A Multilevel Correction Method for Convection-Diffusion Eigenvalue Problems
We propose a multilevel correction method for the convection-diffusion eigenvalue problems which is suitable for not only simple but also multiple eigenvalues. And we prove that the accuracy of resulting eigenpair approximations can be improved after each correction step. The scheme is easy to realize with Matlab, and numerical results are satisfactory.
The convection-diffusion eigenvalue problems have important physical background, such as convection-diffusion in fluid mechanics and environmental problems. Thus, finite element methods for solving this problem become an important topic which has attracted the attention of mathematical and physical fields. Research  discussed a priori error estimates, [2–7] the adaptive algorithms,  an adaptive homotopy approach,  two level algorithms,  function value recovery algorithms, [11, 12] extrapolation methods, and so forth. This paper turns to discuss finite element multilevel discretization based on Lin-Xie correction [13, 14].
Lin and Xie [13, 14] introduced a new type of multilevel correction procedure. Later on, this correction was further developed as well as successfully applied to Steklov eigenvalue problems , Helmholtz transmission eigenvalue problems , and so forth.
In this paper, we apply the method of Lin and Xie to convection-diffusion eigenvalue problems to obtain a multilevel method, which can be described as follows: (1) construct a coarsest finite element space and solve the primal and dual eigenvalue problems in the space; (2) solve two associated boundary value problems in an augmented space by using the previous obtained eigenvalue multiplying the corresponding eigenfunction as the load vector; (3) combine the coarsest finite element space with the obtained eigenfunction approximations in step (2) to obtain a new finite element space and solve the primal and dual eigenvalue problems again on the space. Then return to step (2) for next cycle. And the method is suitable for simple and multiple eigenvalues. What is more, we prove the scheme can reach the optimal order as same as solving the corresponding boundary value problem. Our scheme is easy to realize under the package of iFEM  with Matlab, and the numerical results are satisfactory.
Consider the convection-diffusion eigenvalue problems as follows:where is a polyhedral bound domain. We denote the complex Sobolev space and with norm where is a norm in complex space .
We assume is a bounded and measurable real function on and has a positive lower bound, is a vector of real or complex functions on , and there exist two positive constants such that
The variational form of (1) is given by: Find (complex plane), , satisfying
Let be a mesh of . For each element , let be the diameter of and ; is a ball contained in and . We further assume that is a regular-shape mesh (see Section 17 of Chapter 3 in ): there exists a constant such that if the quantity approaches zero.
Let be finite element space over consisting of continuous piecewise polynomials of degree less than or equal . The finite element approximation of (6) is given by: Find , , such that
It is shown in Section 8 in  that (4) and (5) show that there are two linear bounded operators and satisfyingFrom  we also know is a compact operator; then (6) and (8) have the following equivalent operator form (10) and (11), respectively. Consider
The corresponding adjoint problem of (1) isThe variational form and discrete variational form of (12) are given by: Find , satisfying Find , satisfyingNote that the primal and dual eigenvalues are connected via and .
From , for (13) and (14) we know that (4) and (5) imply that there are two linear operators and , satisfyingEquations (13) and (14) have the following equivalent operator form (16) and (17), respectively. ConsiderObviously we can easily prove is the adjoint operator of in the sense of inner product .
In this paper, let be an eigenvalue of (6) with the algebraic multiplicity and the ascent is 1. Let be the eigenvalue of (8) which converges to . Let be the space spanned by all eigenfunctions corresponding to the eigenvalue of . Let be the space spanned by all generalized eigenfunctions corresponding to eigenvalue of that converge to .
We assume that for any We define that denotes the finite element projection operator of byAnd we define byObviouslyFor any , by (18), we haveDefine and as
Lemma 1. The following estimates hold:
Proof. See .
Where , , , and are some positive constants independent of .
For two linear spaces and , we define We define the gaps between and in asand in as
We can likewise define the gaps between and in and in , respectively, as
For the eigenpair approximations by the finite element method, there exist the following error estimates (see P.699 in ).
Lemma 2. The eigenpair approximations and have the following error estimates:Here and hereafter are some positive constants depending on but independent of the mesh size .
3. One Correction Step with Multigrid Method
Firstly, we define the coarse linear finite element space on the generated mesh with the mesh size . Then we define a sequence of triangulations of domain determined as follows. Suppose is given and let be obtained from according to regular refinement (produce subelements) such that where is an integer and indicates the refinement index and always is 2 in numerical experiments. Based on this sequence of meshes, we construct the corresponding linear finite element spaces such that
Assume we have obtained the eigenpair approximations and for , where eigenvalues are the approximations of eigenvalue of (6) and are the approximations of eigenvalue of (13) and is a basis of and a basis of , and the definition of see Algorithm 3.
Now we introduce a correction step to improve the accuracy of the current eigenpair approximations and .
Algorithm 3. One correction step.
Step 1. For , solve the following equations.
Find and such thatStep 2. Define a new finite element space:and solve the following eigenvalue problems.
Find such thatFind such thatWe output and output a basis of with and a basis of with .
We denote the two steps of Algorithm 3 by Here denotes the coarse finite element space, are given eigenpair approximations, and denotes the computing space.
Note that the primal and dual eigenvalues are connected via .
We adopt the following assumption. (A0) Suppose that there are with and with , and there exists positive constant independent of such that , , and and have a positive lower bound uniformly with respect to , respectively.
When is a simple eigenvalue (), it is clear that (A0) is valid. When , we can also prove that if the distance from to () (in ) has a positive lower bound uniformly with respect to and satisfies the corresponding condition, then (A0) holds.
Theorem 5. Assume the condition (A0) holds, and there exist two numbers , such that the given eigenpairs and in Algorithm 3 have the following error estimates:Then after correction steps, the resultant eigenpair approximations and have the following error estimates:where , , and , , are positive constants independent of ; and are determined by (23) and (24), respectively.
Proof. Since is a basis of and , thus, from (45), we know there exist a basis of and such thatFor any , , we havethus Then from , we haveSince has a positive lower bound uniformly with respect to and (52)–(54) hold, has a positive lower bound uniformly with respect to ; that is, there exists such that which together with (57) yields Thus we getEquation (60) shows that there exists a constant independent of such thatWe set , for . From equalities (6), (19), and (39) and inequalities (5), (51), and (52), for , the following estimates hold: Then we havefor . According to (63) and the error estimate of finite element projectionwe havefor .
Now we estimate the error for the eigenpairs of problem (42). Based on the error estimate theory of finite element method for eigenvalue problems (see, e.g., [1, 14] and Lemma 2), (32), (33), and (65), and the definition of the space , we deduce thatwhereFrom (66) and (67), we can obtain (46) and (48).
Similarly, we can deduce that, for any , ,and there exists a constant independent of such thatAccording to the equalities (13), (20), and (40), inequalities (5), (53), and (54), and the error estimate of finite element projection , we can likewise haveWe can likewise have the following estimates:whereFrom (71) and (72), we can obtain (47) and (49). The estimate (50) can be derived by (36), (46), and (47), and the proof concludes.
4. Multilevel Scheme for the Eigenvalue Problem
In this part, we will give the multilevel scheme based on Algorithm 3. This type of multilevel method can achieve the optimal accuracy which is almost the same to solving the eigenvalue problem directly in the finest finite element space.
Firstly, the sequence of finite element spaces which are defined in Section 3 have the following relations:for , where and are two positive constants independent of and .
Remark 6. The relations (73) and (74) are obviously reasonable, because we can choose . Invariably the upper bound of the estimates holds. We also can obtain the lower bound (see ); are some positive constants independent of .
Algorithm 7 (multilevel scheme).
Implement the following.
Step 1. Construct a sequence of nested finite element spaces such that (38), (73), and (74) hold.
Step 2. Solve the following eigenvalue problems.
Find such that andFind such that andStep 3. For , compute new eigenpair approximations by Algorithm 3: Finally, we obtain eigenpair approximations , , and .
Theorem 8. After implementing Algorithm 7, if the mesh size is small enough such that , and the condition (A0) holds. Then the resultant eigenpair approximations and have the following error estimates:where and .
Proof. At first, by Lemma 2, we haveLet and . From (83)–(86) and Theorem 5 with , for , we haveThen by recursive relation and based on the proof in Theorem 5, (73), (88), and , we haveThe estimate (78) can be obtained by combining (90) and (91). And we can likewise obtain (80). From the proof of Theorem 5 and (78) and (80) we can obtain the desired results (79), (81), and (82).
Remark 9. We can likewise estimate the computational work of Algorithm 7 similar to Section 5 of  and can prove that solving the eigenvalue problem needs nearly the same work as solving the corresponding boundary value problem for Algorithm 7.
5. Numerical Experiments
In this section, we will present two numerical examples of the multilevel scheme by using linear finite elements on uniform triangle meshes. We use MATLAB 2012b under the package of iFEM  to solve Examples 10 and 11 on , respectively. In the numerical examples we give the following notational explainations: : the th finite element eigenvalue by solving the eigenvalue problem directly. : the th finite element eigenvalue by multilevel correction method solving the eigenvalue problem.
Example 10. Consider the convection-diffusion equationwhere and .
The eigenvalues of (92) arewhere . The corresponding adjoint eigenvalue problem has eigenvalues (see ). We obtain that and , so the multiplicity of is 2. For (92) with , , and , the numerical results are shown in Tables 1, 2, and 3, respectively, and we give Figure 1 to present the intuitive trend of the approximations. From them, we can see that the accuracy of and is nearly the same.
Example 11. Consider the convection-diffusion equation (92) where . A reference value for the first eigenvalue of (92) is and the sixth eigenvalue is (see ). From Tables 4–6 and Figure 2, we can get the same accurate approximations as those computed directly when the degrees of freedom are the same, but our running time is decreased. We also see that they are not perfect especially when ; the numerical first eigenvalue doesn’t perform well in approximating process, which is the consequence of the performance of linear algebra routine on this convection dominated problem.
6. Concluding Remarks
Based on [13, 14, 19], in this paper we discuss a multilevel method for the convection-diffusion eigenvalue problems. Theoretical analysis and experimental results show that the approach is easy to carry out and can be used to solve the eigenvalue problems efficiently.
We can replace (39)-(40) in Algorithm 3 by other types of efficient iteration schemes such as local and parallel finite element algorithms based on two-grid discretizations, which was first introduced by Xu and Zhou  and it has been applied successfully to eigenvalue problems (see, e.g., [22–24]). The multilevel method discussed here can also be extended to the general nonsymmetric elliptic eigenvalue problems (including Helmholtz transmission eigenvalue problems). These will be investigated in our future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors cordially thank the editor and the referees for their valuable comments and suggestions and thank Jiayu Han for his help that lead to the improvement of this paper. This work is supported by the National Science Foundation of China (Grant nos. 11201093 and 11161012) and Science and Technology Foundation of Guizhou Province of China (Grant nos. LH7061 and LKS06).
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