Numerical Investigations on Several Stabilized Finite Element Methods for the Stokes Eigenvalue Problem
Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and it is a favorite method among these methods for the Stokes eigenvalue problem.
It is well known that numerical approximation of eigenvalue problems plays an important role in the analysis of the stability of nonlinear PDEs. Meanwhile, they are wildly used in many application areas: structural mechanics and fluid mechanics (see ). Thus, development of an efficient and effective computational method for investigating these problems has practical significance and has drawn the attention of many peoples. At the time of writing, numerous works are devoted to these problems (see [2–9], and the references cited therein). Yin et al.  derived a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem on rectangular mesh. Chen and Lin  proposed the stream function-vorticity-pressure method for the eigenvalue problem. Rate of convergence estimates were derived for the approximation of eigenvalues and eigenvectors by Mercier et al. . Xu and Zhou  proposed a two-grid discretization scheme for solving eigenvalue problems.
Mixed finite element methods are a natural choice for solving fluid mechanics equations because these equations naturally appear in mixed form in terms of velocity and pressure [14, 15]. In the analysis and practice of employing mixed finite element methods in solving the Stokes equations, the inf-sup condition has played an important role because it ensures a stability and accuracy of the underlying numerical schemes. A pair of finite element spaces that are used to approximate the velocity and the pressure unknowns are said to be stable if they satisfy the inf-sup condition. Intuitively speaking, the inf-sup condition is something that enforces a certain correlation between two finite element spaces so that they both have the required properties when employed for the Stokes equations. However, due to computational convenience and efficiency in practice, some mixed finite element pairs which do not satisfy the inf-sup condition are also popular. Thus, much attention has been paid to the study of the stabilized method for the Stokes problem.
Recent studies have focused on stabilization techniques, which include penalty [16–18], regular , multiscale enrichment , and local Gauss integration method [21, 22]. There exist a lot of theoretical results for the stabilized mixed finite element methods for the Stokes equations, and the comparisons between them are also given (see [17, 19, 20, 22–24], and the references cited therein). In this paper, we mainly focus on the Stokes eigenvalue problem solved by these stabilized finite element methods based on the lowest equal-order finite element space pair. Moreover, we present the comparisons between these methods for the considered problem.
The remainder of this paper is organized as follows. In the Section 2, we introduce the studied Stokes eigenvalue problem, the notations, and some well-known results used throughout this paper. Then several stabilized mixed finite element methods are reviewed, and their key stabilization techniques are recalled in Section 3. Comparisons between these stabilized methods are performed numerically in Section 4. Finally, we end with some short conclusions in Section 5.
Let be a bounded, convex, and open subset of with a Lipschitz continuous boundary . We consider the Stokes eigenvalue problem as follows. Find , such that where represents the velocity vector, the pressure, the viscosity, and the eigenvalue.
We will introduce the following Hilbert spaces: The spaces , , are equipped with the -scalar product and -norm or . The space is endowed with the usual scalar product and the norm . Standard definitions are used for the Sobolev spaces , with the norm , . We will write for and for .
We define the continuous bilinear forms and on and , respectively, by and a generalized bilinear form on by
With the above notations, the variational formulation of problem (2.1) reads as follows. Find with , such that for all ,
Moreover, the bilinear form satisfies the inf-sup condition  for all : where is a positive constant depending only on . Therefore, the generalized bilinear form satisfies the continuity property and inf-sup condition: where and are the positive constants depending only on .
3. Stabilized Mixed Finite Element Methods
For , we introduce the finite-dimensional subspaces , which are characterized by , a partitioning of into triangles with the mesh size , assumed to be uniformly regular in the usual sense.
Then we define where represents the space of linear functions on .
Remark 3.1 (Nonconforming finite element space). Denote the boundary edge by and the interior boundary by . Set the centers of and by and , respectively. The nonconforming finite element space for the velocity will be taken to be where is the set of all polynomials on of degree less than 1. Note that the nonconforming finite element space is not a subspace of . In this nonconforming case, the pair of finite element spaces is ; that is, the conforming space is still used for pressure.
Moreover, the discrete mixed finite element formulation for the Stokes eigenvalue problem reads: find with , such that for all ,
Next, we denote by the array of the velocity and by the array of the pressure. It is easy to see that (3.3) can be written in matrix form: where the matrices , , and are deduced in the usual manner, using the bases for and , from the bilinear forms , , and , respectively, and is the transpose of matrix .
Remark 3.2. In the nonconforming case, we define the bilinear forms and a generalized bilinear form on by
Accordingly, the discrete nonconforming formulation for the Stokes eigenvalue problem is: find with , such that for all ,
Note that the lowest equal-order pair does not satisfy the discrete inf-sup condition: where the constant is independent of , and is the discrete energy seminorm in the nonconforming case. In order to fulfill this condition, several ways have been used to stabilize the lowest equal-order finite element space pair.
Method 1 (Penalty method). Find with , such that for all , where is a penalty parameter. The performance of this method obviously depends on the choice of the penalty parameter . Then the matrix form of (3.9) can be expressed as where the matrix is deduced in the usual manner, using the base for , from . Because the coefficient matrix of (3.10) is usually large and sparse, it is not easy to compute the numerical solution using a direct method. In general, one can use the Uzawa algorithm. By some simple calculation, we get where and is a positive constant.
Method 2 (Regular method). Find with , such that for all , where is a stabilization parameter and . The regular method uses a simple way to stabilize the mixed finite element approximation without a loss of accuracy. In fact, it can be treated the regular method as a special Douglas-Wang's scheme . Then the matrix form of this stabilized version can be expressed as where additional blocks and correspond to the following respective terms: As (3.11),we also have
Method 3 (Multiscale enrichment method). Find with , such that for all , where , are the stabilization parameters, , is the normal outward vector, is normal derivative operator, and denotes the jump of across . This stabilized method includes the usual Galerkin least squares stabilized terms on each finite element and positive jump terms at interelement boundaries. Moreover, a direct algebraic manipulation leads to the matrix form where the matrix is deduced in the usual manner, using the bases for , from the term As (3.11), we have
Method 4 (Local Gauss integration method). Find with , such that for all , where is defined by where indicates a local Gauss integral over that is exact for polynomials of degree , . In particular, the trial function must be projected to piecewise constant space when . This stabilization technique is free of stabilization parameters and does not require any calculation of high-order derivatives, a specification of any mesh-dependent parameter or edge-based data structures. Then the corresponding matrix form of this stabilized method is where the matrix is deduced in the usual manner, using the bases for , from the term . As (3.11), we get
Remark 3.3. It is well known that if is well chosen, then and converge, respectively, to and with a rate of convergence based on (3.11), (3.15), (3.18), and (3.22). From these equations, we can find that Method 1 converges faster than Method 4. Methods 2 and 3, whose coefficient matrices are not symmetric, may cost more time to converge.
Remark 3.4. By using the regularity assumptions and well-established techniques for eigenvalue approximation [1, 8, 10, 12], the theoretical convergence rates should be of order and for the velocity in the - and -norms, respectively, of order for the pressure in the -norm, and of order for the eigenvalue by using all these stabilized methods.
4. Numerical Experiments
In this section we numerically compare the performance of the various stabilized mixed finite element methods discussed in the previous section. In all experiments, the algorithms are implemented using public domain finite element software  with some of our additional codes.
Let the computation be carried out in the region . We consider the Stokes eigenvalue problem in the case of the viscidity , and it will be numerically solved by the stabilized mixed methods on uniform mesh (see Figure 1). Here, we just consider the first eigenvalue of the Stokes eigenvalue problem for the sake of simplicity. The exact solution of this problem is unknown. Thus, we take the numerical solution by the standard Galerkin method (- element) computed on a very fine mesh (6742 grid points) as the “exact” solution for the purpose of comparison. Here, we take as the first exact eigenvalue.
As we know, the stabilized term of the regular and multiscale enrichment methods must be controlled by carefully designed stabilization parameters (i.e., ), whose optimal values are often unknown. Hence, in Figures 2 and 3 we show the effect on the error of varying , , and on a fixed mesh for the regular and multiscale enrichment approximations, respectively. An interesting thing can be observed that these two methods have an analogous convergence pattern with respect to the parameter and . Because they involve a similar stabilization term with respect to these two parameters, we note that the errors can become large with some values of the stabilization parameters.
Results gotten from the penalty, regular, multiscale enrichment, local Gauss integration, and nonconforming local Gauss integration methods are presented in Tables 1–5, respectively. Here, we choose , , , and . Because they can deal with the considered problem well. From Tables 1, 2, 3, 4, and 5, we can see that these methods work well and keep the convergence rates just like the theoretical analysis except the multiscale enrichment method. Meanwhile, it can be seen that the penalty method requires the least CPU-time, which validates the analysis in Remark 3.3. As expected, we have an interesting observation that the error of the nonconforming local Gauss integration method is better than the conforming version, which is not surprising since the degrees of freedom of the nonconforming method are nearly three times than that of conforming one on uniform mesh (see Figure 1). Hence, it is natural that the nonconforming local Gauss integration method is more accurate and costs more CPU-time.
Besides, to show the stability and efficiency of these methods for the considered problem, we present the velocity streamlines and the pressure contours with in Figures 5, 6, 7, 8, and 9. Meanwhile, we present the results by the standard Galerkin method (- element) computed on a very fine mesh (6742 grid points) for the purpose of comparison in Figure 4. From Figures 5(a)–9(a), five resolved vortices are captured, which is consistent with that in Figure 4(a). For the pressure, the penalty method is divergent from Figure 5(b), although it costs the least time. From Figures 6(b)-9(b), the nonconforming local Gauss integration method shows the best numerical stability.
In this paper we have presented several stabilized mixed finite element methods in solving the Stokes eigenvalue problem based on the lowest equal-order finite element space pair. By being compared numerically, we get some conclusions as follows.(i)The stability and efficiency of the penalty method depend on the penalty parameter. The smaller this parameter, the more stable the method. However, if this parameter is too small, the condition number of the system matrix arising from this method will become too large to solve. (ii)The performance of the regular and multiscale enrichment method heavily depends on the choice of the stabilization parameters, which is a difficult task in reality. Meanwhile, a poor choice of these stabilization parameters can also lead to serious deterioration in the convergence rates. (iii)The local integration method is free of stabilization parameters and shows numerically the best performance among the methods considered for the given problem. (iv)From Tables 4 and 5, we have an interesting observation that the value of by conforming method becomes small to converge to the exact solution and the one by nonconforming method becomes large to converge to the exact solution. We hold that it can obtain more accurate numerical solution by Lagrange interpolation between conforming and nonconforming results based on the same degrees of freedom of these two methods. It may get superconvergence result on this triangular mesh.
The authors would like to thank the editor and reviewers for their valuable comments and suggestions which helped us improve the results of this paper. This work is in parts supported by the NSF of China (no. 10901131, no. 10971166), the National High Technology Research and Development Program of China (863 Program, no. 2009AA01A135), the China Postdoctoral Science Foundation (no. 200801448, no. 20070421155), and the Natural Science Foundation of Xinjiang Province (no. 2010211B04).
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