen correction method and the data of Erturk et al. [69]. However, our present method spends less CPU time than the three-step Newton iteration method and the four-step Oseen correction method (see Table 4 for details, where the number in bracket is the nonlinear iterations count satisfying the stopping criterion for the nonlinear coarse mesh system). Comparing the numerical results by our present method and that from the four-step Oseen correction method, we see that further iteration does not obviously improve the solutions. Figures 6, 7, 8 describe the computed streamlines and isobars, which are also in good agreement with those in the literature (cf. [69, 70]). This test case illustrated the effectiveness of the proposed method.

Table 4 CPU time in second spent by the methods for the lid-driven cavity flow
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Fig. 6
figure 6

Computed streamlines (left) and isobars (right) for the lid-driven cavity flow at \(Re=1000\)

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Fig. 7
figure 7

Computed streamlines (left) and isobars (right) for the lid-driven cavity flow at \(Re=2500\)

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Fig. 8
figure 8

Computed streamlines (left) and isobars (right) for the lid-driven cavity flow at \(Re=5000\)

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5 Conclusions

In this work, we proposed and studied a three-step Oseen correction method for the steady incompressible Navier–Stokes equations at high Reynolds numbers. This method uses a two-grid discretization strategy and a finite element variational multiscale approach based on two local Gauss integrations for stabilizations. It is based on a variational multiscale-stabilized small nonlinear coarse mesh system and two Oseen-linearized fine mesh problems which have the same stiffness matrices with only different right-hand sides. The theoretical analysis shows that, with suitable values of the algorithmic parameters, the proposed method can yield an optimally asymptotic convergence rate. Numerical results demonstrated the effectiveness of the proposed method, which show that the method can yield a convergence rate of the same order as the one-level finite element variational method used directly on the fine mesh, with a substantial reduction in computational time. It can also yield a much better solution with a higher accuracy than the standard stabilized two-grid Oseen correction method.