Abstract

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.

1. 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, [2–7] 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.

2. Preliminaries

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 .

Let ,

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 [18]): 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 [1] that (4) and (5) show that there are two linear bounded operators and satisfyingFrom [1] 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 [1], 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 .

As for the adjoint problems (13) and (14), the definitions of and are analogous to and .

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 [1].

Where , , , and are some positive constants independent of .

We define

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 [1]).

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

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 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 .

Remark 4. The construction of 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. (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.