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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 805685, 18 pages
Spectral Galerkin Method in Space and Time for the 2D -Navier-Stokes Equations
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.
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 .
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  for a derivation of the 2D -Navier-Stokes equations from the 3D Navier-Stokes equations and a relationship between them. As mentioned in , 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  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  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  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  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 .
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.
2.1. 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:
Lemma 1. If , then where are appropriate constants depending only on .
Lemma 2 (see ). Let , then the function defined by belongs to , and therefore also belongs to .
Lemma 3 (see ). 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 ). Let for integers be nonnegative numbers such that If for and for , , then
Lemma 5 (see ). Let and be nonnegative numbers such that Then,
Lemma 6 (see ). 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:
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  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 . 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).
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,