Abstract and Applied Analysis

Volume 2013 (2013), Article ID 805685, 18 pages

http://dx.doi.org/10.1155/2013/805685

Research Article

Spectral Galerkin Method in Space and Time for the 2D -Navier-Stokes Equations

Dao Trong Quyet

Faculty of Information Technology, Le Quy Don Technical University, 100 Hoang Quoc Viet, Cau Giay, Hanoi 10000, Vietnam

Received 20 March 2013; Accepted 20 June 2013

Academic Editor: Chengjian Zhang

Copyright © 2013 Dao Trong Quyet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove the -stability and -error analysis of the spectral Galerkin method in space and time with the implicit/explicit Euler scheme for the 2D -Navier-Stokes equations in bounded domains when the initial data belong to .

1. Introduction

Let be a bounded domain in with sufficiently smooth boundary . In this paper, we study the spectral Galerkin method in space and time for the following 2D -Navier-Stokes equations:

where is the unknown velocity vector, is the unknown pressure, is the kinematic viscosity coefficient, and is the initial velocity.

The -Navier-Stokes equations are a variation of the standard Navier-Stokes equations. More precisely, when we get the usual Navier-Stokes equations. The 2D -Navier-Stokes equations arise in a natural way when we study the standard 3D problem in thin domains. We refer the reader to [1] for a derivation of the 2D -Navier-Stokes equations from the 3D Navier-Stokes equations and a relationship between them. As mentioned in [1], good properties of the 2D -Navier-Stokes equations can lead to an initiate of the study of the Navier-Stokes equations on the thin three-dimensional domain . In the last few years, the existence and long-time behavior of both weak and strong solutions to the 2D -Navier-Stokes equations have been studied extensively (cf. [2–9]). In this paper, we aim to study numerical approximation of the strong solutions to problem (1). To do this, we assume that

(G) such that

 where is the first eigenvalue of the -Stokes operator in (i.e., the operator defined in Section 2.1 below);(F); that is, .

In this paper, in order to study the numerical approximation of strong solutions to the 2D -Navier-Stokes equations we will use the spectral Galerkin method in space and time, which is based on the eigen-subspaces of the -Stokes operator. As mentioned in [10] for the Navier-Stokes equations, this method enables us to avoid solving the fully nonlinear -Navier-Stokes equations on the low-frequency subspace, whereas to obtain the low-frequency component of the numerical solution, the usual multilevel spectral methods and the postprocessing Galerkin methods need to solve the fully nonlinear -Navier-Stokes equations on the low-frequency subspace. In what follows, we will explain the spectral Galerkin method used in the paper. For the related function spaces, we refer the reader to Section 2.1.

Let and be the eigenvectors and eigenvalues of the -Stokes operator. For a fixed integer , let be the orthogonal projection of onto . Then, the spectral Galerkin method in space is defined as follows: find such that

In order to simplify the implementation of the scheme, we restrict ourselves to the semi-implicit Euler scheme applied to the spectral Galerkin method in space. We consider a spectral Galerkin method in space and time with the implicit/explicit Euler scheme: find () such that

where is the time step size and

Here, the linear term is treated implicitly to avoid serve time step limitations, whereas the nonlinear term is kept explicitly so that the corresponding discrete system is easily invertible. It is well known that this type of scheme is only stable under some restriction on the time step size. We will obtain -stability uniform in time stated in Theorem 13, provided that the following condition holds

for some positive constant depending on the data . As mentioned in [11] for the case of 2D Navier-Stokes equations, the stability condition (6) is a significant improvement compared with the results provided by the nonlinear Galerkin method [12] and the multilevel method [13, 14].

We also derive an -error estimate of the numerical solution under the stability condition (6):

where , and denotes a general positive constant depending only on the data . Noting that is a singular factor near .

Compared to He's works [11] on the spectral method of the 2D Navier-Stokes equations, here we have to address some additional difficulties. Firstly, to treat the more general condition , instead of the usual function spaces used for the Navier-Stokes equations, we use the function spaces which are defined suitably for the -Navier-Stokes equations (see Section 2.1 for details). Secondly, we have to deal with the term in the equation, which only appears for the -Navier-Stokes equations. It is worthy noticing that when , we of course recover the results for the Navier-Stokes equations in [11].

The paper is organized as follows. In the next section, we recall some results on function spaces and inequalities for the nonlinear terms related to the -Navier-Stokes equations, and some discrete Gronwall inequalities are frequently used later. In Section 3, we prove several estimates for the strong solution and the Galerkin approximate solutions of problem (1). In Section 4, we study the error analysis of the spectral Galerkin method in space. Stability and error analysis of the spectral Galerkin method in space and time are discussed in the last section.

2. Preliminaries

2.1. Function Spaces and Operators

Let and be endowed, respectively, with the inner products

and norms , . Thanks to assumption , the norms and are equivalent to the usual ones in and in .

Let

Denote by the closure of in , and denote by the closure of in . It follows that , where the injections are dense and continuous. We will use for the norm in , and for duality pairing between and .

We now define the trilinear form by

whenever the integrals make sense. It is easy to check that if , then

Hence

Set by , by . Denote , then and , for all , where is the ortho-projector from onto . Consequently, there exists an orthogonal basis of consisting of the eigenvectors of :

Furthermore, we can also define the th power of for all . The space is the Hilbert space when equipped with the scalar product and norm , where and denote the scalar product and norm in . In particular, and .

Let . Then, the following estimates hold:

Using the Hölder inequality, the Ladyzhenskaya inequality (when ):

and the interpolation inequalities, as in [15, 16], one can prove the following.

Lemma 1. If , then

where are appropriate constants depending only on .

Lemma 2 (see [3]). Let , then the function defined by

belongs to , and therefore also belongs to .

Lemma 3 (see [4]). Let , then the function defined by

belongs to , and hence also belongs to . Moreover,

Since

we have

2.2. Discrete Gronwall Inequalities

Hereafter, we will frequently use the following modified discrete Gronwall lemmas.

Lemma 4 (see [12]). Let for integers be nonnegative numbers such that

If

for and

for , , then

Lemma 5 (see [13]). Let and be nonnegative numbers such that

Then,

Lemma 6 (see [11]). Let and for integers be nonnegative numbers such that

Suppose that , for all , and set . Then,

Moreover, if

and for all , then

3. Existence and Some Estimates of Strong Solutions

In this section, we will prove some estimates for the strong solution and the Galerkin approximate solutions of problem (1). First, with the operators defined in Section 2.1, one can write this problem as follows:

Definition 7. For given, a strong solution of problem (1) is a function for all such that , and satisfies (35) in for a.e. .

Theorem 8. Suppose that and . Then, problem (1) has a unique strong solution satisfying the following estimates for all ,

where , , , , and is a generic positive constant depending only on the data . Moreover, all above estimates are also valid for the Galerkin approximate solutions of problem (35).

Proof. We refer to [3] for the proof of existence and uniqueness of the strong solution and estimates (36)–(38). We now prove (39)–(40).

First, we take the scalar product of (35) with and , respectively, and add the resulting relations to obtain

Using Lemmas 1 and 3 and Cauchy's inequality, we have

By combining these inequalities with (41), we get

Integrating (43) from to and using (36)–(38), we obtain, after multiplying by , that

In view of (44), there exists a sequence such that

Now, differentiating (35) with respect to yields

We take the scalar product (46) with to obtain

By Lemma 3, we have

Using Lemma 1 and Cauchy's inequality, we get

Multiplying the last inequality by , we have

Therefore, integrating (50) from to , letting , and using (44) and (45), one finds, after multiplying by , that

Moreover, in view of (35), (36)–(38), and (51), we see that

Also, in view of (51), there exists a sequence such that

We again take the scalar product (46) with to obtain

By Lemma 3 and Cauchy's inequality, we have

Using Lemma 1, Cauchy's inequality, and Young's inequality, we obtain

Multiplying the last inequality by , we have

Integrating (57) from to , letting , and using (36)–(38), (51), and (53), we obtain, after a final multiplication by ,

Using Lemmas 1 and 3 and (46), we deduce that

Combining (44), (51), (52), (58), and (59) yields (39) and (40).

Finally, we observe that the problem for the approximate solution is similar to problem (35), and

so satisfies the same estimates as those for the strong solution of problem (35).

4. Spectral Galerkin Method in Space

For a fixed integer , let be the orthogonal projection of onto and . Then, every solution of problem (1) can be decomposed uniquely into

Now, we apply and to (35) to obtain

and the initial conditions , .

Using Theorem 8 and the property of , we arrive at the following estimates of :

We now define the spectral Galerkin method as follows: find such that

with the initial condition .

In order to give an analysis of the error in the -norm, we begin with a technical result concerning a dual linearized -Navier-Stokes problem which is a similar problem to that used in [17]. We consider, for any given and , the dual problem: find such that

for all with . It is easy to see that (67) is a well-posed problem and has a unique solution .

Next, we prove a regularity result of problem (67).

Lemma 9. If , then the solution of problem (67) satisfies

Proof. Taking in (67), we obtain

Using Lemma 3, we have

Using Lemma 1 and Cauchy's inequality, we get

Then, we have

Multiplying this inequality by and using (14), we obtain

Multiplying this inequality by yields

Integrating (74) from to and noting that , we obtain, after multiplying by , that

Moreover, inserting into (67), we get

Using Lemma 3, (14), and Cauchy's inequality, we have

Therefore,

Multiplying this inequality, by and using (14), we obtain

Using Lemma 1, Cauchy's inequality and Young's inequality, we have

Combining the above estimates with (79) yields

Integrating this inequality from to and using (75), we have

Taking in (67), then using (14), Lemmas 1 and 3, we obtain

Hence,

Integrating (84) from to and using (82), (75), and Theorem 8, we complete the proof.

Lemma 10. If , then the error satisfies

Proof. We set and subtract (66) from (62) to obtain

with . Taking the scalar product of (86) with , we obtain

Using Lemma 3, we get

Multiplying the last inequality by and using (14), we obtain

Due to Lemma 1 and Cauchy's inequality, we have

Combining (89) with the above estimate yields

Multiplying (91) by yields

Integrating this inequality from to and using (64), Theorem 8, we obtain, after a final multiplication by , that

which is (85).

Lemma 11. If , then the error satisfies the following bound:

Proof. Take and in (67) to obtain

Multiplying (86) by , we have

Adding (96) and (95), we get

Using Lemma 1, we have

Hence, we deduce from (97) that

Integrating (99) from to and noting that , we obtain

Using (64), (65), Theorem 8, and Lemmas 9 and 10, we deduce from (100) that

Now, multiplying (91) by , we have

Integrating (102) from to and using Theorem 8, we obtain, after a final multiplication by ,

Using (65) and (101) in (103) yields

which is (94).

Finally, by combining Lemma 11 with (64) and using Theorem 8, we get the following error estimate.

Theorem 12. If , then the error satisfies the following bound:

5. Spectral Galerkin Method in Space and Time

5.1. Stability Analysis

In this subsection, we consider the semi-implicit Euler scheme applied to the spatially discrete spectral Galerkin approximation, show the stability of this scheme, and establish some preliminaries related to the error analysis uniform in time.

We consider the semi-implicit Euler scheme and define recursively a solution such that

for with the initial condition , where is a time step such that for some integer and

In order to derive the -bound on the error , we will begin with a time discrete duality argument which is similar to the one used in [11, 17]. We consider the dual scheme correponding to scheme (106): for any fixed and , , find , such that

with an initial condition .

The following theorem provides the -stability of scheme (106).

Theorem 13. Under the assumptions of Theorem 8, if and satisfy the following condition:

then the semi-implicit Euler scheme is the -stability; that is,

Proof. Clearly, scheme (106) defines a unique sequence . Now, we will prove (111)–(115).

Taking the scalar product of (106) with , we obtain

Using Lemma 3 and Cauchy's inequality, we have

Hence,

By (19), we have

where

Substituting those into (118), we get

Next, by taking the scalar product of (106) with , we obtain

Using Lemma 3 and Cauchy's inequality, we have

Substituting into (122), we get

Due to (17)–(19), Cauchy's inequality, and Young's inequality, we have

By combining (124) with the above estimates, we obtain

On the other hand, from (17)–(19), Cauchy's inequality, and Young's inequality, we have

Combining (124) with those estimates, we obtain

Next, from (106) we have

Using (11), Lemmas 1 and 3, and Cauchy's inequality, we get

Therefore, we have

Combining this inequality with (128) yields

Moreover, we deduce from (106) that

where is defined by

Taking the scalar product of (133) with , we obtain

By Lemma 3, we get

Using Lemma 1 and (19), we have

By combining (136) with the above estimates, we arrive at

Multiplying (138) by and noting that , we obtain

Now, we will prove (111)–(115) by induction. Obviously, (111)–(115) are true for . Assuming that (111)–(115) hold for , we need to prove (111)–(115) for . In view of (110) and the inductive assumption, we obtain

Summing (121) from to and to , respectively, and using (140), we obtain (111) and (112) with after a final multiplication by . Noting that , using (140) in (121), and then applying Lemma 5 with

we obtain (113) for .

Furthermore, setting

in (126), using (111)–(113) and (140), we obtain

with

for all and

for all . Applying Lemma 4 to (143), we obtain (114) for .

Applying again Lemma 4 to (132) with

and using (111)–(114) and (140), we obtain

Applying Lemma 5 to (139) with

and using (114), (140), and (147) yields

Finally, due to (17)–(19), we have

Combining these inequalities with (106) and (140) yields

Using (113), (114) and (149) in this inequality yields

Combining this inequality with (149) and (147) implies (115) for .

The following lemma provides the stability of scheme (109).

Lemma 14. If and satisfy (110), then problem (109) admits a unique solution satisfying

Proof. In view of (17), (110), and (114), we can prove that the following bilinear form

is elliptic. Hence, problem (109) has a unique solution for . Next, by taking in (109) and using (11), we have

Using Lemma 3, we get

Using (17) and Cauchy's inequality, we have

Combining above inequalities and using (14), we obtain

Summing (158) from to , noting that , and using Theorem 13, we arrive at

for all . Let

in (159) to obtain

with . Applying Lemma 6 to this inequality yields

Moreover, by taking in (109), we have

Using Lemma 3 and Cauchy's inequality, we deduce that

Therefore,

Using (17)–(19) and Cauchy’s inequality, we have

Combining above inequality, noting that , we obtain

for all . Summing (167) from to , we obtain

Combining (168) with (162) and using Theorem 13, we complete the proof.

5.2. Error Analysis

In this subsection, we will establish the - and -error estimates uniform in time for the fully discrete spectral Galerkin method with the explicit time discretization for the nonlinear term. To do this, by integrating (35) from to , we obtain

Subtracting (106) from (153) and setting , we have

with and

To derive a bound on , we need to provide the following estimates of .

Lemma 15. Under the assumptions of Theorem 13, the error satisfies the following bounds:

Proof. In view of (17)–(19), Lemma 3, and Theorem 8, we deduce from (171) that

By Theorem 8, (175), and the fact that , we have

For , summing (176) from to , we have

For , we sum (177) from to and use Theorem 8 to obtain

which, together with (179), gives (172).

Finally, multiplying (178) by , ; , and noting that , , we obtain

Observing again (170) with and using Theorem 8 yields

For , summing (181) from to and using Theorem 12, we obtain

Combining (182) and (183) with (179) yields (173) and (174).

Now, we prove the following error estimate.

Lemma 16. Under the assumptions of Theorem 13, one has

Proof. Taking the scalar product of (170) with , using (11), Lemma 3 and noting that , we have

Using (17)–(19), we get

Hence, by combining the above inequalities with (185) and using Theorem 8, (110) and (114), we obtain

for all . Summing (187) from to and using Lemma 15, we obtain

We set

in (188) to obtain

with . Applying Lemma 6 to (190) and using Lemma 15, we deduce that

Multiplying (191) by , we get (184).

Lemma 17. Under the assumptions of Theorem 13, one has

Proof. Replacing by in (170) and taking the scalar product of (170) with , we obtain

Next, setting , , in (109), we get

Adding (193) to (194), we arrive at

From (17) and (18), noting that , one finds that

Combining (195) with the above estimates yields

with . Summing (197) from to and using Lemma 15, we obtain

Applying Lemmas 15 and 16 and Theorem 8 in (198), we get

Now, multiplying (187) by and noting that

we arrive at

Summing (201) from to and using (199) and (174), we obtain, after a final multiplication by , the estimate (192).

Finally, combining Theorems 12, 8, and 13 with Lemma 17, we obtain the error estimate of the numerical solution .

Theorem 18. Under the assumptions of Theorem 13, the following error estimates holds:

Acknowledgment

This work is supported by a grant from the Le Quy Don Technical University.

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