A Stabilized Low Order Finite-Volume Method for the Three-Dimensional Stationary Navier-Stokes Equations

This paper proposes and analyzes a stabilized finite-volume method FVM for the threedimensional stationary Navier-Stokes equations approximated by the lowest order finite element pairs. The method studies the new stabilized FVM with the relationship between the stabilized FEM FEM and the stabilized FVM under the assumption of the uniqueness condition. The results have three prominent features in this paper. Firstly, the error analysis shows that the stabilized FVM provides an approximate solution with the optimal convergence rate of the same order as the usual stabilized FEM solution solving the stationary Navier-Stokes equations. Secondly, superconvergence results on the solutions of the stabilized FEM and stabilized FVM are derived on theH1-norm and the L2-norm for the velocity and pressure. Thirdly, residual technique is applied to obtain the L2-norm error for the velocity without additional regular assumption on the exact solution.


Introduction
Recently, the development of stable mixed FEMs is a fundamental component in the search for the efficient numerical methods for solving the Navier-Stokes equations governing the flow of an incompressible fluid by using a primitive variable formulation.The object of this work is to analyze the stabilized finite volume method for solving the three-dimensional stationary Navier-Stokes equations.
The importance of ensuring the compatibility of the component approximations of velocity and pressure by satisfying the so-called inf-sup condition is widely understood.The numerous mixed finite elements satisfying the inf-sup condition have been proposed over the years.However, elements not satisfying the inf-sup condition may also work well.So far, the most convenient choice of the finite element space from an implementational point of view would be the elements of the low polynomial order in the velocity and the pressure with an identical degree distribution for both the velocity and the pressure.
This paper focuses on the stabilized method called local polynomial pressure projection for the three-dimensional Navier-Stokes equations 1-5 .The proposed method is characterized by the following features.First, the method does not require approximation of derivatives, specification of mesh-dependent parameters, edge-based data structures, and a nonstandard assembly procedure.Second, this method is completely local at the element level.
On the other hand, FVM has become an active area in numerical analysis.The most attractive things are that FVM can keep local conservation and have the advantages of FVM and finite difference methods.The FVM is also termed the control volume method, the covolume method, or the first-order generalized difference method.Nowadays, it is difficult in analyzing FVM to obtain L 2 -norm error estimates because trail functions and test functions are derived from different spaces.Many papers were devoted to its error analysis for secondorder elliptic and parabolic partial differential problems 6-10 .Error estimates of optimal order in the H 1 -norm are the same as those for the linear FEM 9, 11 .Error estimates of optimal order in the L 2 -norm can be obtained as well 8, 9 .Moreover, the FVM for generalized Stokes problems was studied by many people 11-13 .They analyzed this method by using a relationship between it and the FEM and obtained its error estimates through those known for the latter method.Also, it still requires H 3 smoothness assumption of the exact solution to obtain O h 2 error bound in most previous literatures.However, for the Stokes problems only the finite elements that satisfy the discrete inf-sup condition have been studied.
The work of 14, 15 for the two-dimensional stationary Stokes equations is extended in this paper for the three-dimensional stationary Navier-Stokes equations approximated by lowest equal-order finite elements.Following the abstract framework of the relationship between the stabilized FEM and stabilized FVM 14, 15 , the stabilized FVM is studied, and the optimal error estimate of the stabilized FVM is obtained for the three-dimensional stationary Navier-Stokes equations relying on the uniqueness condition.As far as known, there still requires much research on FVM results 16 about the velocity in L 2 -norm and superconvergence result between FEM solution and FVM solution of the three-dimensional Navier-Stokes equations.
The remainder of the paper is organized as follows.In Section 2, an abstract functional setting of the three-dimensional Navier-Stokes problem is given with some basic assumptions.In Section 3, the stability of the stabilized FVM is analyzed and provided by Brouwer's fixed-point theorem.In Section 4, the optimal error estimates of the stabilized finite volume approximation for the three-dimensional stationary Navier-Stokes equations are obtained.

FVM Formulation
Let Ω be a bounded domain in R 3 , assumed to have a Lipschitz-continuous boundary Γ and to satisfy a further condition stated in A1 below.The three-dimensional stationary Navier-Stokes equations are considered as follows: where ν > 0 is the viscosity, u u 1 x , u 2 x , u 3 x represents the velocity vector, p p x the pressure, and f f 1 x , f 2 x , f 3 x the prescribed body force.In order to introduce a variational formulation, we set 17

2.2
As mentioned above, a further assumption on Ω is presented.A1 Assume that Ω is regular so that the unique solution v, q ∈ X, M of the steady Stokes problem for a prescribed g ∈ Y exists and satisfies where c > 0 is a general constant depending on Ω.Here and after, • i and | • | i denote the usual norm and seminorm of the Sobolev space We denote by •, • the inner product on L 2 Ω or Y .The space H 1 0 Ω and X are equipped with their equivalent scalar product and norm 17

2.5
It is well known 18 that for each v ∈ X there hold the following inequalities: where γ is a positive constant depending only on Ω.
The continuous bilinear form a Also, the trilinear term is defined by Mathematical Problems in Engineering and satisfies b u, v, w ≤ c 0 ∇u 0 ∇v 0 ∇w 0 .

2.9
Then the mixed variational form of 2.1a -2.1c is to seek u, p ∈ X, M such that The existence and uniqueness results are classical and can be found in 18-20 .
We introduce the finite-dimensional subspace X h , M h ⊂ X, M , which is characterized by τ h with mesh scale h, a partitioning of Ω into tetrahedron or hexahedron, assumed to be regular in the usual sense see 20-22 .Here, the space X h , M h satisfies the following approximation properties.For each v ∈ D A , p ∈ H 1 Ω , there exist approximations together with the inverse inequality The stable and accurate finite element approximational solution of 2.10 requires that X h , M h satisfies the discrete inf-sup condition sup where β is positive constant independent of h.
The main purpose of this paper is to study a stabilized FVM for the stationary 3D Navier-Stokes equations.We follow 23, 24 to obtain the dual partition K h .We first choose an arbitrary point Q in the interior of each tetrahedron K and then connect Q with the barycenters Q ijk of its 2D faces ΔP i P j P k by straight lines see Figure 1 .On each face ΔP i P j P k , we connect by straight lines Q ijk with the middle points of the segments P i P j , P j P k , and P k P i , respectively.Then the contribution of K to the control volume K of a vertex P of K is the volume surrounding P by these straight lines, for example, the contribution from one simplex to the control volume K with the interfaces γ 12 and γ 13 .
Then, the dual finite element space can be constructed for the FVM as follows: Obviously, the dimensions of X h and X h are the same.Furthermore, there exists an invertible linear mapping Γ h : X h → X h such that for with where {φ j } indicates the basis for the finite element space X h , and {χ j } denotes the basis for the finite volume space X h that are the characteristic functions associated with the dual partition K h :

2.17
The above idea of connecting the trial and test spaces in the Petrov-Galerkin method through the mapping Γ h was first introduced in 25, 26 in the context of elliptic problems.Furthermore, the mapping Γ h satisfies the following properties 26 .

2.19
where h K is the diameter of the element K.
Multiplying 2.1a by Γ h v h ∈ X h and integrating over the dual elements K ∈ K h , 2.1b by q h ∈ M h and over the primal elements K ∈ K h , and applying Green's formula, we define the following bilinear forms for the FVM: where n is the unit normal outward to ∂ K j and these terms are well posed.
As noted above, this paper forces on a class of unstable velocity-pressure pairs consisting of the lowest equal-order finite elements where R i K , i 0, 1 represent piecewise constant range and continuous range on set K, R i , i 0,1 are spaces of polynomials, the maximum degree of which is bounded uniformly with respect to K ∈ τ h and h.The corresponding stabilized FEM is formulated as follows 3 : Also, the corresponding stabilized FVM is defined for the solution u h , p h ∈ X h , M h as follows:

where C h u h , p h ; v h , q h A u h , Γ h v h D Γ h v h , p h d u h , q h
G p h , q h .

2.24
Obviously, the bilinear form G •, • can be defined by the following symmetry form: 1 G p, q p − Π h p, q − Π h q .

2.25
Note that

2.26
Here, the operator Π h satisfies the following properties: 1, 4

2.29
In particular, the L 2 -projection operator Π h can be extended to the vector case.This section concentrates on the study of a relationship between the FEM and FVM for the Stokes equations.

Lemma 2.2. It holds that 11-13
with the following properties:

2.32
Based on detailed results on existence, uniqueness, and regularity of the solution for the FVM 2.23 , the following result establishes its continuity and weak coercivity.

2.33
Moreover, where β is independent of h.

Stability
In this section, we analyze the results of FVM for the three-dimensional stationary Navier-Stokes equations.Firstly, we are now in a position to show the well-posedness of system 2.23 Theorem 3.1 stability .For each h > 0 such that 23 admits a solution u h , p h ∈ X h , M h .Moreover, if the viscosity ν > 0, the body force f ∈ Y , and the mesh size h > 0 satisfy Proof.For fixed f ∈ Y , we introduce the set

3.5
Mathematical Problems in Engineering 9

A P h w h , Γ h v h D Γ h v h , p h d T h w h , q h G p h , q h b w h , T h w h , Γ
where T h w h , p h :≡ T 1 w h , T 2 p h u h , p h .We will prove that T h maps B M into B M .First, taking v h , q h u h , p h ∈ X h , M h in 3.6 , and using 2.12 and 2.19 , we see that for for all

3.11
Mathematical Problems in Engineering which, together with 3.10 , gives Since the mapping T h is well defined, it follows from Brouwer's fixed-point theorem that there exists a solution to system 2.23 .
To prove uniqueness, assume that u 1 , p 1 and u 2 , p 2 are two solutions to 2.23 .Then we see that which shows that e 0 by 3.15 ; that is, u 1 u 2 .Next, applying 3.3 to 3.13 and 2.34 yields that p 1 p 2 .Therefore, it follows that 2.23 has a unique solution.

4.5
Similarly, by Lemma 2.1 and 2.25 , we have

4.6
Combining the above inequalities with 4.3 gives In the same argument, it follows from 2.34 that 4.6 -4.8 , and using a triangle inequality completes the proof of Theorem 4.1.
As noted above, it is still difficult to achieve an optimal error estimate for the velocity in the L 2 -norm for the three-dimensional stationary Navier-Stokes equations.Here, the following dual problem is proposed and analyzed: Because of convexity of the domain Ω, this problem has a unique solution that satisfies the regularity property 18  Combining all these inequalities with 4.15 yields 4.13 .
In this paper, we have obtained optimal and convergent results of the stabilized mixed finite volume method for the stationary Navier-Stokes equations approximated by the low order finite elements.Furthermore, we could apply the same technique presented to develop and obtain the corresponding results of other stabilized mixed finite volume methods in two or three dimensions.

Figure 1 :
Figure 1: Control volumes in three-dimensional case.

Theorem 4.1 optimal
error and superconvergent results .Assume that h > 0 satisfies 3.2 and f ∈ Y and ν > 0 satisfy 3.2 .Let u, p ∈ X, M and u h , p h ∈ X h , M h be the solution of 2.10 and 2.23 , respectively.Then it holds Φ − Φ h d e, Ψ − Ψ h − d Φ − Φ h , η − G η, Ψ h G p, Ψ h a e, Φ h − A e, Γ h Φ h − d Φ h , η − D Γ h Φ h , η b u; e, Φ − Γ h Φ h b e; u, Φ − Γ h Φ h b e; e, Γ h Φ h a e, Φ − Φ h d e, Ψ − Ψ h − d Φ − Φ h , η − G η, Ψ h G p,Ψ h b e; e, Γ h Φ Obviously, we deduce from Theorem 3.1, 2.27 -2.29 , 4.11 , the inverse inequality 2.12 , and the Cauchy inequality that a e, Φ − Φ h d e, Ψ − Ψ h 12Theorem 4.2 optimal L 2 -error for the velocity .Let u, p be the solution of 2.1a -2.1c and let u h , p h be the solution of 4.3 .Then, under the assumptions of Theorem 4.1, it holdsu − u h 0 ≤ Ch 2 u 2 p 1 f 1 .4.13Proof.Multiplying 2.1a and 2.1b by Γ h Φ h ∈ X h and Ψ h ∈ M h and integrating over the dual elements K and the primary elements K, respectively, and adding the resulting equations to 2.23 with v h , q h Φ h , Ψ h , we see thatA e, Γ h Φ h D Γ h Φ h , η d e, Ψ h G η, Ψ h b e; u, Γ h Φ h b u; e, Γ h Φ h − b e; e, Γ h Φ h G p, Ψ h , h b u; e, Φ − Γ h Φ h b e; u, Φ − Γ h Φ h f − u • ∇ u, Φ h − Γ h Φ h .4.15