Two-Level Stabilized Finite Volume Methods for Stationary Navier-Stokes Equations

Abstract

We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths.

1. Introduction

We consider a two-level method for the resolution of the nonlinear system arising from finite volume discretizations of the equilibrium, incompressible Navier-Stokes equations: −𝜈Δ𝑢+(𝑢⋅∇)𝑢+∇𝑝=𝑓inΩ,∇⋅𝑢=0inΩ,𝑢=0in𝜕Ω, (1.1)(1.2)(1.3) where 𝑢=(𝑢₁(𝑥),𝑢₂(𝑥)) is the velocity vector, 𝑝=𝑝(𝑥) is the pressure, 𝑓=𝑓(𝑥) is the body force, 𝜈>0 is the viscosity of the fluid, and Ω⊂ℝ², the flow domain, is assumed to be bounded, to have a Lipschitz-continuous boundary 𝜕Ω, and to satisfy a further condition stated in (H1).

Finite volume method is an important numerical tool for solving partial differential equations. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer, and petroleum engineering. The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. Usually, the former one is called finite volume method [1, 2], MAC (marker and cell) method [3], or cell-centered method [4], and the latter one is called finite volume element method (FVE) [5–7], covolume method [8], or vertex-centered method [9, 10]. We refer to the monographs [11, 12] for general presentations of these methods. The most important property of FVE is that it can preserve the conservation laws (mass, momentum, and heat flux) on each control volume. This important property, combined with adequate accuracy and ease of implementation, has attracted more people to do research in this field.

On the other hand, the two-level finite element strategy based on two finite element spaces on one coarse and one fine mesh has been widely studied for steady semilinear elliptic equations [13, 14] and the Navier-Stokes equations [15–22]. For the finite volume element method, Bi and Ginting [23] have studied two-grid finite volume element method for linear and nonlinear elliptic problems; Chen et al. [24] have applied two-grid methods for solving a two-dimensional nonlinear parabolic equation using finite volume element method. Chen and Liu [25] have also studied this method for semilinear parabolic problems. However, to the best of our knowledge, there is no two-level finite volume convergence analysis for the Navier-Stokes equations in the literature.

In this paper we aim to combine FVE method based on 𝑃₁−𝑃₀ macroelement with two-level strategy to solve the two-dimensional Navier-Stokes (1.1)–(1.3). The heart of the analysis is the use of a transfer operator to connect finite volume and finite element estimations which will lead to more difficult term to estimate. We choose the two-grid spaces as two conforming finite element spaces 𝑋_𝐻 and 𝑋_ℎ on one coarse grid with mesh size 𝐻 and one fine grid with mesh size ℎ≪𝐻. We propose two algorithms of two-level method for resolving the nonlinearity in the stabilized finite volume approximation of the problem (1.1)–(1.3): the simple and Newton algorithms. First we prove that the simple two-level stabilized finite volume solution (𝑢^ℎ,𝑝^ℎ) is the following error estimate:𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝐶ℎ+𝐻². (1.4) Second we prove that the Newton two-level stabilized finite volume solution (𝑢^ℎ,𝑝^ℎ) is the following error estimate:𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝐶ℎ+𝐻³logℎ^1/2, (1.5) where 𝐶 denotes some generic constant which may stand for different values at its different occurrences.

Hence, the two-level algorithms achieve asymptotically optimal approximation as long as the mesh sizes satisfy ℎ=𝑂(𝐻²) for the simple two-level stabilized finite volume solution and ℎ=𝑂(𝐻³|logℎ|^1/2) for the Newton two-level stabilized finite volume solution. As a result, solving the nonlinear Navier-Stokes equations will not be much more difficult than solving one single linearized equation.

The rest of this paper is organized as follows. In the next section, we introduce some notations and construct a FVE scheme. In Section 3 we recall same preliminary estimates of the stabilized finite volume approximations. Finally the two-level FVE algorithms and the improved error estimates are presented and established in Section 4.

2. Finite Volume Scheme

2.1. Notations

We will use ‖⋅‖_𝑚 and |⋅|_𝑚 to denote the norm and seminorm of the Sobolev space (𝐻^𝑚(Ω))^𝑑, 𝑑=1,2. Let 𝐻¹₀(Ω) be the standard Sobolev subspace of 𝐻¹(Ω) of functions vanishing on 𝜕Ω. We introduce the following notations:𝑋=𝐻¹₀(Ω)²,𝑌=𝐿²₀(Ω)=𝑞∶𝑞∈𝐿²(Ω),_Ω𝑞=0. (2.1) The scalar product and norm in 𝑌 are denoted by the usual 𝐿²(Ω) inner product (⋅,⋅) and ‖⋅‖₀, respectively. As mentioned above, we need a further assumption on Ω.

H1
Assume that Ω is regular so that the unique solution (𝑣,𝑞)∈𝑋×𝑌 of the steady Stokes problem −Δ𝑣+∇𝑞=𝑔,∇⋅𝑣=0;𝑣|_𝜕Ω=0 (2.2) for a prescribed 𝑔∈(𝐿²(Ω))² exists and satisfies ‖𝑣‖₂+𝑞₁≤𝐶𝑔₀, (2.3) where 𝐶>0 is a constant depending on Ω.

The weak formulation of the problem (1.1)–(1.3) is to find (𝑢,𝑝)∈𝑋×𝑌 such that𝑎()()=𝑑𝑢,𝑣−𝑑𝑣,𝑝+𝑏𝑢,𝑢,𝑣𝑓,𝑣,∀𝑣∈𝑋,𝑢,𝑞=0,∀𝑞∈𝑌, (2.4) where the bilinear forms 𝑎(⋅,⋅), 𝑑(⋅,⋅) and the trilinear form 𝑏(⋅,⋅,⋅) are given by𝑎(𝑢,𝑣)=𝜈(∇𝑢,∇𝑣)=𝜈_Ω𝑑=∇𝑢∶∇𝑣𝑑𝑥,∀𝑢,𝑣∈𝑋,𝑣,𝑞∇⋅𝑣,𝑞=_Ω𝑏()=))+1𝑞∇⋅𝑣𝑑𝑥,∀𝑢,𝑞∈𝑋×𝑌,𝑢,𝑤,𝑣((𝑢⋅∇𝑤,𝑣2))=1((∇⋅𝑢𝑤,𝑣2))−1((𝑢⋅∇𝑤,𝑣2((𝑢⋅∇𝑣),𝑤)∀𝑢,𝑣,𝑤∈𝑋. (2.5) Introducing the generalized bilinear form on (𝑋×𝑌)² by𝐵𝑢,𝑝;𝑣,𝑞=𝑎(𝑢,𝑣)−𝑑𝑣,𝑝+𝑑𝑢,𝑞, (2.6) we can rewrite (2.4) in a compact form: find (𝑢,𝑝)∈𝑋×𝑌 such that𝐵𝑢,𝑝;𝑣,𝑞+𝑏(𝑢,𝑢,𝑣)=𝑓,𝑣,∀𝑣,𝑞∈𝑋×𝑌. (2.7)

Let 𝒯_ℎ be a quasi-uniform triangulation of Ω with ℎ=maxℎ_𝐾, where ℎ_𝐾 is the diameter of the triangle 𝐾∈𝒯_ℎ. We assume that the partition 𝒯_ℎ has been obtained from a macrotriangular partition Λ_ℎ by joining the sides of each element of Λ_ℎ. Every element 𝐾∈𝒯_ℎ must lie in exactly one macroelement 𝒦, which implies that macroelements do not overlap. For each 𝒦, the set of interelement edges which are strictly in the interior of 𝒦 will be denoted by Γ_𝒦, and the length of an edge 𝑒∈Γ_𝒦 is denoted by ℎ_𝑒.

Based on this triangulation, let 𝑋_ℎ be the standard conforming finite element subspace of piecewise linear velocity,𝑋_ℎ=𝑣∈𝐶(Ω)∩𝑋∶𝑣∣_𝐾islinear,∀𝐾∈𝒯_ℎ;𝑣∣_𝜕Ω=0, (2.8) and let 𝑌_ℎ be the piecewise constant pressure subspace𝑌_ℎ=𝑞∈𝑌∶𝑞∣_𝐾isconstant,∀𝐾∈𝒯_ℎ. (2.9) It is well known that the standard 𝑃₁−𝑃₀ element does not satisfy the inf-sup condition and cannot be applied to problem (1.1)–(1.3) directly. But a locally stabilized method based on the macroelement can be used to yield adequate approximations [6].

In order to describe the FVE method for solving problem (1.1)–(1.3), we will introduce a dual partition 𝒯^∗_ℎ based upon the original partition 𝒯_ℎ whose elements are called control volumes. We construct the control volumes in the same way as in [5, 26]. Let 𝑧_𝐾 be the barycenter of 𝐾∈𝒯_ℎ. We connect 𝑧_𝐾 with line segments to the midpoints of the edges of 𝐾, thus partitioning 𝐾 into three quadrilaterals 𝐾_𝑧, 𝑧∈𝑍_ℎ(𝐾), where 𝑍_ℎ(𝐾) are the vertices of 𝐾. Then with each vertex 𝑧∈𝑍_ℎ=∪_{𝐾∈𝒯ℎ}𝑍_ℎ(𝐾), we associate a control volume 𝑉_𝑧, which consists of the union of the subregions 𝐾_𝑧, sharing the vertex 𝑧. Thus we finally obtain a group of control volumes covering the domain Ω, which is called the dual partition 𝒯^∗_ℎ of the triangulation 𝒯_ℎ. We denote by 𝑍⁰_ℎ the set of interior vertices.

We call the partition 𝒯^∗_ℎ regular or quasi-uniform if there exists a positive constant 𝐶>0 such that𝐶⁻¹ℎ²≤meas𝑉_𝑧≤𝐶ℎ²,𝑉_𝑧∈𝒯^∗_ℎ. (2.10) If the finite element triangulation 𝒯_ℎ is quasi-uniform, then the dual partition 𝒯^∗_ℎ is also quasi-uniform [23].

2.2. Construction of the FVE Scheme

We formulate the FVE method for the problem (1.1)–(1.3) as follows: given a 𝑧∈𝑍⁰_ℎ and 𝐾∈𝒯_ℎ, integrating (1.1) over the associated control volume 𝑉_𝑧 and (1.1) over the element 𝐾 and applying Green’s formula, we obtain an integral conservation form−𝜈_𝜕𝑉𝑧∇𝑢𝑛𝑑𝑠+_𝜕𝑉𝑧𝑝𝑛𝑑𝑠+_𝑉𝑧𝑢⋅∇𝑢𝑑𝑥=_𝑉𝑧𝑓𝑑𝑥,∀𝑧∈𝑍⁰_ℎ,_𝐾∇⋅𝑢𝑑𝑥=0,∀𝐾∈𝒯_ℎ, (2.11)(2.12) where 𝑛 denotes the unit outer normal vector to 𝜕𝑉_𝑧 (Figure 1).

(a)

(b)

Figure 1 

Left-hand side: a sample region with blue lines indicating the corresponding control volume 𝑉𝑧. Right-hand side: a triangle 𝐾 partitioned into three subregions 𝐾𝑧.

Let 𝐼^∗_ℎ∶𝑋_ℎ→𝑋^∗_ℎ be the transfer operator defined by𝐼^∗_ℎ𝑣=_{𝑧∈𝑍0ℎ}𝑣(𝑧)𝜒_𝑧,∀𝑣∈𝑋_ℎ, (2.13) where𝑋^∗_ℎ=𝑣=𝑣₁,𝑣₂∈𝐿²(Ω)²∶𝑣_𝑖∣_𝑉𝑧isconstant,𝑖=1,2∀𝑧∈𝑍⁰_ℎ, (2.14) and 𝜒_𝑧 is the characteristic function of the control volume 𝑉_𝑧. The operator 𝐼^∗_ℎ satisfies [26]𝐼^∗_ℎ𝑣₀≤𝐶‖𝑣‖₀,∀𝑣∈𝑋_ℎ. (2.15) Now for an arbitrary 𝐼^∗_ℎ𝑣, we multiply (2.11) by 𝑣(𝑧) and sum over all 𝑧∈𝑍⁰_ℎ to get𝑎_ℎ𝑢,𝐼^∗_ℎ𝑣−𝑑_ℎ𝐼^∗_ℎ𝑣,𝑝+𝑏_ℎ𝑢,𝑢,𝐼^∗_ℎ𝑣=𝑓,𝐼^∗_ℎ𝑣,∀𝑣∈𝑋_ℎ. (2.16) Here 𝑎_ℎ∶𝑋×𝑋_ℎ→ℝ, 𝑑_ℎ∶X_ℎ×𝑌→ℝ and 𝑏_ℎ∶𝑋×𝑋×𝑋_ℎ→ℝ are defined by𝑎_ℎ𝑢,𝐼^∗_ℎ𝑣=−𝜈_{𝑧∈𝑍0ℎ}𝑣(𝑧)_𝜕𝑉𝑧𝑑∇𝑢𝑛𝑑𝑠,_ℎ𝐼^∗_ℎ𝑣,𝑝=_{𝑧∈𝑍0ℎ}𝑣(𝑧)_𝜕𝑉𝑧𝑏𝑝𝑛𝑑𝑠,_ℎ𝑢,𝑢,𝐼^∗_ℎ𝑣=_{𝑧∈𝑍0ℎ}𝑣(𝑧)_𝑉𝑧(𝑢⋅∇)𝑢𝑑𝑥. (2.17) We also define the trilinear forms 𝑏̃(⋅,⋅,⋅) and 𝑏(⋅,⋅,⋅) on 𝑋×𝑋×𝑋_ℎ by𝑏̃𝑢,𝑣,𝐼^∗_ℎ𝑤=(𝑢⋅∇)𝑣,𝐼^∗_ℎ𝑤+12(∇⋅𝑢)𝑣,𝐼^∗_ℎ𝑤,𝑏𝑢,𝑣,𝑤−𝐼^∗_ℎ𝑤=(𝑢⋅∇)𝑣,𝑤−𝐼^∗_ℎ𝑤+12(∇⋅𝑢)𝑣,𝑤−𝐼^∗_ℎ𝑤. (2.18) To formulate the discrete problem so as to eliminate any such potential difficulties, we rewrite (2.16) as follows:𝑎_ℎ𝑢,𝐼^∗_ℎ𝑣−𝑑_ℎ𝐼^∗_ℎ𝑣,𝑝+𝑏̃𝑢,𝑢,𝐼^∗_ℎ𝑣=𝑓,𝐼^∗_ℎ𝑣,∀𝑣∈𝑋_ℎ. (2.19) We multiply (2.12) by 𝑞∈𝑌_ℎ and sum over all 𝐾∈𝒯_ℎ: then, we obtain𝑏𝑢,𝑞=0,∀𝑞∈𝑌_ℎ. (2.20) Now we rewrite (2.19) and (2.20) to a variational form similar to finite element problems. The locally stabilized FVE scheme is to find (𝑢_ℎ,𝑝_ℎ)∈𝑋_ℎ×𝑌_ℎ such that𝑎_ℎ𝑢_ℎ,𝐼^∗_ℎ𝑣_ℎ−𝑑_ℎ𝐼^∗_ℎ𝑣_ℎ,𝑝_ℎ+𝑏̃𝑢_ℎ,𝑢_ℎ,𝐼^∗_ℎ𝑣_ℎ=𝑓,𝐼^∗_ℎ𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝑑𝑢_ℎ,𝑞_ℎ−𝛽𝑐_ℎ𝑝_ℎ,𝑞_ℎ=0,∀𝑞_ℎ∈𝑌_ℎ, (2.21) where𝑐_ℎ𝑝,𝑞=_{𝒦∈Λℎ𝑒∈Γ𝒦}ℎ_𝑒𝑒𝑝_𝑒𝑞_𝑒𝑑𝑠 (2.22) is a stabilized form defined on (𝐻¹(Ω)+𝑌_ℎ)², [⋅]_𝑒 is the jump operator across the edge 𝑒, and 𝛽>0 is the local stabilization parameter. It is trivial that 𝑐_ℎ(𝑝,𝑞_ℎ)=𝑐_ℎ(𝑝_ℎ,𝑞)=𝑐_ℎ(𝑝,𝑞)=0, forall𝑝,𝑞∈𝐻¹(Ω), forall𝑝_ℎ,𝑞_ℎ∈𝑌_ℎ.

A general framework for analyzing the locally stabilized formulation (2.21) can be developed using the notion of equivalence class of macroelements. As in Stenberg [27] each equivalence class, denoted by ℰ𝒦, contains macroelements which are topologically equivalent to a reference macroelement 𝒦.

Let 𝐵_ℎ𝑢_ℎ,𝑝_ℎ;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ=𝑎_ℎ𝑢_ℎ,𝐼^∗_ℎ𝑣_ℎ−𝑑_ℎ𝐼^∗_ℎ𝑣_ℎ,𝑝_ℎ+𝑑𝑢_ℎ,𝑞_ℎ,𝐵_ℎ𝑢_ℎ,𝑝_ℎ;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ=𝐵_ℎ𝑢_ℎ,𝑝_ℎ;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ+𝛽𝑐_ℎ𝑝_ℎ,𝑞_ℎ. (2.23)(2.24) We can rewrite (2.21) in a compact form: find (𝑢_ℎ,𝑝_ℎ)∈𝑋_ℎ×𝑌_ℎ such that𝐵_ℎ𝑢_ℎ,𝑝_ℎ;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ+𝑏̃𝑢_ℎ,𝑢_ℎ,𝐼^∗_ℎ𝑣_ℎ=𝑓,𝐼^∗_ℎ𝑣_ℎ,∀𝑣_ℎ,𝑞_ℎ∈𝑋_ℎ×𝑌_ℎ. (2.25)

3. Technical Preliminaries

This section considers preliminary estimates which will be very useful in the error estimates of two-level finite volume solution (𝑢^ℎ,𝑝^ℎ).

The following lemma gives the boundedness of the trilinear form 𝑏(⋅,⋅,⋅).

Lemma 3.1 (see [21]). The following estimates hold: 𝑏()(),|𝑏(≤1𝑢,𝑤,𝑣=−𝑏𝑢,𝑣,𝑤𝑢,𝑤,𝑣)|2𝐶₀‖𝑢‖₀^1/2|𝑢|₁^1/2|𝑤|₁‖𝑣‖₀^1/2|𝑣|₁^1/2+‖𝑤‖₀^1/2‖𝑤‖₁^1/2‖𝑣‖₁^1/2|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∈𝐻²(Ω)²|𝑏(∩𝑋,𝑤∈𝑌,𝑢,𝑣,𝑤)|≤𝐶logℎ^1/2|𝑢|₁|𝑣|₁‖𝑤‖₀,∀𝑢,𝑣,𝑤∈𝑋_ℎ. (3.1) Here and after 𝐶_𝑖, 𝑖=1,2, and 𝐶 are positive constants depending only on the data (𝜈,𝑓,Ω).

The existence and uniqueness results of (2.7) can be found in [28, 29].

Theorem 3.2. Assume that 𝜈>0 and 𝑓∈𝑌 satisfy the following uniqueness condition: 𝑁1−¹𝜈²𝑓₋₁>0, (3.2) where 𝑁₁=sup_{𝑢,𝑣,𝑤∈𝑋}𝑏(𝑢,𝑣,𝑤)|𝑢|₁|𝑣|₁|𝑤|₁. (3.3) Then the problem (2.7) admits a unique solution (𝑢,𝑝)∈(𝐻¹₀(Ω)²∩𝑋,𝐻¹(Ω)∩𝑌) such that |𝑢|₁≤1𝜈𝑓₋₁,‖𝑢‖₂+𝑝₁≤𝐶𝑓₀. (3.4)

In [30] the following lemma was proved, which shows that the finite volume element bilinear forms 𝑎_ℎ(⋅,𝐼^∗_ℎ⋅) and 𝑑_ℎ(𝐼^∗_ℎ⋅,⋅) are equal to the finite element ones, respectively.

Lemma 3.3. For any 𝑢_ℎ,𝑣_ℎ∈𝑋_ℎ, and 𝑞_ℎ∈𝑌_ℎ, one has 𝑎_ℎ𝑢_ℎ,𝐼^∗_ℎ𝑣_ℎ=𝑎𝑢_ℎ,𝑣_ℎ,𝑑_ℎ𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ=𝑑𝑣_ℎ,𝑞_ℎ. (3.5)

The following theorem establishes the weak coercivity of (2.24) [6, 31].

Theorem 3.4. Given a a stabilization parameter 𝛽≥𝛽₀, suppose that every macroelement 𝒦∈Λ_ℎ belongs to one of the equivalence classes ℰ𝒦 and that the following macroelement connectivity condition is valid: for any two neighboring macroelements 𝒦₁ and 𝒦₂ with ∫_{𝒦1∩𝒦2}≠0, there exists 𝑣∈𝑋_ℎ such that supp𝑣⊂𝒦₁∩𝒦_{2𝒦1∩𝒦2}𝑣⋅𝑛𝑑𝑠≠0. (3.6) Then 𝛼₁𝑢_ℎ1+𝑝_ℎ0≤sup_{(𝑣ℎ,𝑞ℎ)∈𝑋ℎ×𝑌ℎ}𝐵_ℎ𝑢,𝑝;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ𝑣_ℎ1+𝑞_ℎ0, (3.7) for all (𝑢_ℎ,𝑝_ℎ)∈𝑋_ℎ×𝑌_ℎ, and 𝐵_ℎ𝑢,𝑝;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ≤𝛼₂𝑢_ℎ1+𝑝_ℎ0𝑣_ℎ1+𝑞_ℎ0𝑣_ℎ,𝑞_ℎ∈𝑋_ℎ×𝑌_ℎ, (3.8) where 𝛼₁,𝛼₂>0 are constants independent of ℎ and 𝛽, 𝛽₀ is some fixed positive constant, and 𝑛 is the out-normal vector.

Next, we establish the existence and the uniqueness of FVE scheme (2.25), by the fixed-point theorem, in the following.

Theorem 3.5 (see [6]). Suppose the assumptions of Theorems 3.2 and 3.4 hold, and the body force 𝑓 satisfies the following uniqueness condition 1−4𝑁𝜈²𝑓₋₁>0. (3.9) Then the variation problem (2.25) admits a unique solution (𝑢_ℎ,𝑝_ℎ)∈(𝑋_ℎ×𝑌_ℎ) such that 𝑢_ℎ1≤1𝜈𝑓₋₁,𝑝_ℎ0≤𝛼₂𝛼₁𝑓₋₁+4𝛼₂𝑁𝛼₁𝜈²𝑓₋₁, (3.10) where 𝑁=max𝐶𝑁₁,𝑁₂,𝑁₂=sup_{𝑢,𝑣,𝑤∈𝑋}𝑏̃𝑢,𝑣,𝐼^∗_ℎ𝑤|𝑢|₁|𝑣|₁|𝑤|₁. (3.11)

For the error estimate, we introduce the Galerkin projection (𝑅_ℎ,𝑄_ℎ)∶𝑋×𝑌→𝑋_ℎ×𝑌_ℎ defined by𝐵_ℎ𝑅_ℎ𝑣,𝑞,𝑄_ℎ𝑣,𝑞;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ=𝐵_ℎ𝑣,𝑞;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ (3.12) for each (𝑣,𝑞)∈𝑋×𝑌 and all (𝑣_ℎ,𝑞_ℎ)∈𝑋_ℎ×𝑌_ℎ. We obtain the following results by using the standard Galerkin finite element [6, 17].

Theorem 3.6. Under the assumptions of Theorems 3.2 and 3.4, the projection (𝑅_ℎ,𝑄_ℎ) satisfies 𝑣−𝑅_ℎ𝑣,𝑞₁+𝑞−𝑄_ℎ𝑣,𝑞₀≤𝐶|𝑣|₁+𝑞₀, (3.13) for all (𝑣,𝑞)∈𝑋×𝑌 and 𝑣−𝑅_ℎ𝑣,𝑞₀+ℎ𝑣−𝑅_ℎ𝑣,𝑞₁+𝑞−𝑄_ℎ𝑣,𝑞₀≤𝐶ℎ²‖𝑣‖₂+𝑞₁, (3.14) for all (𝑣,𝑞)∈(𝐷(𝐴)∩𝑋)×(𝐻¹(Ω)∩𝑌).

Then the optimal error estimates can be obtained as follows.

Theorem 3.7 (see [6, 32]). Under the assumptions of Theorems 3.2, 3.4, 3.5, and 3.6, the solution (𝑢_ℎ,𝑝_ℎ) of (2.25) satisfies 𝑢−𝑢_ℎ0+ℎ𝑢−𝑢_ℎ1+𝑝−𝑝_ℎ0≤𝐶ℎ². (3.15)

4. Two-Level FVE Algorithms and Its Error Analysis

In this section, we will present two-level stabilized finite volume element algorithm for (1.1)–(1.3) and derive some optimal bounds for errors. The idea of the two-level method is to reduce the nonlinear problem on a fine mesh into a linear system on a fine mesh by solving a nonlinear problem on a coarse mesh. The basic mechanisms are two quasi-uniform triangulations of Ω, 𝒯_𝐻, and 𝒯_ℎ, with two different mesh sizes 𝐻 and ℎ(ℎ≪𝐻), and the corresponding solutions spaces (𝑋_𝐻,𝑌_𝐻) and (𝑋_ℎ,𝑌_ℎ), which satisfy (𝑋_𝐻,𝑌_𝐻)⊂(𝑋_ℎ,𝑌_ℎ) and will be called the coarse and the fine spaces, respectively. Now find (𝑢^ℎ,𝑝^ℎ) as follows.

Algorithm 4.1 (Simple two-level stabilized FVE approximation). We have the following steps:

Step 1. On the coarse mesh 𝒯_𝐻, solve the stabilized Navier-Stokes problem.
Find (𝑢_𝐻,𝑝_𝐻)∈𝑋_𝐻×𝑌_𝐻 such that, forall(𝑣_𝐻,𝑞_𝐻)∈𝑋_𝐻×𝑌_𝐻, 𝐵_𝐻𝑢_𝐻,𝑝_𝐻;𝐼^∗_𝐻𝑣_𝐻,𝑞_𝐻+𝑏̃𝑢_𝐻,𝑢_𝐻,𝐼^∗_𝐻𝑣_𝐻=𝑓,𝐼^∗_𝐻𝑣_𝐻. (4.1)

Step 2. On the fine mesh 𝒯_ℎ, solve the stabilized linear Stokes problem.
Find (𝑢^ℎ,𝑝^ℎ)∈𝑋_ℎ×𝑌_ℎ such that, forall(𝑣_ℎ,𝑞_ℎ)∈𝑋_ℎ×𝑌_ℎ, 𝐵_ℎ𝑢^ℎ,𝑝^ℎ;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ+𝑏̃𝑢_𝐻,𝑢_𝐻,𝐼^∗_ℎ𝑣_ℎ=𝑓,𝐼^∗_ℎ𝑣_ℎ. (4.2)

Next, we study the convergence of (𝑢^ℎ,𝑝^ℎ) to (𝑢,𝑝) in some norms. For convenience, we set 𝑒=𝑅_ℎ(𝑢,𝑝)−𝑢^ℎ and 𝜂=𝑄_ℎ(𝑢,𝑝)−𝑝^ℎ.

Theorem 4.2. Under the assumptions of Theorems 3.2, 3.4, 3.5, and 3.6 for H and h, the simple two-level stabilized FVE solution (𝑢^ℎ,𝑝^ℎ) satisfies the following error estimates: 𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝐶ℎ+𝐻². (4.3)

Proof. Subtracting (4.2) from (2.25) and using the Galerkin projection (3.12), it is easy to see that 𝐵_ℎ𝑒,𝜂;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ+𝑏𝑢,𝑢−𝑢_𝐻,𝑣_ℎ+𝑏𝑢−𝑢_𝐻,𝑢,𝑣_ℎ−𝑏𝑢−𝑢_𝐻,𝑢−𝑢_𝐻,𝑣_ℎ+𝑏𝑢−𝑢_𝐻,𝑢−𝑢_𝐻,𝑣_ℎ−𝐼^∗_ℎ𝑣_ℎ−𝑏𝑢−𝑢_𝐻,𝑢,𝑣_ℎ−𝐼^∗_ℎ𝑣_ℎ−𝑏𝑢,𝑢−𝑢_𝐻,𝑣_ℎ−𝐼^∗_ℎ𝑣_ℎ+𝑏𝑢,𝑢,𝑣_ℎ−𝐼^∗_ℎ𝑣_ℎ=𝑓,𝑣_ℎ−𝐼^∗_ℎ𝑣_ℎ, (4.4) for all (𝑣_ℎ,𝑞_ℎ)∈𝑋_ℎ×𝑌_ℎ. Due to (3.7), Lemma 3.1, (3.15), and (4.4), we have 𝛼₁|𝑒|₁+𝜂₀≤sup_{(𝑣ℎ,𝑞ℎ)∈𝑋ℎ×𝑌ℎ}𝐵_ℎ𝑢,𝑝;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ𝑣_ℎ1+𝑞_ℎ0≤𝐶‖𝑢‖₂𝑢−𝑢_𝐻0+𝐶𝑢−𝑢_𝐻²₁+𝐶ℎ‖𝑢‖₂𝑢−𝑢_𝐻1𝐻+𝐶ℎ≤𝐶²+ℎ, (4.5) which, along with (3.14), yields 𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝑢−𝑅_ℎ𝑢,𝑝₁+𝑝−𝑄_ℎ𝑢,𝑝₀+|𝑒|₁+𝜂₀≤𝐶ℎ+𝐻². (4.6)

Algorithm 4.3 (The Newton two-level stabilized FVE approximation). We have the following steps:

Step 1. On the coarse mesh 𝒯_𝐻, solve the stabilized Navier-Stokes problem.
Find (𝑢_𝐻,𝑝_𝐻)∈𝑋_𝐻×𝑌_𝐻 by (4.1).

Step 2. On the fine mesh 𝒯_ℎ, solve the stabilized linear Stokes problem.
Find (𝑢^ℎ,𝑝^ℎ)∈𝑋_ℎ×𝑌_ℎ such that, forall(𝑣_ℎ,𝑞_ℎ)∈𝑋_ℎ×𝑌_ℎ, 𝐵_ℎ𝑢^ℎ,𝑝^ℎ;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ+𝑏_ℎ𝑢^ℎ,𝑢_𝐻,𝐼^∗_ℎ𝑣_ℎ+𝑏_ℎ𝑢_𝐻,𝑢^ℎ,𝐼^∗_ℎ𝑣_ℎ=𝑓,𝐼^∗_ℎ𝑣_ℎ+𝑏_ℎ𝑢_𝐻,𝑢_𝐻,𝐼^∗_ℎ𝑣_ℎ. (4.7)

Now, we will study the convergence of the Newton two-level stabilized finite element solution (𝑢^ℎ,𝑝^ℎ) to (𝑢,𝑝) in some norms. To do this, let us set 𝑒=𝑅_ℎ(𝑢,𝑝)−𝑢^ℎ, 𝐸=𝑢−𝑅_ℎ(𝑢,𝑝), and 𝜂=𝑄_ℎ(𝑢,𝑝)−𝑝^ℎ.

Theorem 4.4. Under the assumptions of Theorems 3.2, 3.4, 3.5, and 3.6 for 𝐻 and ℎ, the Newton two-level stabilized FVE solution (𝑢^ℎ,𝑝^ℎ) satisfies the following error estimates: 𝑢−𝑢^ℎ₁+𝑝−𝑝^ℎ₀≤𝐶ℎ+logℎ^1/2𝐻³. (4.8)

Proof. Subtracting (4.7) from (2.25), using the Galerkin projection (3.12) and taking (𝑣_ℎ,𝑞_ℎ)=(𝑒,𝜂), we get 𝐵_ℎ𝑒,𝜂;𝐼^∗_ℎ𝑒,𝜂+𝑏(𝐸,𝑢,𝑒)+𝑏𝑅_ℎ−𝑢_𝐻,𝑢−𝑢_𝐻,𝑒+𝑏𝑢_𝐻−,𝐸,𝑒𝑏𝑅_ℎ−𝑢_𝐻,𝑅_ℎ−𝑢_𝐻,𝑒−𝐼^∗_ℎ𝑒−𝑏𝑅_ℎ,𝑅_ℎ−𝑢,𝑒−𝐼^∗_ℎ𝑒−𝑏𝑅_ℎ,𝑢,𝑒−𝐼^∗_ℎ𝑒−𝑏𝑒,𝑢_𝐻,𝑒+𝑏𝑒,𝑢_𝐻,𝑒−𝐼^∗_ℎ𝑒+𝑏𝑢_𝐻,𝑒,𝑒−𝐼^∗_ℎ𝑒=𝑓,𝑒−𝐼^∗_ℎ𝑒. (4.9) Using Lemma 3.1 and Theorems 3.2, 3.5, and 3.7, we obtain 𝑏(𝐸,𝑢,𝑒)+𝑏𝑢_𝐻,𝐸,𝑒+𝑓,𝑒−𝐼^∗_ℎ𝑒≤𝐶|𝑢|₁+𝑢_𝐻1|𝐸|₁|𝑒|₁+𝑓₀𝑒−𝐼^∗_ℎ𝑒₀≤𝐶ℎ|𝑒|₁,𝑏𝑅_ℎ−𝑢_𝐻,𝑢−𝑢_𝐻,𝑒=𝑏𝑅_ℎ−𝑢_𝐻,𝑒,𝑢−𝑢_𝐻≤𝐶logℎ^1/2𝑅_ℎ−𝑢_𝐻1|𝑒|₁𝑢−𝑢_𝐻0≤𝐶logℎ^1/2𝑅_ℎ−𝑢₁+𝑢−𝑢_𝐻1|𝑒|₁𝑢−𝑢_𝐻0≤𝐶logℎ^1/2𝐻³|𝑒|₁.𝑏𝑅_ℎ−𝑢_𝐻,𝑅_ℎ−𝑢_𝐻,𝑒−𝐼^∗_ℎ𝑒≤𝐶logℎ^1/2𝑅_ℎ−𝑢_𝐻1𝑅_ℎ−𝑢_𝐻1𝑒−𝐼^∗_ℎ𝑒₀≤𝐶logℎ^1/2𝐻³|𝑒|₁.𝑏𝑅_ℎ,𝑅_ℎ−𝑢,𝑒−𝐼^∗_ℎ𝑒≤𝑁𝑅_ℎ1𝑅_ℎ−𝑢₁|𝑒|₁≤𝐶ℎ|𝑒|₁,𝑏𝑅_ℎ−𝑢,𝑒−𝐼^∗_ℎ𝑒≤𝐶𝑅_ℎ1‖𝑢‖₂𝑒−𝐼^∗_ℎ𝑒₀≤𝐶ℎ|𝑒|₁≤𝐶ℎ|𝑒|₁,𝜈|𝑒|²₁−𝑏𝑒,𝑢_𝐻,𝑒−𝑏𝑒,𝑢_𝐻,𝑒−𝐼^∗_ℎ𝑒−𝑏𝑢_𝐻,𝑒,𝑒−𝐼^∗_ℎ𝑒≥𝜈|𝑒|²₁−3𝑁𝑢_𝐻1|𝑒|²₁≥𝜈1−3𝑁𝜈²𝑓₋₁|𝑒|²₁. (4.10)(4.11)(4.12)(4.13)(4.14)(4.15) Combining (4.10)–(4.15) with (4.9) yields |𝑒|₁≤𝐶ℎ+logℎ^1/2𝐻³. (4.16) Thanks to (3.7), (4.9), Theorems 3.2 and 3.5, and estimates (4.10)–(4.14) and (4.16), we have 𝜂₀≤𝛼₁⁻¹sup_{(𝑣ℎ,𝑞ℎ)∈𝑋ℎ×𝑌ℎ}𝐵_ℎ𝑒,𝜂;𝐼^∗_ℎ𝑣_ℎ,𝑞_ℎ𝑣_ℎ1+𝑞_ℎ0≤𝐶ℎ+log|ℎ|^1/2𝐻³+|𝑒|₁≤𝐶ℎ+log|ℎ|^1/2𝐻³. (4.17) Combining (4.16) and (4.17) with (3.14) and Theorem 3.2 yields (2.13).