An Oseen Two-Level Stabilized Mixed Finite-Element Method for the 2D/3D Stationary Navier-Stokes Equations

Abstract

We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e., 𝑄₁−𝑃₀ and 𝑃₁−𝑃₀). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh size H, a large general Stokes equation on the fine mesh with mesh size ℎ=𝑂(𝐻)². The Oseen two-level stabilized finite-element method provides an approximate solution (𝑢^ℎ,𝑝^ℎ) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh size h. Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations.

1. Introduction

There are numerous works devoted to the development of efficient stable mixed finite-element methods for solving the Navier-Stokes equations. It is a well-known fact that the lowest-order conforming elements spaces 𝑃₁−𝑃₀ (linear velocity, constant pressure) and 𝑄₁−𝑃₀ (bilinear velocity, constant pressure), which are the most attractive choice from an implementation point of view, fail to satisfy the inf-sup condition [1]. In order to make full use of these finite-element spaces, some kinds of stabilized finite-element methods for the two- or three-dimensional Stokes flows and two dimensional Navier-Stokes flows appeared in [2–13]. However, these stabilized methods depend on the stabilization parameters, the derivatives of the pressure, the edge-based data structures, and nested mesh. Examples include the local and global pressure jump formulations for the bilinear-constant pair where the constraint is relaxed by using the jumps of the pressure across element interfaces. Moreover, there is no satisfactory way to choose the optimal parameters for any given mesh.

The idea of the stabilized finite-element method based on the local pressure projection is derived from [14] for the Stokes equations. This method differs from existing stabilization techniques and avoids approximation of derivatives, specification of mesh-dependent parameters, interface boundary data structures, and evaluated locally at the element level. This stabilized technique has been extended to solve the two-dimensional Navier-Stokes equation by Wang et al. [15].

In the last two decades, two-level strategy has been studied for efficiently solving the nonlinear partial differential and sometimes also some linear problems. The basic idea of two-level method is to compute an initial approximation on a very coarse mesh (involving the solution of a very small number of nonlinear equations). Moreover, the fine structures are captured by solving one linear system. If we choose the appropriate proportion between the coarse and fine scale, then the two-level stabilized methods have the same convergence rate as the usual stabilized finite-element methods. Some ideas of two-level method can be found in the works of Xu [16, 17], Layton and Tobiska [18], Layton [19], Layton and Lenferink [20], He et al. [21–24], and Li [25].

Recently, He and Li [22] and Li [25] have combined simplified and Newton’s two-level iterative techniques with different stabilized finite-element methods for solving the 2D Navier-Stokes problems. The method we study in this paper is to combine the new stabilized finite-element method in [14] with the two-level method based on the Oseen iterative technique by the lowest-order conforming finite-element pairs 𝑄₁−𝑃₀ and 𝑃₁−𝑃₀ for solving the 2D/3D stationary Navier-Stokes problems. We present theoretical analysis under the assumption of the uniqueness condition. The results of Theorem 4.1 in Section 4 show that if we choose the coarse mesh scale 𝐻 and the fine mesh scale ℎ satisfying ℎ=𝑂(𝐻²), the method we study is of the convergence rate of the same order as the usual stabilized finite-element method. However, our method is more simple and efficient. The numerical experiments further confirm the theoretical results.

This paper is organized as follows. In Section 2, an abstract functional setting for the steady Navier-Stokes equations is given, together with some basic notations and assumptions. Section 3 is to recall the stabilized finite-element method based on the local pressure projection [14] for the steady Stokes equations. we borrow the well-posedness and the optimal error estimate of the stabilized finite-element method for the steady Navier-Stokes equations in [15]. In Section 4, the uniform stability and convergence of the Oseen two-level stabilized finite method are proved. In Section 5, a series of numerical experiments is given to illustrate the theoretical results.

2. Functional Setting of the Navier-Stokes Equations

Let Ω be a bounded domain in 𝑅^𝑑(𝑑=2,3), assumed to have a Lipschitz continuous boundary Γ and to satisfy a further condition stated in (𝐻₀) below. The steady incompressible Navier-Stokes equations are considered as follows:()1−𝜈Δ𝑢+𝑢⋅∇𝑢+2(div𝑢)𝑢+∇𝑝=𝑓inΩ,div𝑢=0inΩ,𝑢=0onΓ,_Ω𝑝𝑑𝑥=0. (2.1)(2.2) Here 𝑢∶Ω→𝑅^𝑑 and 𝑝∶Ω→𝑅 are the velocity and pressure, 𝜈>0 is the viscosity, and 𝑓 represents the body forces.

For the mathematical setting of problem (2.1)-(2.2), we introduce the following Sobolev spaces: 𝑋=𝐻¹₀(Ω)^𝑑,𝑌=𝐿²(Ω)^𝑑,𝑀=𝐿²₀(Ω)=𝑞∈𝐿²(Ω),_Ω𝑞(𝑥),𝐷(𝐴)𝑑𝑥=0=𝐻²(Ω)^𝑑∩𝑋. (2.3) We make a regularity assumption on the Stokes problem as follows.

Assumption 2.1 (𝐻₀). For a given 𝑔∈𝑌 and the Stokes problem −Δ𝑣+∇𝑞=𝑔,div𝑣=0inΩ,𝑣|_Γ=0, (2.4) we assume that (𝑣,𝑞) satisfy the following regularity result: ‖𝑣‖₂+𝑞₁≤𝜅𝑔₀, (2.5) where ‖⋅‖_𝑖 is the norm of the Sobolev space 𝐻^𝑖(Ω) or 𝐻^𝑖(Ω)^𝑑,𝑖=0,1,2, as appropriate, and 𝜅 is a positive constant depending only on Ω, which may stand for different value at its different occurrences. Subsequently, the positive constants 𝜅 and 𝑐 (with or without a subscript) will depend only on the data (𝜈,Ω,𝑓). Because the norm and seminorm are equivalent on 𝐻¹₀(Ω)^𝑑, we use the same notation ‖𝑢‖₁ for them.

Now, the bilinear forms 𝑎(⋅,⋅) and 𝑑(⋅,⋅) on 𝑋×𝑋 and 𝑋×𝑀 are defined, respectively, by 𝑎(𝑢,𝑣)=𝜈(∇𝑢,∇𝑣),∀𝑢,𝑣∈𝑋,𝑑𝑣,𝑞=𝑞,div𝑣,∀𝑣,𝑞∈(𝑋,𝑀). (2.6) Also, a generalized bilinear form ℬ((⋅,⋅);(⋅,⋅)) on (𝑋,𝑀)×(𝑋,𝑀) is defined by 𝐵𝑢,𝑝;𝑣,𝑞=𝑎(𝑢,𝑣)−𝑑𝑣,𝑝+𝑑𝑢,𝑞. (2.7) Then there hold the following estimates for the bilinear term 𝐵((⋅,⋅);(⋅,⋅)) ([2–4]): ℬ𝑢,𝑝;𝑢,𝑝=𝜈‖𝑢‖²₁,ℬ𝑢,𝑝;𝑣,𝑞≤𝛼‖𝑢‖₁+𝑝₀‖𝑣‖₁+𝑞₀,sup_{(𝑣,𝑞)∈(𝑋,𝑀)}ℬ𝑢,𝑝;𝑣,𝑞‖𝑣‖₁+𝑞₀≥𝛽‖𝑢‖₁+𝑝₀, (2.8) for all (𝑢,𝑝),(𝑣,𝑞)∈(𝑋,𝑀) and constants 𝛼,𝛽>0. Moreover, we define the trilinear form 𝑏()=))+1𝑢,𝑣,𝑤((𝑢⋅∇𝑣,𝑤2))=1((div𝑢𝑣,𝑤2))−1((𝑢⋅∇𝑣,𝑤2((𝑢⋅∇)𝑤,𝑣),∀𝑢,𝑣,𝑤∈𝑋. (2.9) By the above notations and Hölder’s inequality, there hold the following estimates:𝑏()()|𝑏(≤1𝑢,𝑣,𝑤=−𝑏𝑢,𝑤,𝑣,∀𝑢∈𝑋,𝑣,𝑤∈𝑋,𝑢,𝑣,𝑤)|2𝑐₀‖𝑢‖₀^1/2‖𝑢‖₁^1/2‖𝑣‖₁‖𝑤‖₀^1/2‖𝑤‖₁^1/2+‖𝑣‖₀^1/2‖𝑣‖₁^1/2‖𝑤‖₁|𝑏(+|𝑏(+|𝑏(,∀𝑢,𝑣,𝑤∈𝑋,𝑢,𝑣,𝑤)|𝑣,𝑢,𝑤)|𝑤,𝑢,𝑣)|≤𝑐₁‖𝑢‖₁‖𝑣‖₂‖𝑤‖₀,∀𝑢∈𝑋,𝑣∈𝐷(𝐴),𝑤∈𝑌. (2.10)(2.11)(2.12) Also, the Poincare inequality holds: ‖𝑣‖₀≤𝛾₀‖𝑣‖₁, (2.13) where 𝑐₀,𝑐₁, and 𝛾₀ are positive constants depending only on Ω.

For a given 𝑓∈𝑌, the variational formulation of problem (2.1)-(2.2) reads as follows: find (𝑢,𝑝)∈(𝑋,𝑀) such that ℬ𝑢,𝑝;𝑣,𝑞+𝑏(𝑢,𝑢,𝑣)=𝑓,𝑣,∀𝑣,𝑞∈(𝑋,𝑀). (2.14) The following existence and uniqueness result is classical [26, 27].

Theorem 2.2. Assume that 𝜈>0 and 𝑓∈𝑌 satisfy the following uniqueness condition: 𝑐1−⁰𝛾²₀𝜈²𝑓₋₁>0. (2.15) Then the variational problem (2.14) admits a unique solution (𝑢,𝑝)∈(𝐷(𝐴)∩𝑋,𝐻¹(Ω)∩𝑀) such that ‖𝑢‖₁≤𝛾₀𝜈𝑓₋₁,‖𝑢‖₂+𝑝₁≤𝜅𝑓₀. (2.16)

3. The Stabilized Finite-Element Method

In this section, we focus on the stabilized method proposed by [14] for the Stokes equations. Let ℎ be a real positive parameter tending to zero. Finite-element subspace (𝑋_ℎ,𝑀_ℎ) of (𝑋,𝑀) is characterized by 𝜏_ℎ=𝜏_ℎ(Ω), a partitioning of Ω into triangles, quadrilaterals, tetrahedrons, or hexahedrons 𝐾, assumed to be regular in the usual sense. The set of all interelement boundaries will be denoted by Γ_ℎ, and the norm will be endowed as follows:‖𝑢‖_Γℎ=_{𝛾𝑓∈Γℎ𝛾𝑓}𝑢²𝑑𝑆^1/2. (3.1) In this paper, the finite element of velocity is defined by setting 𝑅₁(𝐾)=𝑃₁(𝐾)𝑄,if𝐾istriangularortetrahedron,₁(𝐾),if𝐾isthequadrilateralorhexahedron. (3.2) Then, the finite-element pairs are coupled as follows: 𝑋_ℎ=𝑣∈𝑋;𝑣_𝑖𝐾∈𝑅₁(𝐾),𝑀,𝑖=1,…,𝑑_ℎ=𝑞∈𝑀∶𝑞_𝐾∈𝑃₀(𝐾),∀𝐾∈𝜏_ℎ. (3.3)

Also, well-known approximation results are presented as follows [26, 27].(H1) Approximation properties𝑣−𝐼_ℎ𝑣₀+ℎ^1/2𝑣−𝐼_ℎ𝑣_Γℎ+ℎ𝑣−𝐼_ℎ𝑣₁≤𝜅ℎ²‖𝑣‖₂,∀𝑣∈𝐷(𝐴),𝑞−𝐽_ℎ𝑞₀≤𝜅ℎ𝑞₁,∀𝑞∈𝐻¹(Ω)∩𝑀. (3.4)(H2) Inverse inequality 𝑣_ℎ1≤𝜅ℎ⁻¹𝑣_ℎ0,∀𝑣_ℎ∈𝑋_ℎ,𝑞_ℎΓℎ≤𝜅ℎ^−1/2𝑞_ℎ0,∀𝑞_ℎ∈𝑀_ℎ. (3.5)

Here, 𝐼_ℎ∶𝐷(𝐴)→𝑋_ℎ is the interpolation operator, and 𝐽_ℎ∶𝐻¹(Ω)∩𝑀→𝑀_ℎ is the 𝐿²-orthogonal projection.

Note that neither of these methods are stable in the standard Babuška-Brezzi sense; 𝑃₁−𝑃₀ triangle or tetrahedron “locks” on regular grids (since there are more discrete incompressibility constraints than velocity degrees of freedom), and the 𝑄₁−𝑃₀ quadrilateral or hexahedron is the most infamous example of unstable mixed method [28]. Obviously, the following Lemma 3.1 manifest the insufficiency of the Babuška-Brezzi condition.

Lemma 3.1 (see [14]). There exist positive constants 𝜅₁ and 𝜅₂ whose values are independent of ℎ and such that sup_{𝑣ℎ∈𝑋ℎ}∫_Ω𝑝_ℎ∇⋅𝑣_ℎ𝑑Ω𝑣_ℎ1≥𝜅₁𝑝_ℎ0−𝜅₂ℎ^1/2[𝑝_ℎ]_Γℎ,∀𝑝_ℎ∈𝑀_ℎ. (3.6)

In order to counterbalance the terms ℎ^1/2‖[𝑝_ℎ]‖_Γℎ in (3.6) and compensate the inf-sup deficiency of the lowest-order conforming finite-element pairs, the bilinear form 𝐺(⋅,⋅) can be defined: 𝐺𝑝,𝑞=𝑝−Π𝑝,𝑞−Π𝑞, (3.7) where the local pressure projection Π∶𝐿²(Ω)→𝑅₁.

Lemma 3.2 (see [14]). There exist positive constants 𝐶₁ and 𝐶₂ whose values are independent of ℎ and such that 𝐶₂ℎ^1/2[𝑝_ℎ]_Γℎ≤𝑝_ℎ−Π𝑝_ℎ0,∀𝑝_ℎ∈𝑀_ℎ. (3.8) Furthermore, it holds that sup_{𝑣ℎ∈𝑋ℎ}∫_Ω𝑝_ℎ∇⋅𝑣_ℎ𝑑Ω𝑣_ℎ1≥𝐶₁𝑝_ℎ0−𝐶₂𝑝_ℎ−Π𝑝_ℎ0,∀𝑝_ℎ∈𝑀_ℎ. (3.9)

With the above notations and the choice of the velocity-pressure finite-element spaces (𝑋_ℎ,𝑀_ℎ)⊂(𝑋,𝑀), we define the stabilized bilinear form as follows:ℬ_ℎ𝑢_ℎ,𝑝_ℎ;𝑣,𝑞=𝑎𝑢_ℎ,𝑣−𝑑𝑣,𝑝_ℎ+𝑑𝑢_ℎ,𝑞+𝐺𝑝_ℎ,𝑞,∀𝑣,𝑞∈𝑋_ℎ,𝑀_ℎ. (3.10) Then the stabilized discrete variational formulation of the Navier-Stokes problem (2.14) reads as follows: find (𝑢_ℎ,𝑝_ℎ)∈(𝑋_ℎ,𝑀_ℎ) such that, for all (𝑣,𝑞)∈(𝑋_ℎ,𝑀_ℎ), ℬ_ℎ𝑢_ℎ,𝑝_ℎ;𝑣,𝑞+𝑏𝑢_ℎ,𝑢_ℎ,𝑣=𝑓,𝑣. (3.11) The stability of the stabilized finite-element method based on the local pressure projection for the Stokes equation is presented as follows.

Theorem 3.3 (see [2, 14]). Let (𝑋_ℎ,𝑀_ℎ) be the lowest-order conforming finite-element spaces. Assume that Π𝑝₀≤𝜅𝑝₀,∀𝑝∈𝐿²(Ω),𝑝−Π𝑝₀≤𝜅ℎ𝑝₁,∀𝑝∈𝐻¹(Ω), (3.12) hold. Then there exists a positive constant 𝛽, independent of ℎ and satisfying ℬ_ℎ𝑢,𝑝;𝑣,𝑞≤𝛼‖𝑢‖₁+𝑝₀‖𝑣‖₁+𝑞₀,∈(),𝛽𝑢,∀𝑢,𝑝𝑣,𝑞𝑋,𝑀_ℎ1+𝑝_ℎ0≤sup_{𝑣ℎ,𝑞ℎ∈𝑋ℎ,𝑀ℎ}ℬ_ℎ𝑢_ℎ,𝑝_ℎ;𝑣_ℎ,𝑞_ℎ𝑣_ℎ1+𝑞_ℎ0,∀𝑢_ℎ,𝑝_ℎ∈𝑋_ℎ,𝑀_ℎ. (3.13)(3.14) Applying Brouwer's fixed-point theorem, the following theorem can be directly given.

Theorem 3.4 (see [2, 15]). Under the assumptions of Theorems 2.2 and 3.3, problem (3.11) admits a unique solution (𝑢_ℎ,𝑝_ℎ)∈(𝑋_ℎ,𝑀_ℎ) satisfying 𝑢_ℎ1≤𝛾₀𝜈𝑓₀,𝑝_ℎ0≤𝛽⁻¹𝑐₀𝜈⁻²𝛾³₀𝑓²₀+𝛾𝑓₀. (3.15)

For simpleness, the Stokes projection (𝑅_ℎ,𝑄_ℎ)∶(𝑋,𝑀)→(𝑋_ℎ,𝑀_ℎ) is defined byℬ_ℎ𝑅_ℎ𝑣,𝑞,𝑄_ℎ𝑣,𝑞;𝑣_ℎ,𝑞_ℎ=ℬv,𝑞;𝑣_ℎ,𝑞_ℎ,∀𝑣_ℎ,𝑞_ℎ∈𝑋_ℎ,𝑀_ℎ, (3.16) for all (𝑣,𝑞)∈(𝑋,𝑀), and satisfies [2, 15] 𝑣−𝑅_ℎ(𝑣,𝑞)₀+ℎ𝑣−𝑅_ℎ𝑣,𝑞₁+𝑞−𝑄_ℎ𝑣,𝑞₀≤𝜅ℎ²‖𝑣‖₂+𝑞₁,∀𝑣,𝑞∈𝐷(𝐴),𝐻¹(Ω)∩𝑀. (3.17)

Thus, we can easily obtain the following optimal error estimate for the steady Navier-Stokes equations.

Theorem 3.5 (see [15]). Under the assumptions of Theorems 2.2 and 3.3, one has 𝑢−𝑢_ℎ0+ℎ𝑢−𝑢_ℎ1+𝑝−𝑝_ℎ0≤𝜅ℎ². (3.18)

4. Oseen Two-Level Stabilized Finite-Element Method

In this section, we will present Oseen two-level stabilized finite-element method and derive optimal bound of the errors. Let 𝐻 and ℎ≪𝐻 be two real positive parameters tending to 0. Also, a coarse mesh partition 𝜏_𝐻(Ω) of Ω is made as in Section 3, and a fine mesh partition 𝜏_ℎ(Ω) is generated by a mesh refinement process to 𝜏_𝐻(Ω). The conforming finite-element space pais (𝑋_ℎ,𝑀_ℎ) and (𝑋_𝐻,𝑀_𝐻)⊂(𝑋_ℎ,𝑀_ℎ) based on the partition 𝜏_ℎ(Ω) and 𝜏_𝐻(Ω), respectively, are constructed in Section 3. The Oseen two-level stabilized finite approximations is applied by using the lowest finite-element pairs 𝑃₁−𝑃₀ and 𝑄₁−𝑃₀ as follows.

Step I. Solve the nonlinear Navier-Stokes problem on a coarse mesh: find (𝑢_𝐻,𝑝_𝐻)∈(𝑋_𝐻,𝑀_𝐻) such that ℬ_ℎ𝑢_𝐻,𝑝_𝐻;𝑣_𝐻,𝑞_𝐻+𝑏𝑢_𝐻,𝑢_𝐻,𝑣_𝐻=𝑓,𝑣_𝐻,∀𝑣_𝐻,𝑞_𝐻∈𝑋_𝐻,𝑀_𝐻. (4.1)

Step II. Solve the linear Stokes problem on a fine mesh: find (𝑢_ℎ,𝑝_ℎ)∈(𝑋_ℎ,𝑀_ℎ) such that ℬ_ℎ𝑢^ℎ,𝑝^ℎ;𝑣_ℎ,𝑞_ℎ+𝑏𝑢_𝐻,𝑢^ℎ,𝑣_ℎ=𝑓,𝑣_ℎ,∀𝑣_ℎ,𝑞_ℎ∈𝑋_ℎ,𝑀_ℎ. (4.2)

Obviously, the Oseen two-level stabilized finite-element method admits a unique solution by applying the Lax-Milgram theorem. For convenience, we set 𝑒_ℎ=𝑢_ℎ−𝑢^ℎ,𝜂_ℎ=𝑝_ℎ−𝑝^ℎ, then a sequence of error estimates for the Oseen two-Level errors 𝑢−𝑢^ℎ and 𝑝−𝑝^ℎ are derived as follows.

Theorem 4.1. Under the assumptions of Theorems 2.2 and 3.3, the Oseen two-level stabilized finite-element scheme (4.1)-(4.2) admits a unique solution (𝑢^ℎ,𝑝^ℎ)∈(𝑋_ℎ,𝑀_ℎ) such that 𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝜅ℎ+𝐻². (4.3)

Proof. Subtracting (3.11) from (4.2) yields that 𝑎𝑒_ℎ,𝑣_ℎ−𝑑𝑣_ℎ,𝜂_ℎ+𝑑𝑒_ℎ,𝑞_ℎ+𝐺𝜂_ℎ,𝑞_ℎ+𝑏𝑢_ℎ−𝑢_𝐻,𝑢_ℎ,𝑣_ℎ+𝑏𝑢_𝐻,𝑒_ℎ,𝑣_ℎ=0. (4.4) Using (2.10) and taking (𝑣_ℎ,𝑞_ℎ)=(𝑒_ℎ,𝜂_ℎ), then 𝜈𝑒_ℎ²₁+𝐺𝜂_ℎ,𝜂_ℎ=𝑏𝑢_𝐻−𝑢_ℎ,𝑢_ℎ,𝑒_ℎ. (4.5) For the trilinear terms, by (2.11), (2.12), (2.16), and (3.18), we estimate the following 𝑏𝑢_𝐻−𝑢_ℎ,𝑢_ℎ,𝑒_ℎ=𝑏𝑢_𝐻−𝑢_ℎ,𝑢_ℎ−𝑢,𝑒_ℎ+𝑏𝑢_𝐻−𝑢_ℎ,𝑢,𝑒_ℎ≤𝜅𝑢_𝐻−𝑢_ℎ1𝑢−𝑢_ℎ1+‖𝑢‖₂𝑢_ℎ−𝑢_𝐻0𝑒_ℎ1≤𝜅𝐻²𝑒_ℎ²₁≤𝜈2𝑒_ℎ²₁+𝜅𝐻⁴. (4.6) Thanks to (3.14), (4.6), and the relation ℎ<𝐻, it follows that ‖𝑒‖₁+𝜂₀≤𝛽⁻¹sup_{(𝑣ℎ,𝑞ℎ)∈(𝑋ℎ,𝑀ℎ)}ℬ_ℎ𝑒,𝜂;𝑣_ℎ,𝑞_ℎ𝑣_ℎ1+𝑞_ℎ0≤𝑐𝐻². (4.7) Then, we deduce from a triangle inequality, (3.18), and (4.7) that 𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝑒_ℎ1+𝑢−𝑢_ℎ1+𝜂_ℎ0+𝑝−𝑝_ℎ0≤𝜅ℎ+𝐻². (4.8)

5. Numerical Examples

In this section, we concentrate on the performance of the one-level finite-element method and Oseen two-level finite element method described in this paper.

In order to implement these methods presented in this paper, a suitable choice of local pressure projection Π is important. The simplest way to accomplish this is to use standard finite-element projection or interpolation operators.

From a practical viewpoint, the main factors in the choice of Π are simplicity and locality, that is, computation of its action must be done at the element level using only standard nodal data structures. We recall the technique presented in Pavel Bochev [14], which is different from references [29, 30]. For a given node 𝑁_𝑖 in 𝜏_ℎ, let Ω_𝑖 denote its dual volume. Given a function 𝑞∈𝐿²(Ω), let 𝑞_𝑖 be the constant function on Ω_𝑖 that minimizes the functional𝐽_𝑖𝑞=12Ω_𝑖𝑞_𝑖−𝑞²𝑑Ω. (5.1) Then setΠ𝑞=^𝑁nodes_𝑖=1𝑞_𝑖𝑁_𝑖(𝑥)∈𝑅₁, (5.2) where 𝑁_𝑖(𝑥) denotes the nodal basis of 𝑅₁ and 𝑁_nodes is the number of nodes in 𝜏_ℎ. The action of the operator defined in (5.2) can be computed locally at the element level and has the same properties as the usual Clement interpolation, that is, (3.12) is satisfied. For 𝑞∈𝑞_ℎ∈𝑅₀, the functional in (5.1) further simplifies to𝐽_𝑖𝑞_ℎ=_𝐾∩Ω_𝑖≠0𝑉_𝑖(𝐾)𝑞_𝑖−𝑞_𝑘ℎ², (5.3) where 𝑞_𝑘ℎ is the constant value of 𝑞_ℎ on 𝐾 and 𝑉_𝑖(𝐾) is the volume fraction of the element 𝐾 that belongs to the dual cell Ω_𝑖 associated with the node 𝑁_𝑖. Minimization of 𝐽_𝑖 then yields the formula 𝑞_𝑖=∑_𝐾∈Ω_𝑖𝑉_𝑖(𝐾)𝑞_𝑘ℎ∑_𝐾∈Ω_𝑖𝑉_𝑖(𝐾); (5.4) that is, the nodal values of Π𝑞_ℎ are area weighted averages of the surrounding constant pressure values of 𝑞_ℎ.

For the purpose of numerical comparisons, we consider the spatial domain in 𝑅² as [0,1]×[0,1] and viscosity 𝜈=1.0. The velocity and pressure are designed on the same uniform triangulation of Ω. The exact solution is given by 𝑢=𝑢₁𝑥,𝑦,𝑢₂(),𝑢𝑥,𝑦,𝑝𝑥,𝑦=10𝑥−0.5₁𝑥,𝑦=10𝑥²(1−𝑥)²𝑦,𝑢1−𝑦1−2𝑦₂𝑥,𝑦,=−10𝑥(1−𝑥)(1−2𝑥)𝑦²1−𝑦², (5.5) and 𝑓 is determined by (2.1).

For simplicity, the unit square is divided into 𝑁² small squares, where 𝑁 dedicates the number of the partition in each direction. When 𝐾 is the square, 𝑉_𝑖(𝐾)=1/𝑁². Then the formula (5.4) can be given as follows: 𝑞_𝑖=∑_𝐾∈Ω_𝑖𝑞_𝑘ℎ𝑛_𝑘, (5.6) where 𝑛_𝑘 is the number of elements in Ω_𝑖.

To establish a reference point for the evaluation of the possible impact from the Oseen two-level stabilized finite-element method, we compute the one-level stabilized finite-element method for the steady Navier-Stokes equations on the fine mesh. The Oseen two-level stabilized finite-element solutions (𝑢^ℎ,𝑝^ℎ) are obtained by solving the solution (𝑢_𝐻,𝑝_𝐻) of the stationary Navier-Stokes equations on a coarse mesh and then solving a linear system on a fine mesh with mesh size ℎ.

Table 1 shows the relative energy error estimates of the numerical velocity and pressure for these methods. In particular, the results of the one-level stabilized finite-element method are presented on the first line; those of the Oseen two-level stabilized finite-element method with relationship ℎ=𝑂(𝐻²) on second line. From Table 1, we find that these two methods have a convergence rate of the same order, as shown in Theorem 4.1. However, the Oseen method is probably two times faster than the one-level stabilized method.

Table 1 

Comparison of the one-level stabilized method with the Oseen two-level stabilized method.

In conclusion, the Oseen two-level stabilized finite-element method is an efficient and potential method for solving the 2D/3D steady Navier-Stokes equation. It is also suitable to solve some practical engineering problems arising in the fluid dynamics. Furthermore, the methods can help to solve the two-dimensional and three-dimensional nonstationary incompressible viscous flows which will be discussed in our further work.

Acknowledgments

This work was supported by the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant no. KM201110772019), the National High Technology Research and Development Program of China (Grant no. 2010AA012301), the National Natural Science Foundation of China (Grant nos. 11071193, 641075062, 40821092), the Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (Grant no. PHR201006116), the National Basic Research Program (Grant no. 2010CB951001, 2010CB428403) and Beijing Natural Science Foundation (Grant no. 1122015). The author would like to thank the referees for their valuable comments and suggestions.