A Multilevel Correction Method for Convection-Diffusion Eigenvalue Problems

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 [1] discussed a priori error estimates, [2–7] the adaptive algorithms, [8] an adaptive homotopy approach, [9] two level algorithms, [10] 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 aswell as successfully applied to Steklov eigenvalue problems [15], Helmholtz transmission eigenvalue problems [16], 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 [17] with Matlab, and the numerical results are satisfactory.


Introduction
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 [1] discussed a priori error estimates, [2][3][4][5][6][7] the adaptive algorithms, [8] an adaptive homotopy approach, [9] two level algorithms, [10] 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 [15], Helmholtz transmission eigenvalue problems [16], 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 [17] with Matlab, and the numerical results are satisfactory.
Let  ℎ = {} be a mesh of Ω ⊂ R  .For each element  ∈  ℎ , let ℎ  be the diameter of  and   = sup{diam();  is a ball contained in } and ℎ = max{ℎ  :  ∈  ℎ }.We further assume that  ℎ is a regular-shape mesh (see Section 17 of Chapter 3 in [18]): there exists a constant  such that if the quantity ℎ approaches zero.
Let  ℎ ⊂  be finite element space over  ℎ consisting of continuous piecewise polynomials of degree less than or equal .The finite element approximation of ( 6) is given by: It is shown in Section 8 in [1] that (4) and (5) show that there are two linear bounded operators  :  2 (Ω) →  and From [1] we also know  :  2 (Ω) →  2 (Ω) 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) is The variational form and discrete variational form of ( 12) are given by: Note that the primal and dual eigenvalues are connected via  =  * and  ℎ =  * ℎ .From [1], for (13) and (14) we know that (4) and (5) imply that there are two linear operators  * :  2 (Ω) →  and Equations ( 13) and ( 14) have the following equivalent operator form ( 16) and ( 17), respectively.Consider Obviously 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  ,ℎ be the eigenvalue of (8) which converges to   .Let (  ) be the space spanned by all eigenfunctions corresponding to the eigenvalue   of .Let  ℎ (  ) be the space spanned by all generalized eigenfunctions corresponding to eigenvalue  ,ℎ of  ℎ that converge to   .
We define For two linear spaces  and , we define We define the gaps between (  ) and  ℎ (  ) in ‖ ⋅ ‖ 1 as and in ‖ ⋅ ‖ 0 as We can likewise define the gaps between  * ( *  ) and  * ℎ ( *  ) in ‖ ⋅ ‖ 1 and in ‖ ⋅ ‖ 0 , respectively, as For the eigenpair approximations by the finite element method, there exist the following error estimates (see P.699 in [1]).have the following error estimates: Here and hereafter   are some positive constants depending on   but independent of the mesh size ℎ.

One Correction Step with Multigrid Method
Based on [13,14,19] we introduce one correction step to improve the accuracy of the given eigenvalue and eigenfunction approximations.Firstly, we define the coarse linear finite element space   on the generated mesh   with the mesh size .Then we define a sequence of triangulations  ℎ  of domain Ω determined as follows.Suppose  ℎ 1 =   is given and let  ℎ +1 be obtained from  ℎ  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 = are the approximations of eigenvalue   of ( 6) and are the approximations of eigenvalue  *  of (13) and , and the definition of  ,ℎ  see Algorithm 3. Now we introduce a correction step to improve the accuracy of the current eigenpair approximations Algorithm 3. One correction step.
Remark 4. The construction of  ,ℎ +1 is proposed by Lin and Xie [13,14], which is significant and crucial for designation of a new multilevel method.
We adopt the following assumption.
Remark 9. We can likewise estimate the computational work of Algorithm 7 similar to Section 5 of [14] and can prove that solving the eigenvalue problem needs nearly the same work as solving the corresponding boundary value problem for Algorithm 7.

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 [17] to solve Examples 10 and 11 on Ω = (0, 1) 2 and Ω = (−1, 1) 2 \ [0, 1] 2 , respectively.In the numerical examples we give the following notational explainations:  ,ℎ : the th finite element eigenvalue by solving the eigenvalue problem directly.  ,ℎ : the th finite element eigenvalue by multilevel correction method solving the eigenvalue problem.

Example 11. Consider the convection-diffusion equation (92)
where Ω = (−1, 1) 2 \ [0, 1] 2 .A reference value for the first eigenvalue of ( 92) is ( 2 1 +  2 2 )/4 + 9.639724 and the sixth eigenvalue is ( 2 1 +  2 2 )/4 + 41.47451 (see [3]).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 b = (10, 0)  ; 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.

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 [21] and it  has been applied successfully to eigenvalue problems (see, e.g., [22][23][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.