The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem.

This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. Specifically, the spatial discretization adopts the traditional triangle P1-P0 trial function pair, combined with macro element form to ensure local stability. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and we can indeed obtain the optimal theoretic and numerical order error estimation of velocity and pressure. This helps to enrich the corresponding theoretical results.

1. Introduction

Recently, due to the characteristics of simple implementation, the finite volume method has been widely used in many scientific research and engineering fields. It has obtained many ideal numerical simulation and calculation results, and often used to solve the complex engineering calculation problems well. Nevertheless, compared with a wide range of application scenarios, its theoretical analysis, such as stability and convergence analysis, is far behind, which inevitably shadow benefits of the finite volume method, so it needs to be studied continuously. Among them, the theoretical analysis of Navier Stokes process is one of the important field.

Finite volume method is an effective method to solve differential equations. In the past several decades, the calculation methods used to solve Navier-Stokes problems have developed rapidly. The results are richer and richer, but to our dismay, the theoretical analysis of the algorithm is still insufficient [1,2]. As we all know, based on the current advanced computing equipment, simple numerical methods are easy to distribute and suitable for large-scale computing, which makes them the hope of solving complex problems, such as incompressible Navier-Stokes equations. Among them, using simple coordinated low-order elements and local stabilization method is a good choice [3,4].

However, it is well known that low order coordinated finite volume function pairs, such as , are unstable for numerical solution of Navier-Stokes equations. The common way to overcome this shortage is to use local stabilization technique, that is, add “macro element condition” to improve the stability of the algorithm. This kind of low order method has been widely analyzed and applied, and has been proved to be effective in practice [5,6,7,8,9]. The basic idea of this method was first proposed by Boland and Nicolaides, and has been vigorously developed since then. The recent work of Wen [10] and Li [11] paves the way for the numerical analysis of Stokes and Navier Stokes problems. In addition, He [12] and Li [13] have given the locally stable finite volume method for partial spatial discretization of Navier-Stokes problems, which has a good effect except some fully discrete results.

This paper continues to analyze the convergence results of using FVM to solve two-dimensional time-dependent Navier-Stokes equations, so as to enrich the relevant theories. Here we will still focus on pairs. For this purpose, let’s assume is the uniform and regular triangulation of . It should be reminded that the finite element space here does not have the inf-sup condition for , so a similar skill in the paper [11,12,13] is required. Because these papers mainly discuss the spatial discrete case, this paper studies a approximation based on time-discretization is Euler semi-implicit and space-discretization is Locally stable FVM.

For brevity, this paper assumes is a regular triangulation that satisfies the general regular condition [14,15]. Let the mesh size is h and the time step is , the theoretical results of the optimal order convergence of the fully discrete FVM based on the low order coordinated finite element local stabilization are as follows:

For such a finite volume solution obtained by the fully discrete locally stabilized FVM, we derive in this paper the following order error estimates: where .

The rest of the paper is organized as follows. Section 2 introduces some basic concepts and function definitions related to Navier-Stokes problem and the locally stable FVM. Some basic results are prepared in Section 3. Section 4 mainly analyzes the error estimation of time semi-discretization based on Euler semi-implicit scheme. Section 5 proves the error estimations of time and spatial fully discretization. Section 6 contains some numerical results and a summary of the article is included in Section 7.

2. Foundation of Finite Volume Method for the Navier-Stokes Problem

This article consider the follow non-stationary Navier-Stokes equations where be a bounded domain in assumed to have a common continuous boundary (stronger than Lipschitz continuity) [14,16]; are the velocity vector and is the pressure, is the body force, is the initial velocity, and is the viscosity.

For the convenience of analysis of problem (3), we introduce the following common abbreviations where V is the closed subset of X and H is denote the closed subset of Y, respectively. The spaces , are endowed with the common norm denoted by and , respectively. The norms of the Hilbert space and X are

For more information about the above marks, we refer the reader to [14,16,17]. We also need to denote as the Laplace operator and as the Stokes operator, where P is the -orthogonal projection of Y onto H.

It is known [17] that where , is positive constant depending only on . C, like the quantities , appear subsequently, is a positive constant depending on .

This paper uses following kind of continuous bilinear forms and on and , respectively, and a trilinear form on

With the above notations, the general variational formulation of problem (3) is: get satisfy: for all .

In order to get the fully dispersed error estimates, we need the following smoothness.

Assume some continuity of for all

Now, we consider the fully discrete locally stabilized FVM for two-dimensional time-dependent incompressible Navier-Stokes Equation (3). For convenience, let a partitioning of into triangles satisfied the regular in the usual sense (see [5,14]). are the corresponding finite element subspace of . is the set of all interelement boundaries.

In order to define the finite volume method for Equation (3), We need introduce a popular configuration of dual partition for : the interior point is chosen to the barycenter of element , and the midpoint on side of . See Figure 1. This type of dual partition is locally regular if is locally regular.

Figure 1

The finite volume partition of geometric region.

Corresponding to Hilbert space , we define the following finite element velocity subspaces and the pressure subspace

The finite volume dual of velocity is Let interpolation operator :. The dimensional reduction and are defined as follows:

The finite volume forms of velocity on is, where is the unit outnormal vector. The finite volume form of pressure on is defined as To facilitate the analysis, we need the following two trilinear forms. The last time difference part is The finite volume form of the right side is For the convenience of reading, we introduce the following generalized form

In this paper, the norms are defined as following: where is the area of (see Figure 2).

Figure 2

The area of partition of triangular.

To describe the locally stabilized formulation of the non-stationary Navier-Stokes problem, we use the classic not overlap macroelement partitioning  [18]. For every macroelement in , the set of interelement(small finite element) edges is denoted by , and the length of an edge is denoted by .

With the above definitions, a locally stabilized formulation of the non-stationary Navier-Stokes problem (3) can be stated as follows.

Locally stabilized finite volume formulation for non-stationary Navier-Stokes: Find where for all

In order get the regularity of the above definitions, we need the following stability results [5,8,9,12].

For any two neighboring macroelements Then for all where

3. Technical Preliminaries

The main task of this section is to prepare many basic estimates which will help the error analyses for the finite volume solution .

Since the bilinear forms and are coercive on , they generate invertible operators and respectively through the condition:

Moreover, we also need the discrete gradient operators:

Firstly, we have the following classical properties (see [8,19]) where . For the triangular element, It follows from (4), (5) and (13) that [20,21]

As for the trilinear forms and ,we can deduce the following results.

If

Since the following discrete analogue of the Sobolev inequality holds [17] If , we have combining the above formula with (14) and (16), we can get the first formula in (15).

To prove the others in (15), we need the follow discrete results [17,22], namely for any , For any , we have [9] which together with (14), (17) imply (15).  □

Similar to the results in [12], we also need to define the projection operator as Due to Theorem 2, we know that is well defined and have the properties [12]: for all .

Beside, we need the specific result in He et al. [12].

Under the assumptions of Theorems 1 and 2, for all

Since our error analysis for the time discretization depends heavily on these regularity estimates, we then provide some smoothness estimates of . The main idea is similar to the work in He et al. [9,23].

Under the assumptions of Theorems 3, the finite volume solution for all

Note from (13) and (14) we can deduce the following estimates: and from Theorems 1 and 3, we have for all . Then using the similar method in [9], we can get these estimates (20).  □

Finally, in order to get the upper bounders of velocity and pressure in the time related case, we state the classical Gronwall lemma used in [24].

Let Then,

The following is dual Gronwall lemma.

Given integer Then,

4. Error Estimates for Semi-Discrete for Time Depended Navier-Stokes Equations

In this section we consider the time discretization of the locally stabilized and get some useful estimates. Let T time to stop calculation and N the time corresponding step. So we have

For the first part, we need to analysis the errors of finite element Original case. It’s well known that the common Euler semi-implicit scheme applied to the spatially discrete problem (9) can be described as: for all , where is starting value and To deduce the discretization error , we integrate and differentiate (9), respectively, to get for all .

Subtracting (25) from (26) and using (27) and the relation: for all for some Hilbert space F, we have for all , where and

Under the assumptions of Theorem 3 the error

By using (14), (15) and (29), we derive Applying Theorem 4 in (39), we obtain Utilizing Theorem 4, if we sum (32) from to , we can derive the first inequality in (30) directly.

Next, we deduce from (19) and (29) that Because We can deduce from (33) and Theorem 4 to get for . Summing (35) from to , we derive the second inequality in (30).

As for the last one, Deriving from (14) and (29), we have Hence, Formula (36) and Theorem 4 imply that Similarly, summing (37) from to and using Theorem 4 deduces for i = 0,1,2, which yields the third one in (30).  □

Now, let’s discuss the second part: the error of time discrete duality argument corresponding to (25). Firstly, the dual problem corresponding to (25) usually describes as: find such that, for all where .

For the dual part, we also need the existence, uniqueness and regularity of problem (38), so we introduce the following results:

With the similar method in He [9], we have the following two lemmas.

Assume that the assumptions of Theorem 3 is valid and Then,

Assume that the assumptions of Theorem 3 are valid and that

5. Error Analysis for Time and Spacial Discrete

In this section we proof the upper bounds for the error in and norms, and deduce the last optimal order estimates. Firstly, we have

Assume that the assumptions of Theorem 3 are valid and

Setting in (28) and using (14) we obtain From (14) and (15), we deduce that Combining (43) with the above estimate yields Summing (44) from to , we obtain Applying Lemmas 3 and 6 in (45) and by (14) yields (43).  □

Under the assumptions of Lemma 7, we have

We derive from (28) that for all . Setting in (47), we obtain From (5) and Lemma 1, we get that Combining (48) with the above estimates and using Lemma 7 yields Summing (48) from to and applying Lemmas 5 and 7, we obtain Combining the above estimate with (14) which yields(46).  □

Under the assumptions of Lemma 7, we have that for all

Setting in (38) and in (28) we obtain Adding (52) to (51), we obtain It follows from (14) and (15) that Combining (53) with the above estimates yields with . Summing (54) from to and applying Lemma 6, we obtain Applying Lemmas 4, 5, 7 and 8 in (55), we get and

Now, multiplying (55) by and noting we get Summing (57) from to and applying Lemmas 3 and 5, we deduce that Applying Lemma 4 and (56) in (58), we have this and (56) yield (50).  □

Under the assumptions of Lemma 7, the error

Multiplying (49) by noting and using Theorem 4, we have that Summing (61) from to and applying Lemmas 4 and 9, we have  □

Under the assumptions of Lemma 7, the error

From Theorem 2, (5), (14) and (28), we deduce that Hence

Summing (64) for n from to and applying Lemmas 4, 9 and 10, we deduce that which yields (63).  □

Under the assumptions of Lemma 7, the error for all

Integration by parts directly can show Summing (67) from to , using (19) and noting , we obtain Combining (68) with (14), Theorem 3, Lemmas 7 and 9 yields the first estimate in (66).

Moreover, by (5), (9), (11), (14) and (20), we deduce that Hence, we obtain from Theorems 1 and 3 that Combining (63) with (65)–(72) yields for all .

Besides, noting that Recalling (Lemma 4.3) in [12], we have where Using (9), (11), (14) and (75), we obtain It follows from (5), (7), (18)–(20), and the above estimate that Substituting (76) into (74) and using (73) and Theorem 1, we get Combining (77) with (65), we have This and (69) yield (66).  □

6. Single Numerical Example

In this section, some numerical results are computed to test the rationality of the theoretical analysis. Because it is difficult to obtain the analytical solution of the general problem governed by the Navier-Stokes equation, we show the relevant numerical results through an example with analytical solution for simplicity. So we consider the following model problem in the unit square area . Here this example might as well takes as 0.01. Only the velocities and pressure are given here. The right term f of the equations can be obtained by bringing the relationship between and into the NS equations, and the initial values of and can be obtained by bringing into the calculation.

Now, consider a unit square domain with an exact solution given by f is determined by (3). It can be verified that such satisfy the non divergence condition.

For simplicity, we can record the time and spatial discretization of the problem as follows: where the matrices in (80) correspond to the differential operators: , and I is the identity matrix. The right-hand side contains the source term.

To make the next iterations less complex, here are a few new notations. Let , , then we can further record the above equation as follows:

Besides, in order to improve the calculation efficiency, we can generally adopt Newton iterative method to solve the above nonlinear problems. The typical calculation steps are as follows:

It is worth noting that since Newton iterative method requires high initial values, we need to use the following Picard method to obtain the initial values of Newton iterative method:

In this way, we can finally transform the time-dependent Navier Stokes problem into a large-scale linear system of equations and solve it through such links: Euler time discretization → finite volume space discretization → Newton iterative transformation → Picard format transformation → large linear equations → solving equations.

Based on the above description, we can get some simulation results (). The following Figure 3 is the result of one iteration based on the initial value. The time step here is , and the spatial grid is divided into two congruent triangles on the basis of equidistant rectangular grid. We only pay attention to that only of the data in both X and Y directions are selected.

Figure 3

Preliminary calculated velocity and pressure.

The following Figure 4 is the result after 1000 iterations with time step . Here, the reference value of streamline is the same as that when .

Figure 4

The calculated velocity and pressure at .

Compared Figure 2 with Figure 3, it can be seen from Figure 3 that the streamline and flow field are weakened accordingly. It is easy to understand that since the interpretation constructed here is decaying, both velocity and pressure show a decaying trend. This also shows that our numerical method maintains strong robustness, and the calculation results are more intuitive.

The following table shows the error order of velocity and pressure after 1000 iterations with grids selected under the condition of .

From the above Table 1, we can see the optimal convergence rate, almost 2 for velocities and 1 for pressure are really obtained, which confirm the numerical analysis above.

Table 1

Convergence order of spatial solution the FVM (, ).

h	∥A˜h12(u−u_h)∥_0∥A˜h12u∥_0	∥u−u_h∥_0∥u∥_0	∥p−p_h∥_0∥p∥_0
1/20	0.0699111	0.0035712	0.0771281
1/40	0.0394281	0.0009511	0.0440669
1/201/40	1.773	3.754	1.750
1/80	0.0209534	0.0002638	0.0241342
1/401/80	1.718	3.605	1.673

The following Figure 5 shows the error curve calculated based on spatial grid with different time steps: (`dt’ in Figure 5 is ). It can be seen from here that the initial error tends to increase, but the error also decreases with the decrease of flow field energy, which shows that the time iteration is stable.

Figure 5

Velocity error of different time steps.

The following Table 2 shows the error ratio of different time steps: = 0.01, 0.005, 0.0025, 0.00125:

Table 2

Numerical results of the FVM (, ).

t	1	2	3	4	5	6	7	8
Δt=0.01Δt=0.005	1.9513	1.9513	1.9456	1.9424	1.9404	1.9390	1.9380	1.9373
Δt=0.05Δt=0.0025	1.9253	1.9253	1.9225	1.9210	1.9200	1.9193	1.9188	1.9184
Δt=0.0025Δt=0.00125	1.9126	1.9126	1.9112	1.9104	1.9099	1.9096	1.9093	1.9092

The error ratio curve in the Figure 6 below shows the convergence of one section of time and is relatively stable.

Figure 6

1 order convergence ratio in time direction.

Table 2 and Figure 6 tell us the optimal convergence rate of time is 1 which is consistent with the theoretical analysis.

Due to time constraints, our numerical results only show these. It is also worth noting that if the solution does not decay but increases, and if the growth rate is fast, the error of numerical results will increase with the increase of numerical calculation time. At that time, the method may not converge or inefficient. This requires a little attention in specific applications.

7. Conclusions

After detailed theoretical analysis, this article finally proves that if we use the finite volume method based on element to approximate the non-stationary Navier-Stokes equation, we can achieve the follow optimal numerical error estimation:

The optimal error estimate (83) shows that the time discretization of Euler method is 1 order and the space discretization is 2 order in this space-time full discretization finite volume method, which is consistent with the theoretical optimal order error estimation of element in solving differential equations. Although the proof process is challenging and cumbersome, the optimal result is also obvious and certain. However, this work is still helpful to reveal some special aspects of the finite volume method which is different from finite element and other methods in solving complex differential equations. Therefore, it is helpful to improve the corresponding numerical analysis theory.

In addition, with the continuous research and exploration of using neural network to solve differential equations recently [25,26,27], although very gratifying numerical results have been obtained, but the effectiveness and convergence of neural network-based methods for solving differential equations are still unclear, and there is a lack of theory. However, we know gradually that the key point of neural network in the calculation of differential equations is to use low-order functions for numerical approximation, which is similar to the basic principle of using low-order continuous functions to discretize and approximate differential equations, except that the former is global approximation while the latter is piecewise approximation. Therefore, improving and enriching the low-order function approximation theory also provides some important reference in further understanding the neural network in solving differential equations efficiently, It is worth studying.