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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 651808, 27 pages
Stabilized Multiscale Nonconforming Finite Element Method for the Stationary Navier-Stokes Equations
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China
Received 22 May 2012; Accepted 8 August 2012
Academic Editor: Rudong Chen
Copyright © 2012 Tong Zhang et al. 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 consider a stabilized multiscale nonconforming finite element method for the two-dimensional stationary incompressible Navier-Stokes problem. This method is based on the enrichment of the standard polynomial space for the velocity component with multiscale function and the nonconforming lowest equal-order finite element pair. Stability and existence uniqueness of the numerical solution are established, optimal-order error estimates are also presented. Finally, some numerical results are presented to validate the performance of the proposed method.
As the development of science and technology, finite element method has became an important and powerful tool for the complex fluid problems, such as for the Navier-Stokes equations. It is well known that the pressure and velocity pairs satisfy the discrete Inf-Sup condition  that plays the key role for simulating the Navier-Stokes equations. However, some unstable mixed finite element pairs which violate the so-called Inf-Sup condition are also popular, see [2–4]. In order to overcome this restriction, various of stabilized methods have been proposed, including the bubble condensation-based methods , pressure projection method (PPM) [6–8], the local Gauss integration method (LGIM) [9–11], multiscale method [12, 13], macroelement stabilized method [3, 14], and so on. Most of these stabilized methods necessarily need to introduce the stabilization parameters either explicitly or implicitly. In addition, some of these techniques are conditionally stable or are of suboptimal accuracy. Therefore, the development of mixed finite element methods free from stabilization parameters has become increasingly important.
In 2005, Franca et al. gave a new multiscale method for the reaction-diffusion equation in . The chief characteristic of their method is to use the Petrov-Galerkin approach to split the solution into two parts, and the trial function space is enriched with an unstable bubble-like function, which is the solution to a local problem. Later, Barrenechea and Valentin  considered the relationship between the enriched multiscale method and stabilized techniques for generalized Stokes problem based on the - pair. By enriching the velocity space with an unusual bubble function, Araya et al. established the convergence for the Stokes problem in , their method is different from usual residual free bubble method in , in which one should choose local basis functions to enrich the standard finite element spaces by solving some local problem analytically. Furthermore, the method proposed in  can also be used to treat the unsteady reaction-diffusion problem (see ).
Compared with conforming finite element method, the nonconforming finite element methods are more popular due to their simplicity and small support sets of basis functions. Crouzeix and Ravizrt in  used the nonconforming piecewise linear velocity and a piecewise constant pressure to solve the Stokes equations. In this paper, motivated by the ideas of [13, 15, 16], we will use the Petrov-Galerkin approach based on the nonconforming velocity space to handle with the steady Navier-Stokes equations. The main differences between [13, 15, 16, 19] and this work lie in the following: (i) the finite element spaces of velocity are different; it is nonconforming element in this paper; (ii) the treated problems are different; we consider the nonlinear problem; (iii) the finite element pairs are different; the - pair is used in this paper.
The outline of this work is arranged as follows. In the following section, the abstract functional setting for steady Navier-Stokes equations is recalled. Section 3 is devoted to derive the general form of enriched multiscale method based on the - pair. After providing the stability and existence uniqueness for the approximation solution, the optimal error estimates are established in Section 4. In Section 5, Some numerical results are presented to verify the established theoretical analysis. Finally, Some conclusions are made in Section 6.
Let be an open bounded domain of with Lipschitz continuous boundary and satisfy a further condition stated in below. The incompressible stationary Navier-Stokes equations with the homogeneous Dirichlet boundary condition are where represents the velocity, the pressure, the prescribed body force, the viscosity coefficient.
In order to introduce the variational formulation for problem (2.1), we set
The standard notations of Sobolev space are used. To simplify, we use instead of as and for . The spaces are endowed with the usual -scalar product and -norm . The spaces and are equipped with the scalar product and the norm , , .
Define is the operator associated with the Navier-Stokes problem, it is positive self-adjoint operator from onto .
Introducing the bilinear operator and defining a trilinear form on as follows:
The variational formulation of problem (2.1) reads as: find such that for all where
Clearly, the bilinear forms and are continuous on and , respectively. Moreover, also satisfies (see ): where is a positive constant depending only on .
It is easy to verify that satisfies the following important properties for all , (see ): where is a constant. Here and below, the letter (with or without subscript) denotes a generic positive constant, depending at most on the data , and . Furthermore, the following estimates about are hold [1, 20]: for all , , and for all ,, .
As mentioned above, a further assumption about is needed (see ).
Assume that is regular so that the unique solution of the steady Stokes equations for a prescribed exists and satisfies
Under the assumption of , if is of or is a two-dimensional convex polygon, it has been shown that (see ) where is a positive constant only depending on .
Theorem 2.1. Assume that and satisfy the following uniqueness condition: Then problem (2.5) admits a unique solution with such that
3. Enriched Nonconforming Finite Element Method
Let be a regular triangulation of into element , that is, , where is the area of the element and is the diameter of ; the mesh parameter is given by . Denote the boundary segment and the interior boundary by and , respectively. Let and be the sets of and . The centers of and are indicated by and , respectively. The finite element spaces investigated in this paper are the following mixed finite element spaces: where is the set of line polynomials on , and noting that the nonconforming finite element space is not a subspace of . Defining the energy norm The finite element spaces and satisfy the following approximation property (see [4, 21]): for , there are two approximations and such that and the compatibility conditions hold for all and : where denotes the jump of the function across the boundary .
Set and . Then for all , , , the discrete bilinear forms are For the nonconforming space , we define a local operator satisfying Then the local operator satisfies (see ) The global operator is defined as .
As noted, the choice - is an unstable pair that does not satisfy the discrete Inf-Sup condition. Therefore, we need to introduce the enrichment multiscale method to overcome this restriction.
Let be a finite dimensional space, called multiscale space, such that The discrete weak formulation of the Stokes equations is to find and , such that for all and . Let , we can solve it through the following local problem: where denotes the length of the edge ; the normal outward vector on ;, are the tangential and normal derivative operators, respectively; is the identity matrix. Equation (3.11) is well posed, that is, can be expressed by , , and on each element . For convenience, we define two local operators and by With Green formulation and (3.12), for all , , (3.10) can be rewritten as With the help of (3.13), the enriched nonconforming finite element method for the stationary Navier-Stokes equations (2.1) is rewritten as follows: find such that for all , where
By applying the technique to one used in , we can obtain that , , and . Moreover, if is a piecewise constant, then we have , Define the mesh-dependent norms as follows:
Remark 3.1. The assumption of piecewise constant is made simply to analyze the problem (3.14), but this assumption does not affect the precision of this method, and (3.14) may be implemented as it is presented for a general function . Here, we do not give the detail proof about this fact; readers can visit Appendix B of the paper  for .
Remark 3.2. Generally speaking, the following linear algebra equations can be obtained from the discrete system of original problem: where the matrices and are deduced from the diffusion, convection, and incompressible terms; is the variation of the source term. The norm of matrix gets smaller as the convection increases; therefore, some unnecessary oscillations will be created. In order to eliminate these oscillations, we introduce the stabilized term, in this case, the coefficient matrix of discrete formulation transforms into where is derived from the stabilized term, that is, the term of . As the considered problem has strong convection, in order to obtain a good behavior of matrix , we should choose a proper . In this way, the singularly perturbed problem can be treated effectively. The reason that we treat the convection term not use enriched function technique is to simply the theoretical analysis and computation, and the discrete convection term has no influence about the stabilized term .
Lemma 3.3. Let , then,
Proof. The results follow from the definition of (3.15) and the mesh-dependent norms in each .
Before establishing the stability of scheme (3.14), we introduce the local trace theorem (see ). There exists , independent of , such that
Theorem 3.4. There exist two positive constants , depending on , for all , such that
Proof. It follows from , , inverse inequality, (3.15), and (3.21) that
that is, the continuity result (3.22) holds.
From the properties of the nonconforming finite element given in , for all , there exists a function , such that and Using the Cauchy-Schwartz inequality and (3.25), we have Using (3.21) and inverse inequality, we obtain that Combining (3.26) with (3.27) yields where with , and is chosen small enough. Let Using (3.26) and Lemma 3.3 we have provided that and . Denote Then we have Taking , we obtain the desired result (3.23).
Proof. Let Hilbert space be with the scalar product and norm
and be a nonvoid, convex, and compact subset of defined by
Defining a continuous mapping from into as follows: given for all , find such that Taking , using (2.8)–(2.13) and inverse inequality yields As a consequence, we have Using again (2.17), (3.23), (3.37), and inverse inequality, we arrive at Hence, the two estimates imply , thanks to the fixed point theorem, the mapping has at least one fixed point ; namely, is a numerical solution of problem (3.14).
Next, we shall prove that the problem (3.14) has a unique solution . In fact, if also satisfies (3.14), then for all we have Taking in (3.41) and using again (2.8)–(2.13), Lemma 3.3, it follows that Which, together with the strong uniqueness condition gives . Using again (2.13), (3.23), and (3.41), we obtain which implies .
4. Error Estimates
In order to derive the error estimates of the numerical solution , we introduce the Galerkin projection defined as follows: for all Noting the Theorem 3.4, is well defined.
Proof. From , we have . For all , using (4.1) yields From the definition of , (3.3), combining Theorem 3.4, (4.3), the triangular with inverse inequalities, we arrive at It is easy to check that Combining (4.4), (4.5), and inverse inequality yields In order to derive the estimate in the -norm, we consider the following dual problem with : Based on the assumption of (A1), (4.7)–(4.9) have a unique solution and satisfy Multiplying (4.7) and (4.8) by and , respectively, integrating over , and using (4.3) with , we see that where is the finite element interpolation of in and satisfies (3.3). For each , we define the mean value of and on Note that each interior edge appears twice in the sum of (4.11); and are constants. Then it follows from (4.11) that Combining (3.3) with Lemma 4.1, we deduce that With the help of (4.12), we have Combining the definition of , (4.16), and local trace theorem (3.21) with the standard argument for the nonconforming element (see ), we see that In a similar way, we have By combining (4.13)–(4.15) with (4.17)-(4.18), we deduce that which, together with (4.6). We finish the proof.
Proof. We get the following error equation by combining (2.1) with (3.14), for all
With (4.3), (4.21) can be rewritten as
where and .
From Theorem 3.5 and (4.22), we get that Again, with (2.13), Theorem 2.1, inverse inequality, and Lemma 4.1, we have and using the similar arguments as for (4.17)-(4.18) yields Combining (4.23)–(4.27) with Theorem 3.5, we arrive at Choosing in (4.22), we obtain that Using (2.13), (2.17), Theorem 2.1, and Lemma 3.3, we get Combining (4.24)–(4.28) with (4.29) yields: From (4.28) and (4.31), we obtain that . Furthermore, we finish the proof by combining triangles inequality with Lemma 4.1, (4.28), and (4.31).
Proof. Using the duality argument for a linearized stationary Navier-Stokes problem; for some given and the solution of (2.1), defining by
where is defined as , for all ; multiplying (4.33) and (4.34) by and , respectively; integrating over , from (2.8)–(2.11), it is easily to see that the bilinear form is continuity and coercive, by using the Lax-Milgram’s Lemma, (4.33)–(4.35) have a unique solution .
Multiplying (4.33) and (4.34) by and , respectively, using (2.13) and Theorem 2.1, we have On the other hand, estimating the right term yields By using (4.36) and (4.37), we arrive at Setting and taking the scalar product of (4.33) with in yields Using the Gagliardo-Nirenberg inequality yields With the help of the Agmon’s inequality, we have Furthermore, the following estimates are hold: Combining above inequalities with (4.38) and (4.39), we arrive at Applying the continuous Inf-Sup condition (2.8) yields Combining (4.43)-(4.44) with (2.18), (4.38), we arrive at Taking , multiplying (4.33) and (4.34) by and , respectively, using (4.1) yields Setting in (4.22), and using (4.46), we obtain that