A Multilevel Correction Method for Convection-Diffusion Eigenvalue ...

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 904347, 10 pages http://dx.doi.org/10.1155/2015/904347

Research Article A Multilevel Correction Method for Convection-Diffusion Eigenvalue Problems Zhonglan Peng, Hai Bi, Hao Li, and Yidu Yang School of Mathematics and Computer Science, Guizhou Normal University, GuiYang 550001, China Correspondence should be addressed to Yidu Yang; [email protected] Received 3 February 2015; Accepted 30 March 2015 Academic Editor: Maria Gandarias Copyright © 2015 Zhonglan Peng 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 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: −∇ ⋅ (𝐷 (𝑥) ∇𝑢) + b (𝑥) ⋅ ∇𝑢 = 𝜆𝑢, 𝑢 = 0,

in Ω,

on 𝜕Ω,

(1)

where Ω ⊂ R𝑑 (𝑑 ≥ 2) is a polyhedral bound domain. We denote the complex Sobolev space 𝐻1 (Ω) and 𝐻01 (Ω) = {V ∈ 𝐻1 (Ω), V|𝜕Ω = 0} with norm 1/2

‖V‖1 = (‖V‖20 + ‖∇V‖20 )

,

where ‖ ⋅ ‖0 is a norm in complex space 𝐿 2 (Ω).

(2)

2

Mathematical Problems in Engineering Let 𝑉 = 𝐻01 (Ω),

Find 𝜆∗ ∈ C, 𝑢∗ ∈ 𝑉, ‖𝑢∗ ‖0 = 1, satisfying 𝑎 (V, 𝑢∗ ) = 𝜆∗ 𝑏 (V, 𝑢∗ ) ,

𝑎 (𝑢, V) = ∫ 𝐷 (𝑥) ∇𝑢 ⋅ ∇V + (b (𝑥) ⋅ ∇𝑢) V, Ω

(3) 𝑏 (𝑢, V) = ∫ 𝑢V.

|𝑎 (𝑤, V)| ≤ 𝐶1 ‖𝑤‖1 ‖V‖1 , Re 𝑎 (V, V) ≥ 𝑐1 ‖V‖21 ,

∀V, 𝑤 ∈ 𝑉, ∀V ∈ 𝑉.

(4) (5)

The variational form of (1) is given by: Find 𝜆 ∈ C (complex plane), 𝑢 ∈ 𝑉, ‖𝑢‖0 = 1, satisfying 𝑎 (𝑢, V) = 𝜆𝑏 (𝑢, V) ,

∀V ∈ 𝑉.

(6)

Let 𝜋ℎ = {𝐾} be a mesh of Ω ⊂ R𝑑 . For each element 𝐾 ∈ 𝜋ℎ , let ℎ𝐾 be the diameter of 𝐾 and 𝜌𝐾 = sup{diam(𝑆); 𝑆 is a ball contained in 𝐾} and ℎ = max{ℎ𝐾 : 𝐾 ∈ 𝜋ℎ }. We further assume that 𝜋ℎ is a regular-shape mesh (see Section 17 of Chapter 3 in [18]): there exists a constant 𝜎 such that ℎ𝐾 ≤ 𝜎 ∀𝐾 ∈ ⋃𝜋ℎ , 𝜌𝐾 ℎ

(7)

if the quantity ℎ approaches zero. Let 𝑉ℎ ⊂ 𝑉 be finite element space over 𝜋ℎ consisting of continuous piecewise polynomials of degree less than or equal 𝑠. The finite element approximation of (6) is given by: Find 𝜆 ℎ ∈ C, 𝑢ℎ ∈ 𝑉ℎ , ‖𝑢ℎ ‖0 = 1, such that 𝑎 (𝑢ℎ , V) = 𝜆 ℎ 𝑏 (𝑢ℎ , V) ,

∀V ∈ 𝑉ℎ .

(8)

It is shown in Section 8 in [1] that (4) and (5) show that there are two linear bounded operators 𝑇 : 𝐿 2 (Ω) → 𝑉 and 𝑇ℎ : 𝐿 2 (Ω) → 𝑉ℎ satisfying 𝑎 (𝑇𝑓, V) = 𝑏 (𝑓, V) ,

∀V ∈ 𝑉, ∀𝑓 ∈ 𝐿 2 (Ω) ,

𝑎 (𝑇ℎ 𝑓, V) = 𝑏 (𝑓, V) ,

∀V ∈ 𝑉ℎ , ∀𝑓 ∈ 𝐿 2 (Ω) .

(9)

−1

𝑇𝑢 = 𝜆 𝑢, 𝑇ℎ 𝑢ℎ =

(10)

𝜆−1 ℎ 𝑢ℎ .

(11)

The corresponding adjoint problem of (1) is −∇ ⋅ (𝐷 (𝑥) ∇𝑢∗ ) − ∇ ⋅ (b (𝑥)𝑢∗ ) = 𝜆∗ 𝑢∗ , 𝑢∗ = 0,

on 𝜕Ω.

in Ω,

𝑎 (V, 𝑢ℎ∗ ) = 𝜆∗ℎ 𝑏 (V, 𝑢ℎ∗ ) ,

(12)

The variational form and discrete variational form of (12) are given by:

∀V ∈ 𝑉ℎ .

(14)

Note that the primal and dual eigenvalues are connected via 𝜆 = 𝜆∗ and 𝜆 ℎ = 𝜆∗ℎ . From [1], for (13) and (14) we know that (4) and (5) imply that there are two linear operators 𝑇∗ : 𝐿 2 (Ω) → 𝑉 and 𝑇ℎ∗ : 𝐿 2 (Ω) → 𝑉ℎ , satisfying 𝑎 (V, 𝑇∗ 𝑓) = 𝑏 (V, 𝑓) ,

∀V ∈ 𝑉, ∀𝑓 ∈ 𝐿 2 (Ω) ,

𝑎 (V, 𝑇ℎ∗ 𝑓) = 𝑏 (V, 𝑓) ,

∀V ∈ 𝑉ℎ , ∀𝑓 ∈ 𝐿 2 (Ω) .

(15)

Equations (13) and (14) have the following equivalent operator form (16) and (17), respectively. Consider 𝑇∗ 𝑢∗ = 𝜆∗−1 𝑢∗ ,

(16)

∗ 𝑇ℎ∗ 𝑢ℎ∗ = 𝜆∗−1 ℎ 𝑢ℎ .

(17)

Obviously we can easily prove 𝑇∗ is the adjoint operator of 𝑇 in the sense of inner product 𝑏(⋅, ⋅). In this paper, let 𝜆 𝑖 be an eigenvalue of (6) with the algebraic multiplicity 𝑞 and the ascent is 1. Let 𝜆 𝑖,ℎ be the eigenvalue of (8) which converges to 𝜆 𝑖 . Let 𝑀(𝜆 𝑖 ) be the space spanned by all eigenfunctions corresponding to the eigenvalue 𝜆 𝑖 of 𝑇. Let 𝑀ℎ (𝜆 𝑖 ) be the space spanned by all generalized eigenfunctions corresponding to eigenvalue 𝜆 𝑖,ℎ of 𝑇ℎ that converge to 𝜆 𝑖 . As for the adjoint problems (13) and (14), the definitions of 𝑀∗ (𝜆∗𝑖 ) and 𝑀ℎ∗ (𝜆∗𝑖 ) are analogous to 𝑀(𝜆 𝑖 ) and 𝑀ℎ (𝜆 𝑖 ). We assume that for any 𝑤 ∈ 𝑉 lim inf ‖𝑤 − V‖1 = 0.

ℎ → 0 V∈𝑉ℎ

(18)

We define that 𝑃ℎ denotes the finite element projection operator of 𝑉 → 𝑉ℎ by 𝑎 (𝑤 − 𝑃ℎ 𝑤, Vℎ ) = 0,

From [1] we also know 𝑇 : 𝐿 2 (Ω) → 𝐿 2 (Ω) is a compact operator; then (6) and (8) have the following equivalent operator form (10) and (11), respectively. Consider

(13)

Find 𝜆∗ℎ ∈ C, 𝑢ℎ∗ ∈ 𝑉ℎ , ‖𝑢ℎ∗ ‖0 = 1, satisfying

Ω

We assume 𝐷(𝑥) ∈ 𝐿 ∞ (Ω) is a bounded and measurable real function on Ω and has a positive lower bound, b(𝑥) ∈ 𝑊1,∞ (Ω)𝑑 is a vector of real or complex functions on Ω, and there exist two positive constants 𝐶1 , 𝑐1 such that

∀V ∈ 𝑉.

∀𝑤 ∈ 𝑉, ∀Vℎ ∈ 𝑉ℎ .

(19)

∀𝑤∗ ∈ 𝑉, ∀Vℎ ∈ 𝑉ℎ .

(20)

And we define 𝑃ℎ∗ : 𝑉 → 𝑉ℎ by 𝑎 (Vℎ , 𝑤∗ − 𝑃ℎ∗ 𝑤∗ ) = 0, Obviously 󵄩󵄩 󵄩 󵄩󵄩𝑃ℎ 𝑤󵄩󵄩󵄩1 ≤ 𝐶2 ‖𝑤‖1 , ∀𝑤 ∈ 𝑉, 󵄩󵄩 ∗ ∗ 󵄩󵄩 ∗ 󵄩 ∗󵄩 ∗ 󵄩󵄩𝑃ℎ 𝑤 󵄩󵄩1 ≤ 𝐶2 󵄩󵄩󵄩𝑤 󵄩󵄩󵄩1 , ∀𝑤 ∈ 𝑉.

(21)

For any 𝑤 ∈ 𝑉, by (18), we have 󵄩󵄩 󵄩 󵄩󵄩𝑤 − 𝑃ℎ 𝑤󵄩󵄩󵄩1 󳨀→ 0, as ℎ 󳨀→ 0, 󵄩󵄩 ∗ ∗ ∗󵄩 󵄩󵄩𝑤 − 𝑃ℎ 𝑤 󵄩󵄩󵄩1 󳨀→ 0, as ℎ 󳨀→ 0.

(22)

Mathematical Problems in Engineering

3

Define 𝜂(ℎ) and 𝜂∗ (ℎ) as 𝜂 (ℎ) = 𝜂∗ (ℎ) =

sup

󵄩 󵄩 inf 󵄩󵄩󵄩𝑇𝑓 − V󵄩󵄩󵄩1 ,

(23)

sup

󵄩 󵄩 inf 󵄩󵄩𝑇∗ 𝑓 − V󵄩󵄩󵄩1 . V∈𝑉ℎ 󵄩

(24)

𝑓∈𝐿 2 (Ω),‖𝑓‖0 =1V∈𝑉ℎ

𝑓∈𝐿 2 (Ω),‖𝑓‖0 =1

For the eigenpair approximations by the finite element method, there exist the following error estimates (see P.699 in [1]). 𝑖+𝑞−1

Lemma 2. The eigenpair approximations {𝜆 𝑗,ℎ , 𝑢𝑗,ℎ }𝑗=𝑖 ∗ 𝑖+𝑞−1 }𝑗=𝑖 {𝜆∗𝑗,ℎ , 𝑢𝑗,ℎ

Lemma 1. The following estimates hold: 𝜂 (ℎ) 󳨀→ 0,

𝑎𝑠 ℎ 󳨀→ 0,

𝜂∗ (ℎ) 󳨀→ 0,

𝑎𝑠 ℎ 󳨀→ 0,

󵄩 󵄩 󵄩 󵄩󵄩 ∗ 󵄩󵄩𝑤 − 𝑃ℎ 𝑤󵄩󵄩󵄩0 ≤ 𝐶3 𝜂 (ℎ) 󵄩󵄩󵄩𝑤 − 𝑃ℎ 𝑤󵄩󵄩󵄩1 ,

∀𝑤 ∈ 𝑉,

󵄩 󵄩󵄩 ∗ ∗ ∗󵄩 ∗ ∗ ∗󵄩 󵄩󵄩𝑤 − 𝑃ℎ 𝑤 󵄩󵄩󵄩0 ≤ 𝐶3 𝜂 (ℎ) 󵄩󵄩󵄩𝑤 − 𝑃ℎ 𝑤 󵄩󵄩󵄩1 ,

∀𝑤 ∈ 𝑉.

(32)

Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ (𝜆 𝑖 )) ≤ 𝐶𝑖 𝜂∗ (ℎ) 𝛿ℎ (𝜆 𝑖 ) ,

(33)

Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗ (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝛿ℎ∗ (𝜆∗𝑖 ) ,

(34)

Φ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗ (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝜂 (ℎ) 𝛿ℎ∗ (𝜆∗𝑖 ) ,

(35)

󵄨󵄨 󵄨 󵄨󵄨𝜆 𝑗,ℎ − 𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶𝑖 𝛿ℎ (𝜆 𝑖 ) 𝛿ℎ∗ (𝜆∗𝑖 ) . 󵄨 󵄨

(36)

Proof. See [1]. Where 𝐶2 , 𝐶2∗ , 𝐶3 , and 𝐶3∗ are some positive constants independent of ℎ. We define 𝛿ℎ (𝜆 𝑖 ) = 𝛿ℎ∗ (𝜆∗𝑖 ) =

sup

sup

𝑤∗ ∈𝑀∗ (𝜆∗𝑖 ),‖𝑤∗ ‖0 =1

󵄩 󵄩 inf 󵄩󵄩𝑤∗ − V󵄩󵄩󵄩1 . V∈𝑉ℎ 󵄩

(26)

For two linear spaces 𝑅 and 𝑈, we define ̂ (𝑅, 𝑈) = Θ ̂ (𝑅, 𝑈) = Φ

sup

inf ‖𝑤 − V‖1 ,

sup

inf ‖𝑤 − V‖0 .

𝑤∈𝑅,‖𝑤‖1 =1V∈𝑈

Here and hereafter 𝐶𝑖 are some positive constants depending on 𝜆 𝑖 but independent of the mesh size ℎ.

3. One Correction Step with Multigrid Method

inf ‖𝑤 − V‖1 ,

𝑤∈𝑀(𝜆 𝑖 ),‖𝑤‖0 =1V∈𝑉ℎ

have the following error estimates:

Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ (𝜆 𝑖 )) ≤ 𝐶𝑖 𝛿ℎ (𝜆 𝑖 ) ,

(25)

∗

and

(27)

𝑤∈𝑅,‖𝑤‖0 =1V∈𝑈

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 𝑉𝐻 on the generated mesh 𝜋𝐻 with the mesh size 𝐻. Then we define a sequence of triangulations 𝜋ℎ𝑚 of domain Ω determined as follows. Suppose 𝜋ℎ1 = 𝜋𝐻 is given and let 𝜋ℎ𝑚+1 be obtained from 𝜋ℎ𝑚 according to regular refinement (produce 𝜉𝑑 subelements) such that 1 ℎ𝑚+1 ≈ ℎ𝑚 , 𝜉

We define the gaps between 𝑀(𝜆 𝑖 ) and 𝑀ℎ (𝜆 𝑖 ) in ‖ ⋅ ‖1 as Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ (𝜆 𝑖 )) ̂ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ (𝜆 𝑖 )) , Θ ̂ (𝑀ℎ (𝜆 𝑖 ) , 𝑀 (𝜆 𝑖 ))} = max {Θ (28)

where 𝜉 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

and in ‖ ⋅ ‖0 as Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ (𝜆 𝑖 )) ̂ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ (𝜆 𝑖 )) , Φ ̂ (𝑀ℎ (𝜆 𝑖 ) , 𝑀 (𝜆 𝑖 ))} . = max {Φ (29) We can likewise define the gaps between 𝑀∗ (𝜆∗𝑖 ) and ∗ ∗ 𝑀ℎ (𝜆 𝑖 ) in ‖ ⋅ ‖1 and in ‖ ⋅ ‖0 , respectively, as Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗ (𝜆∗𝑖 )) ̂ (𝑀∗ (𝜆∗ ) , 𝑀∗ (𝜆∗ )) , Θ ̂ (𝑀∗ (𝜆∗ ) , 𝑀∗ (𝜆∗ ))} , = max {Θ 𝑖 ℎ 𝑖 ℎ 𝑖 𝑖 (30) Φ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗ (𝜆∗𝑖 )) ̂ (𝑀∗ (𝜆∗ ) , 𝑀∗ (𝜆∗ )) , Φ ̂ (𝑀∗ (𝜆∗ ) , 𝑀∗ (𝜆∗ ))} . = max {Φ 𝑖 ℎ 𝑖 ℎ 𝑖 𝑖 (31)

(37)

𝑉𝐻 = 𝑉ℎ1 ⊂ 𝑉ℎ2 ⊂ ⋅ ⋅ ⋅ ⊂ 𝑉ℎ𝑛 .

(38)

Assume we have obtained the eigenpair approximations 𝑐 𝑐 𝑐∗ ) ∈ C×𝑉𝐻,ℎ𝑚 , ‖𝑢𝑗,ℎ ‖ = 1 and (𝜆𝑐∗ (𝜆𝑐𝑗,ℎ𝑚 , 𝑢𝑗,ℎ 𝑗,ℎ𝑚 , 𝑢𝑗,ℎ𝑚 ) ∈ C× 𝑚 𝑚 0 𝑐∗ 𝑉𝐻,ℎ𝑚 , ‖𝑢𝑗,ℎ𝑚 ‖0 = 1 for 𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1, where eigenvalues 𝑖+𝑞−1

{𝜆𝑐𝑗,ℎ𝑚 }𝑗=𝑖

are the approximations of eigenvalue 𝜆 𝑖 of (6) and 𝑐∗ 𝑖+𝑞−1 {𝜆 𝑗,ℎ𝑚 }𝑗=𝑖 are the approximations of eigenvalue 𝜆∗𝑖 of (13) 𝑖+𝑞−1 𝑐 𝑐∗ 𝑖+𝑞−1 } is a basis of 𝑀ℎ𝑚 (𝜆 𝑖 ) and {𝑢𝑗,ℎ } a basis and {𝑢𝑗,ℎ 𝑚 𝑗=𝑖 𝑚 𝑗=𝑖 ∗ ∗ of 𝑀ℎ𝑚 (𝜆 𝑖 ), and the definition of 𝑉𝐻,ℎ𝑚 see Algorithm 3.

Now we introduce a correction step to improve the accuracy of the current eigenpair approximations 𝑖+𝑞−1 𝑐 𝑐∗ 𝑖+𝑞−1 {𝜆𝑐𝑗,ℎ𝑚 , 𝑢𝑗,ℎ } and {𝜆𝑐∗ 𝑗,ℎ𝑚 , 𝑢𝑗,ℎ𝑚 }𝑗=𝑖 . 𝑚 𝑗=𝑖 Algorithm 3. One correction step. Step 1. For 𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1, solve the following equations.

4

Mathematical Problems in Engineering ∗ Find 𝑢̂𝑗,ℎ𝑚+1 and 𝑢̂𝑗,ℎ ∈ 𝑉ℎ𝑚+1 such that 𝑚+1 𝑐 𝑎 (̂ 𝑢𝑗,ℎ𝑚+1 , V) = 𝜆𝑐𝑗,ℎ𝑚 𝑏 (𝑢𝑗,ℎ , V) , 𝑚

∀V ∈ 𝑉ℎ𝑚+1 ,

(39)

∗ 𝑐∗ 𝑎 (V, 𝑢̂𝑗,ℎ ) = 𝜆𝑐∗ 𝑗,ℎ 𝑏 (V, 𝑢𝑗,ℎ𝑚 ) , 𝑚+1

∀V ∈ 𝑉ℎ𝑚+1 .

(40)

𝑚

Theorem 5. Assume the condition (A0) holds, and there exist two numbers 𝜀ℎ𝑚 (𝜆 𝑖 ), 𝜀ℎ∗𝑚 (𝜆∗𝑖 ) such that the given eigenpairs 𝑖+𝑞−1

Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑚 (𝜆∗𝑖 )) ≤ 𝐶4 𝜀ℎ∗𝑚 (𝜆∗𝑖 ) ,

𝑉𝐻,ℎ𝑚+1 = 𝑉𝐻 ⊕ span {̂ 𝑢𝑖,ℎ𝑚+1 , . . . , 𝑢̂𝑖+𝑞−1,ℎ𝑚+1 ,

(41)

∗ ∗ , . . . , 𝑢̂𝑖+𝑞−1,ℎ }, 𝑢̂𝑖,ℎ 𝑚+1 𝑚+1

Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑚 (𝜆 𝑖 )) ≤ 𝐶4 𝜂∗ (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) , ∗

Φ (𝑀

and solve the following eigenvalue problems. 𝑐 ) ∈ C × 𝑉𝐻,ℎ𝑚+1 such that Find (𝜆𝑐𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ 𝑚+1 =

𝑐 𝜆𝑐𝑗,ℎ𝑚+1 𝑏 (𝑢𝑗,ℎ , V) , 𝑚+1

∀V ∈ 𝑉𝐻,ℎ𝑚+1 .

𝑚+1

𝑖+𝑞−1

We output {𝜆𝑐𝑗,ℎ𝑚+1 }𝑗=𝑖

𝑐∗ 𝑏 (V, 𝑢𝑗,ℎ ), 𝑚+1

∀V ∈ 𝑉𝐻,ℎ𝑚+1 .

(43)

𝑖+𝑞−1

𝑐 and output a basis {𝑢𝑗,ℎ } 𝑚+1 𝑗=𝑖

𝑐 ‖𝑢𝑗,ℎ ‖ = 𝑚+1 0 𝑐∗ ‖𝑢𝑗,ℎ𝑚+1 ‖0 = 1.

𝑖+𝑞−1 𝑐∗ {𝑢𝑗,ℎ } 𝑚+1 𝑗=𝑖

1 and a basis 𝑀ℎ𝑚+1 (𝜆 𝑖 ) with 𝑀ℎ∗𝑚+1 (𝜆∗𝑖 ) with We denote the two steps of Algorithm 3 by 𝑐 𝑐∗ {𝜆𝑐𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ , 𝑢𝑗,ℎ } 𝑚+1 𝑚+1

of

𝑗=𝑖

(44) Here 𝑉𝐻 denotes the coarse finite element space, 𝑐 𝑐∗ 𝑖+𝑞−1 {𝜆𝑐𝑗,ℎ𝑚 , 𝑢𝑗,ℎ , 𝑢𝑗,ℎ } are given eigenpair approximations, 𝑚 𝑚 𝑗=𝑖 and 𝑉ℎ𝑚+1 denotes the computing space. Note that the primal and dual eigenvalues are connected . via 𝜆𝑐𝑗,ℎ𝑚+1 = 𝜆𝑐∗ 𝑗,ℎ 𝑚+1

Remark 4. The construction of 𝑉𝐻,ℎ𝑚+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. (A0) Suppose that

𝑖+𝑞−1 there are {𝑢𝑗,ℎ𝑚 }𝑗=𝑖 ⊂ 𝑖+𝑞−1 ∗ 1 and {𝑢𝑘,ℎ } ⊂ 𝑚 𝑘=𝑖

𝑀ℎ𝑚 (𝜆 𝑖 )

𝑀ℎ∗𝑚 (𝜆∗𝑖 ) with ‖𝑢𝑗,ℎ𝑚 ‖0 = ∗ with ‖𝑢𝑘,ℎ𝑚 ‖0 = 1, and there exists positive constant 𝑐 ∗ 𝐶󸀠 independent of ℎ𝑚 such that |𝑏(𝑢𝑗,ℎ , 𝑢𝑘,ℎ )| + 𝑚 𝑚 𝑐∗ 󸀠 ∗ |𝑏(𝑢𝑗,ℎ𝑚 , 𝑢𝑘,ℎ𝑚 )| ≤ 𝐶 (𝜂(𝐻) + 𝜂 (𝐻)), (𝑗, 𝑘 = 𝑖, . . . , 𝑖 + 𝑐 ∗ 𝑐∗ 𝑞 − 1, 𝑗 ≠ 𝑘), and |𝑏(𝑢𝑘,ℎ , 𝑢𝑘,ℎ )| and |𝑏(𝑢𝑘,ℎ𝑚 , 𝑢𝑘,ℎ )| 𝑚 𝑚 𝑚 have a positive lower bound uniformly with respect to ℎ𝑚 , respectively. When 𝜆 is a simple eigenvalue (𝑞 = 1), it is clear that (A0) is valid. When 𝑞 > 1, we can also prove that if the 𝑐 𝑐 to span{𝑢𝑗,ℎ , 𝑗 = 𝑘, . . . , 𝑘 + 𝑞 − 1, 𝑗 ≠ distance from 𝑢𝑖,ℎ 𝑚 𝑚 𝑖} (𝑖 = 𝑘, . . . , 𝑘 + 𝑞 − 1) (in ‖ ⋅ ‖𝑏 ) has a positive lower 𝑐∗ satisfies the bound uniformly with respect to ℎ𝑚 and 𝑢𝑖,ℎ 𝑚 corresponding condition, then (A0) holds.

(𝜆∗𝑖 ) ,

(45)

Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑚+1 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝜀ℎ𝑚+1 (𝜆 𝑖 ) ,

(46)

Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑚+1 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝜀ℎ∗𝑚+1 (𝜆∗𝑖 ) ,

(47)

Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑚+1 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝜂∗ (𝐻) 𝜀ℎ𝑚+1 (𝜆 𝑖 ) ,

(48)

Φ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑚+1 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝜂 (𝐻) 𝜀ℎ∗𝑚+1 (𝜆∗𝑖 ) ,

(49)

of

, 𝑉ℎ𝑚+1 ) .

≤

𝐶4 𝜂 (𝐻) 𝜀ℎ∗𝑚

Then after correction steps, the resultant eigenpair approxima𝑖+𝑞−1 𝑖+𝑞−1 𝑐 𝑐∗ } and {𝜆𝑐∗ have the tions {𝜆𝑐𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ 𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ𝑚+1 }𝑗=𝑖 𝑚+1 𝑗=𝑖 following error estimates:

𝑗=𝑖

𝑐 𝑐∗ = Correction (𝑉𝐻, {𝜆𝑐𝑗,ℎ𝑚 , 𝑢𝑗,ℎ , 𝑢𝑗,ℎ } 𝑚 𝑚

(𝜆∗𝑖 ))

𝑓𝑜𝑟 𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1.

𝑖+𝑞−1

𝑖+𝑞−1

(𝜆∗𝑖 ) , 𝑀ℎ∗𝑚

󵄨 󵄨󵄨 𝑐 󵄨󵄨𝜆 𝑗,ℎ − 𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶4 𝜀ℎ𝑚 (𝜆 𝑖 ) 𝜀ℎ∗ (𝜆∗𝑖 ) , 𝑚 󵄨 󵄨 𝑚

(42)

𝑐∗ Find (𝜆𝑐∗ 𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ𝑚+1 ) ∈ C × 𝑉𝐻,ℎ𝑚+1 such that 𝑐∗ 𝑎 (V, 𝑢𝑗,ℎ ) = 𝜆𝑐∗ 𝑗,ℎ 𝑚+1

in Algorithm 3 have

Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑚 (𝜆 𝑖 )) ≤ 𝐶4 𝜀ℎ𝑚 (𝜆 𝑖 ) ,

Step 2. Define a new finite element space:

𝑐 𝑎 (𝑢𝑗,ℎ , V) 𝑚+1

𝑖+𝑞−1

𝑐 𝑐∗ } and {𝜆𝑐∗ {𝜆𝑐𝑗,ℎ𝑚 , 𝑢𝑗,ℎ 𝑗,ℎ𝑚 , 𝑢𝑗,ℎ𝑚 }𝑗=𝑖 𝑚 𝑗=𝑖 the following error estimates:

󵄨󵄨 𝑐 󵄨 󵄨󵄨𝜆 𝑗,ℎ − 𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶𝑖 𝜀ℎ𝑚+1 (𝜆 𝑖 ) 𝜀ℎ∗ (𝜆∗𝑖 ) , (50) 𝑚+1 󵄨 𝑚+1 󵄨 := 𝑞𝐶5 ((𝐶4 /𝑐1 )|𝜆 𝑖 |𝜂∗ (𝐻)𝜀ℎ𝑚 (𝜆 𝑖 ) + where 𝜀ℎ𝑚+1 (𝜆 𝑖 ) ∗ ∗ (𝐶1 /𝑐1 )𝛿ℎ𝑚+1 (𝜆 𝑖 )), 𝜀ℎ𝑚+1 (𝜆 𝑖 ) := 𝑞𝐶5∗ ((𝐶4 /𝑐1 )|𝜆∗𝑖 |𝜂(𝐻)𝜀ℎ∗𝑚 (𝜆∗𝑖 )+ (𝐶1 /𝑐1 )𝛿ℎ∗𝑚+1 (𝜆∗𝑖 )) (𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1), and 𝐶4 , 𝐶5 , 𝐶5∗ are positive constants independent of ℎ𝑚 ; 𝜂∗ (𝐻) and 𝜂(𝐻) are determined by (23) and (24), respectively. 𝑖+𝑞−1

𝑐 Proof. Since {𝑢𝑗,ℎ } 𝑚 𝑗=𝑖 𝑖+𝑞−1 ∗ } {𝑢𝑗,ℎ 𝑚 𝑗=𝑖

exist a basis that

is a basis of 𝑀ℎ𝑚 (𝜆 𝑖 ) and

⊂ 𝑀ℎ∗𝑚 (𝜆∗𝑖 ), thus, from (45), 𝑖+𝑞−1 𝑖+𝑞−1 {𝑢𝑗 }𝑗=𝑖 of 𝑀(𝜆 𝑖 ) and {𝑢𝑗∗ }𝑗=𝑖

we know there ⊂ 𝑀∗ (𝜆∗𝑖 ) such

󵄩󵄩 󵄩󵄩 𝑐 󵄩󵄩𝑢𝑗 − 𝑢𝑗,ℎ 󵄩󵄩 ≤ 𝐶4 𝜀ℎ𝑚 (𝜆 𝑖 ) , 𝑚 󵄩1 󵄩 󵄩󵄩 󵄩󵄩 𝑐 󵄩󵄩𝑢𝑗 − 𝑢𝑗,ℎ 󵄩󵄩 ≤ 𝐶4 𝜂∗ (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) , 𝑚 󵄩0 󵄩 󵄩󵄩 ∗ ∗ 󵄩 󵄩󵄩 ≤ 𝐶 𝜀∗ (𝜆∗ ) , 󵄩󵄩𝑢𝑗 − 𝑢𝑗,ℎ 󵄩 4 ℎ𝑚 𝑖 𝑚 󵄩1 󵄩 󵄩󵄩 ∗ ∗ 󵄩 󵄩󵄩𝑢𝑗 − 𝑢𝑗,ℎ 󵄩󵄩󵄩 ≤ 𝐶4 𝜂 (𝐻) 𝜀ℎ∗ (𝜆∗𝑖 ) . 𝑚 󵄩0 𝑚 󵄩 For any 𝑤 ∈ 𝑀(𝜆 𝑖 ), ‖𝑤‖0 = 1, we have

(51) (52) (53) (54)

𝑖+𝑞−1

𝑤 = ∑ 𝛾𝑗 𝑢𝑗 ,

(55)

𝑗=𝑖

thus 𝑖+𝑞−1

𝑏 (𝑤, 𝑢𝑘∗ ) = ∑ 𝛾𝑗 𝑏 (𝑢𝑗 , 𝑢𝑘∗ ) . 𝑗=𝑖

(56)

Mathematical Problems in Engineering

5

𝑐 ∗ Then from |𝑏(𝑢𝑗,ℎ , 𝑢𝑘,ℎ )| ≤ 𝐶󸀠 (𝜂(𝐻) + 𝜂∗ (𝐻)) (𝑗 ≠ 𝑘), we 𝑚 𝑚 have

Thus we get 𝑖+𝑞−1

󵄨 󵄨 ∑ 󵄨󵄨󵄨𝛾𝑘 󵄨󵄨󵄨 𝑘=𝑖

󵄨󵄨 󵄨󵄨 𝑖+𝑞−1 󵄨󵄨 󵄨 1 󵄨󵄨 󵄨󵄨 ∗ ∗ 󵄨󵄨 󵄨 󵄨 󵄨 𝑏 (𝑤, 𝑢 ) − 𝛾 𝑏 (𝑢 , 𝑢 ) ∑ 󵄨󵄨𝛾𝑘 󵄨󵄨 = 󵄨󵄨 󵄨 󵄨 𝑗 𝑗 𝑘 𝑘 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑏 (𝑢𝑘 , 𝑢𝑘∗ )󵄨󵄨󵄨 󵄨󵄨󵄨 𝑗=𝑘,𝑗=𝑖 ̸ 󵄨 󵄨

𝑖+𝑞−1 1{ 󵄨 󵄨 ≤ 𝑞 {1 + ( ∑ 󵄨󵄨󵄨󵄨𝛾𝑗 󵄨󵄨󵄨󵄨) 𝛿 𝑗=𝑖 {

󵄨󵄨 𝑖+𝑞−1 󵄨󵄨 { 1 𝑐 ∗ 󵄨 ≤ 󵄨󵄨 󵄨󵄨 {1 + 󵄨󵄨󵄨 ∑ 𝛾𝑗 [𝑏 (𝑢𝑗 − 𝑢𝑗,ℎ𝑚 , 𝑢𝑘 ) ∗ 󵄨󵄨𝑗=𝑘,𝑗=𝑖 󵄨󵄨𝑏 (𝑢𝑘 , 𝑢𝑘 )󵄨󵄨 󵄨 ̸ {

⋅ [𝐶4 (𝜂∗ (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) + 𝜂 (𝐻) 𝜀ℎ∗𝑚 (𝜆∗𝑖 ))

+

𝑐 𝑏 (𝑢𝑗,ℎ , 𝑢𝑘∗ 𝑚

+

󵄨󵄨} 𝑐 ∗ 𝑏 (𝑢𝑗,ℎ , 𝑢𝑘,ℎ )] 󵄨󵄨󵄨} 𝑚 𝑚 󵄨

−

∗ 𝑢𝑘,ℎ ) 𝑚

󵄨󵄨 󵄨

󵄨󵄨 󵄨}

} + 𝐶󸀠 (𝜂 (𝐻) + 𝜂∗ (𝐻)) ] } . } Equation (60) shows that there exists a constant 𝐶5 independent of ℎ𝑚 such that 󵄨󵄨󵄨𝛾 󵄨󵄨󵄨 ≤ 𝐶 , 𝑗 = 𝑖, 𝑖 + 1, . . . , 𝑖 + 𝑞 − 1. (61) 󵄨󵄨 𝑗 󵄨󵄨 5 We set 𝛼𝑗 := 𝜆 𝑖 /𝜆𝑐𝑗,ℎ𝑚 , for 𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1. From equalities (6), (19), and (39) and inequalities (5), (51), and (52), for 𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1, the following estimates hold:

𝑖+𝑞−1

{ 1 󵄨󵄨 󵄨󵄨 ∗ ≤ 󵄨󵄨 󵄨󵄨 {1 + ∑ 󵄨󵄨󵄨𝛾𝑗 󵄨󵄨󵄨 [𝐶4 (𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) ∗ 󵄨󵄨𝑏 (𝑢𝑘 , 𝑢𝑘 )󵄨󵄨 𝑗=𝑘,𝑗=𝑖 ̸ { + 𝜂 (𝐻) 𝜀ℎ∗𝑚 (𝜆∗𝑖 )) + 𝐶󸀠 (𝜂 (𝐻) } + 𝜂∗ (𝐻)) ] } . } (57)

󵄩2 󵄩󵄩 󵄩󵄩𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 󵄩󵄩󵄩 󵄩1 󵄩 ≤

1 Re 𝑎 (𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 , 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 ) 𝑐1

≤

1 󵄨󵄨 󵄨 󵄨󵄨𝑎 (𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 , 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 )󵄨󵄨󵄨 󵄨 𝑐1 󵄨

=

1 𝑐1

has a positive lower bound uniformly Since with respect to ℎ𝑚 and (52)–(54) hold, |𝑏(𝑢𝑘 , 𝑢𝑘∗ )| has a positive lower bound uniformly with respect to ℎ𝑚 ; that is, there exists 𝛿 > 0 such that

𝑘 = 𝑖, 𝑖 + 1, . . . , 𝑖 + 𝑞 − 1,

(58)

which together with (57) yields

󵄨󵄨 󵄨󵄨𝑎 (𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 , 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 ) 󵄨 󵄨 − 𝑎 (𝑢𝑗 , 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 )󵄨󵄨󵄨󵄨

𝑐 ∗ , 𝑢𝑘,ℎ )| |𝑏(𝑢𝑘,ℎ 𝑚 𝑚

󵄨󵄨 ∗ 󵄨 󵄨󵄨𝑏 (𝑢𝑘 , 𝑢𝑘 )󵄨󵄨󵄨 ≥ 𝛿,

=

1 𝑐1

󵄨󵄨 󵄨󵄨 󵄨󵄨𝛾𝑘 󵄨󵄨

󵄨 − 𝜆 𝑗 𝑏 (𝑢𝑗 , 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 )󵄨󵄨󵄨󵄨 ≤

1 󵄨󵄨 󵄨󵄨 󵄩󵄩 𝑐 󵄩 󵄩 󵄩 󵄨󵄨𝜆 𝑖 󵄨󵄨 󵄩󵄩󵄩𝑢𝑗,ℎ𝑚 − 𝑢𝑗 󵄩󵄩󵄩󵄩0 󵄩󵄩󵄩󵄩𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 󵄩󵄩󵄩󵄩1 𝑐1

≤

𝐶4 󵄨󵄨 󵄨󵄨 ∗ 󵄩 󵄩 󵄨󵄨𝜆 𝑖 󵄨󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) 󵄩󵄩󵄩󵄩𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 󵄩󵄩󵄩󵄩1 . 𝑐1

(63)

for 𝑗 = 𝑖, . . . , 𝑖+𝑞−1. According to (63) and the error estimate of finite element projection

𝑖+𝑞−1

1{ 󵄨 󵄨 ≤ {1 + ( ∑ 󵄨󵄨󵄨󵄨𝛾𝑗 󵄨󵄨󵄨󵄨) 𝛿 𝑗=𝑖 { ⋅ [𝐶4 (𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) +

(62)

󵄨󵄨 𝑐 𝑐 󵄨󵄨𝛼𝑗 𝜆 𝑗,ℎ 𝑏 (𝑢𝑗,ℎ , 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 ) 𝑚 𝑚 󵄨

Then we have 𝐶4 󵄨󵄨 󵄨󵄨 ∗ 󵄩󵄩 󵄩󵄩󵄩𝛼 𝑢̂ 󵄨𝜆 󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) , 󵄩󵄩 𝑗 𝑗,ℎ𝑚+1 − 𝑃ℎ𝑚+1 𝑢𝑗 󵄩󵄩󵄩1 ≤ 𝑐1 󵄨 𝑖 󵄨

∗

(60)

𝜂 (𝐻) 𝜀ℎ∗𝑚

} + 𝐶󸀠 (𝜂 (𝐻) + 𝜂∗ (𝐻)) ] } . }

(𝜆∗𝑖 ))

(59)

𝐶 󵄩󵄩 󵄩 󵄩󵄩𝑢𝑗 − 𝑃ℎ𝑚+1 𝑢𝑗 󵄩󵄩󵄩 ≤ 1 𝛿ℎ𝑚+1 (𝜆 𝑖 ) 󵄩 󵄩1 𝑐1

(64)

we have 𝐶4 󵄨󵄨 󵄨󵄨 ∗ 𝐶 󵄩󵄩 󵄩󵄩󵄩𝛼 𝑢̂ 󵄨𝜆 󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) + 1 𝛿ℎ𝑚+1 (𝜆 𝑖 ) , 󵄩󵄩 𝑗 𝑗,ℎ𝑚+1 − 𝑢𝑗 󵄩󵄩󵄩1 ≤ 𝑐1 󵄨 𝑖 󵄨 𝑐1 (65)

6

Mathematical Problems in Engineering

for 𝑗 = 𝑖, . . . , 𝑖 + 𝑞 − 1. Now we estimate the error for the eigenpairs 𝑖+𝑞−1 𝑐 } of problem (42). Based on the error {𝜆𝑐𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ 𝑚+1 𝑗=𝑖 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 𝑉𝐻,ℎ𝑚+1 , we deduce that

We can likewise have the following estimates: Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑚+1 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝑞𝐶5∗ (

Φ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑚+1 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝜂̃ (𝐻) 𝜀ℎ∗𝑚+1 (𝜆∗𝑖 ) ,

Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑚+1 (𝜆 𝑖 )) ≤ 𝐶𝑖

(71)

󵄩󵄩 󵄩 󵄩󵄩𝑤 − V𝐻,ℎ𝑚+1 󵄩󵄩󵄩 󵄩 󵄩1 V ∈𝑉 𝐻,ℎ 𝐻,ℎ 𝑤∈𝑀(𝜆 ),‖𝑤‖ =1 𝑚+1 𝑚+1 sup 𝑖

inf

where

0

󵄩󵄩𝑖+𝑞−1 󵄩󵄩 𝑖+𝑞−1 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ 𝛾𝑗 𝑢𝑗 − ∑ 𝛾𝑗 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 󵄩󵄩󵄩 ≤ 𝐶𝑖 sup 󵄩 󵄩󵄩 𝑤∈𝑀(𝜆 𝑖 ),‖𝑤‖0 =1 󵄩 󵄩󵄩 𝑗=𝑖 󵄩󵄩1 𝑗=𝑖 󵄩󵄩𝑖+𝑞−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 𝐶𝑖 sup 󵄩󵄩󵄩󵄩 ∑ 𝛾𝑗 (𝑢𝑗 − 𝛼𝑗 𝑢̂𝑗,ℎ𝑚+1 )󵄩󵄩󵄩󵄩 𝛾𝑗 󵄩 󵄩󵄩 󵄩󵄩 𝑗=𝑖 󵄩1 󵄩 󵄩 ≤ 𝐶𝑖 𝑞𝐶5 max {󵄩󵄩󵄩󵄩𝛼𝑖 𝑢̂𝑖,ℎ𝑚+1 − 𝑢𝑖 󵄩󵄩󵄩󵄩1 , . . . , 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩𝛼𝑖+𝑞−1 𝑢̂𝑖+𝑞−1,ℎ𝑚+1 − 𝑢𝑖+𝑞−1 󵄩󵄩󵄩1 } ≤ 𝐶𝑖 𝑞𝐶5 (

𝜂̃ (𝐻) =

inf

𝑤∈𝑀(𝜆 𝑖 ),‖𝑤‖0 =1V𝐻,ℎ𝑚+1 ∈𝑉𝐻,ℎ𝑚+1

󵄩󵄩 󵄩 󵄩󵄩𝑤 − V𝐻,ℎ𝑚+1 󵄩󵄩󵄩 󵄩 󵄩0

󵄩󵄩 󵄩 󵄩󵄩𝑤 − V𝐻,ℎ𝑚+1 󵄩󵄩󵄩 sup ≤ 𝐶𝑖 𝜂̃ (𝐻) inf 󵄩 󵄩1 𝑤∈𝑀(𝜆 ),‖𝑤‖ =1V𝐻,ℎ𝑚+1 ∈𝑉𝐻,ℎ𝑚+1 ∗

𝑖

0

≤ 𝐶𝑖 𝜂̃∗ (𝐻) 𝜀ℎ𝑚+1 (𝜆 𝑖 ) , (66)

sup

inf

𝑓∈𝑊,‖𝑓‖0 =1V∈𝑉𝐻,ℎ𝑚+1

󵄩󵄩 ∗ 󵄩 ∗ 󵄩󵄩𝑇 𝑓 − V󵄩󵄩󵄩1 ≤ 𝜂 (𝐻) .

(67)

From (66) and (67), we can obtain (46) and (48). Similarly, we can deduce that, for any 𝑤∗ ∈ 𝑀∗ (𝜆∗𝑖 ), ∗ ‖𝑤 ‖0 = 1, 𝑖+𝑞−1

𝑤∗ = ∑ 𝛾𝑗∗ 𝑢𝑗∗ ,

(68)

𝑗=𝑖

and there exists a constant 𝐶5∗ independent of ℎ𝑚 such that 󵄨󵄨 ∗ 󵄨󵄨 󵄨󵄨𝛾𝑗 󵄨󵄨 ≤ 𝐶5∗ , 󵄨 󵄨

𝑗 = 𝑖, 𝑖 + 1, . . . , 𝑖 + 𝑞 − 1.

󵄩 󵄩󵄩 󵄩󵄩𝑇𝑓 − V󵄩󵄩󵄩1 ≤ 𝜂 (𝐻) .

(72)

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 𝑉𝐻 = 𝑉ℎ1 ⊂ 𝑉ℎ2 ⊂ ⋅ ⋅ ⋅ ⊂ 𝑉ℎ𝑛 which are defined in Section 3 have the following relations: 1 𝛿ℎ𝑚+1 (𝜆 𝑖 ) ≈ 𝛿ℎ𝑚 (𝜆 𝑖 ) , 𝜉

(69)

According to the equalities (13), (20), and (40), inequalities (5), (53), and (54), and the error estimate of finite element projection 𝑃ℎ∗𝑚+1 , we can likewise have 𝐶 𝐶 󵄩󵄩 ∗ 󵄩 󵄩󵄩𝛼𝑗 𝑢̂𝑗,ℎ − 𝑢𝑗∗ 󵄩󵄩󵄩 ≤ 4 󵄨󵄨󵄨󵄨𝜆∗𝑖 󵄨󵄨󵄨󵄨 𝜂 (𝐻) 𝜀ℎ∗ (𝜆∗𝑖 ) + 1 𝛿ℎ∗ (𝜆∗𝑖 ) . 𝑚+1 𝑚 󵄩 󵄩1 𝑐1 𝑐1 𝑚+1 (70)

(73)

𝛿ℎ𝑚 (𝜆 𝑖 ) ≤ 𝐶6 𝜉𝛿ℎ𝑚+1 (𝜆 𝑖 ) , 1 𝛿ℎ∗𝑚+1 (𝜆∗𝑖 ) ≈ 𝛿ℎ∗𝑚 (𝜆∗𝑖 ) , 𝜉 𝛿ℎ∗𝑚

where 𝜂̃∗ (𝐻) =

inf

𝑓∈𝑊,‖𝑓‖0 =1V∈𝑉𝐻,ℎ𝑚+1

4. Multilevel Scheme for the Eigenvalue Problem

𝐶4 󵄨󵄨 󵄨󵄨 ∗ 𝐶 󵄨󵄨𝜆 𝑖 󵄨󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) + 1 𝛿ℎ𝑚+1 (𝜆 𝑖 )) , 𝑐1 𝑐1

sup

sup

From (71) and (72), we can obtain (47) and (49). The estimate (50) can be derived by (36), (46), and (47), and the proof concludes.

Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑚+1 (𝜆 𝑖 )) ≤ 𝐶𝑖

𝐶4 󵄨󵄨 ∗ 󵄨󵄨 𝐶 ∗ ∗ ∗ ∗ 󵄨𝜆 󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) + 1 𝛿ℎ𝑚+1 (𝜆 𝑖 )) , 𝑐1 󵄨 𝑖 󵄨 𝑐1

(𝜆∗𝑖 )

≤

𝐶6∗ 𝜉𝛿ℎ∗𝑚+1

(74)

(𝜆∗𝑖 ) ,

for 𝑚 = 1, . . . , 𝑛 − 1, where 𝐶6 and 𝐶6∗ are two positive constants independent of 𝑚 and 𝑛. Remark 6. The relations (73) and (74) are obviously reasonable, because we can choose 𝛿ℎ𝑚 (𝜆 𝑖 ) ≈ ℎ𝑚 , 𝛿ℎ∗𝑚 (𝜆∗𝑖 ) ≈ ℎ𝑚 (𝑚 = 1, . . . , 𝑛). Invariably the upper bound of the estimates 𝛿ℎ𝑚 (𝜆 𝑖 ) ≤ 𝐶7 ℎ𝑚 , 𝛿ℎ∗𝑚 (𝜆∗𝑖 ) ≤ 𝐶7∗ ℎ𝑚 (𝑚 = 1, . . . , 𝑛) holds. We also can obtain the lower bound 𝛿ℎ𝑚 (𝜆 𝑖 ) ≥ 𝑐7 ℎ𝑚 , 𝛿ℎ∗𝑚 (𝜆∗𝑖 ) ≥ 𝑐7∗ ℎ𝑚 (𝑚 = 1, . . . , 𝑛) (see [20]); 𝑐7 , 𝐶7 , 𝑐7∗ , 𝐶7∗ are some positive constants independent of ℎ𝑚 . Algorithm 7 (multilevel scheme). Implement the following. Step 1. Construct a sequence of nested finite element spaces 𝑉ℎ1 , 𝑉ℎ2 , . . . , 𝑉ℎ𝑛 such that (38), (73), and (74) hold. Step 2. Solve the following eigenvalue problems. Find (𝜆 𝑗,ℎ1 , 𝑢𝑗,ℎ1 ) ∈ C × 𝑉ℎ1 such that ‖𝑢𝑗,ℎ1 ‖0 = 1 and 𝑎 (𝑢𝑗,ℎ1 , Vℎ1 ) = 𝜆 𝑗,ℎ1 𝑏 (𝑢𝑗,ℎ1 , Vℎ1 ) ,

∀Vℎ1 ∈ 𝑉ℎ1 .

(75)

Mathematical Problems in Engineering

7

∗ ∗ Find (𝜆∗𝑗,ℎ1 , 𝑢𝑗,ℎ ) ∈ C × 𝑉ℎ1 such that ‖𝑢𝑗,ℎ ‖ = 1 and 1 1 0 ∗ ∗ 𝑎 (Vℎ1 , 𝑢𝑗,ℎ ) = 𝜆∗𝑗,ℎ 𝑏 (Vℎ1 , 𝑢𝑗,ℎ ), 1 1 1

∀Vℎ1 ∈ 𝑉ℎ1 .

(76)

Step 3. For 𝑚 = 1, . . . , 𝑛 − 1, compute new eigenpair 𝑖+𝑞−1 𝑐 approximations {𝜆𝑐𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ } ∈ C × 𝑉𝐻,ℎ𝑚+1 , {𝜆𝑐∗ 𝑗,ℎ𝑚+1 , 𝑚+1 𝑗=𝑖 𝑖+𝑞−1 𝑐∗ } 𝑢𝑗,ℎ 𝑚+1 𝑗=𝑖

∈ C × 𝑉𝐻,ℎ𝑚+1 by Algorithm 3:

𝑐 𝑐∗ , 𝑢𝑗,ℎ } {𝜆𝑐𝑗,ℎ𝑚+1 , 𝑢𝑗,ℎ 𝑚+1 𝑚+1

𝑖+𝑞−1 𝑗=𝑖

𝑐 𝑐∗ = Correction (𝑉𝐻, {𝜆𝑐𝑗,ℎ𝑚 , 𝑢𝑗,ℎ , 𝑢𝑗,ℎ } 𝑚 𝑚

𝑖+𝑞−1 𝑗=𝑖

, 𝑉ℎ𝑚+1 ) . (77)

Finally, we obtain 𝑞 eigenpair approximations 𝑖+𝑞−1 𝑐 𝑐 𝑐∗ 𝑖+𝑞−1 {𝜆 𝑗,ℎ𝑛 , 𝑢𝑗,ℎ } ∈ C × 𝑉𝐻,ℎ𝑛 , {𝜆𝑐∗ ∈ C × 𝑉𝐻,ℎ𝑛 , 𝑗,ℎ𝑛 , 𝑢𝑗,ℎ𝑛 }𝑗=𝑖 𝑛 𝑗=𝑖

Let 𝜀ℎ1 (𝜆 𝑖 ) := 𝛿ℎ1 (𝜆 𝑖 ) and 𝜀ℎ∗1 (𝜆∗𝑖 ) := 𝛿ℎ∗1 (𝜆∗𝑖 ). From (83)–(86) and Theorem 5 with 𝐶4 = 𝐶𝑖 , for 𝑚 = 1, . . . , 𝑛 − 1, we have 𝜀ℎ𝑚+1 (𝜆 𝑖 ) ≤ 𝑞𝐶5 (

𝐶𝑖 󵄨󵄨 󵄨󵄨 ∗ 𝐶 󵄨𝜆 󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) + 1 𝛿ℎ𝑚+1 (𝜆 𝑖 )) , 𝑐1 󵄨 𝑖 󵄨 𝑐1 (88)

𝜀ℎ∗𝑚+1 (𝜆∗𝑖 ) ≤ 𝑞𝐶5∗ (

𝐶𝑖 󵄨󵄨 ∗ 󵄨󵄨 𝐶 ∗ ∗ ∗ ∗ 󵄨𝜆 󵄨 𝜂 (𝐻) 𝜀ℎ𝑚 (𝜆 𝑖 ) + 1 𝛿ℎ𝑚+1 (𝜆 𝑖 )) . 𝑐1 󵄨 𝑖 󵄨 𝑐1 (89)

Then by recursive relation and based on the proof in Theorem 5, (73), (88), and 𝑞𝐶5 (𝐶𝑖 /𝑐1 )|𝜆 𝑖 |𝐶6 𝜉𝜂∗ (𝐻) ≤ 1, we have 𝜀ℎ𝑛 (𝜆 𝑖 ) ≤ 𝑞𝐶5 ( ≤ {𝑞𝐶5

and 𝜆𝑐𝑗,ℎ𝑛 = 𝜆𝑐∗ 𝑗,ℎ .

2

Theorem 8. After implementing Algorithm 7, if the mesh size 𝐻 is small enough such that 𝑞𝐶5 (𝐶𝑖 /𝑐1 )|𝜆 𝑖 |𝐶6 𝜉𝜂∗ (𝐻) ≤ 1, 𝑞𝐶5∗ (𝐶𝑖 /𝑐1 )|𝜆∗𝑖 |𝐶6∗ 𝜉𝜂(𝐻) ≤ 1 and the condition (A0) holds. 𝑖+𝑞−1 𝑐 } Then the resultant eigenpair approximations {𝜆𝑐𝑗,ℎ𝑛 , 𝑢𝑗,ℎ 𝑛 𝑗=𝑖 and

2 𝐶𝑖 󵄨󵄨 󵄨󵄨 ∗ 󵄨󵄨𝜆 𝑖 󵄨󵄨 𝜂 (𝐻)} 𝜀ℎ𝑛−2 (𝜆 𝑖 ) 𝑐1

+ (𝑞𝐶5 )

𝑛

𝑐∗ 𝑖+𝑞−1 {𝜆𝑐∗ 𝑗,ℎ𝑛 , 𝑢𝑗,ℎ𝑛 }𝑗=𝑖

𝐶𝑖 󵄨󵄨 󵄨󵄨 ∗ 𝐶 󵄨𝜆 󵄨 𝜂 (𝐻) 𝜀ℎ𝑛−1 (𝜆 𝑖 ) + 1 𝛿ℎ𝑛 (𝜆 𝑖 )) 𝑐1 󵄨 𝑖 󵄨 𝑐1

have the following error estimates:

+ 𝑞𝐶5 ≤

𝐶𝑖 𝑐1

󵄨󵄨 󵄨󵄨 𝐶1 ∗ 󵄨󵄨𝜆 𝑖 󵄨󵄨 𝜂 (𝐻) 𝛿ℎ𝑛−1 (𝜆 𝑖 ) 𝑐 1

𝐶1 𝛿 (𝜆 ) 𝑐1 ℎ𝑛 𝑖

𝑛−1 𝐶 󵄨 󵄨 𝐶1 {(𝑞𝐶5 𝑖 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 𝜂∗ (𝐻)) 𝛿ℎ1 (𝜆 𝑖 ) 𝑐1 𝑐1 𝑛

Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑛 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝐶8 𝛿ℎ𝑛 (𝜆 𝑖 ) , ∗

(78)

Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑛 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝐶8 𝜂 (𝐻) 𝛿ℎ𝑛 (𝜆 𝑖 ) ,

(79)

Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑛 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝐶8∗ 𝛿ℎ∗𝑛 (𝜆∗𝑖 ) ,

(80)

Φ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗𝑛 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝐶8∗ 𝜂 (𝐻) 𝛿ℎ∗𝑛 (𝜆∗𝑖 ) , 󵄨 󵄨󵄨 𝑐 󵄨󵄨𝜆 𝑗,ℎ − 𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶𝑖 𝐶8 𝐶8∗ 𝛿ℎ𝑛 (𝜆 𝑖 ) 𝛿ℎ∗ (𝜆∗𝑖 ) , 𝑛 󵄨 󵄨 𝑛

+ 𝑞𝐶5 ∑ (𝑞𝐶5 𝑚=2

≤

𝑛−𝑚 𝐶𝑖 󵄨󵄨 󵄨󵄨 ∗ 𝛿ℎ𝑚 (𝜆 𝑖 )} 󵄨󵄨𝜆 𝑖 󵄨󵄨 𝜂 (𝐻)) 𝑐1

𝑛−1 𝐶 𝐶1 󵄨 󵄨 {(𝑞𝐶5 𝑖 𝐶6 𝜉 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 𝜂∗ (𝐻)) 𝛿ℎ𝑛 (𝜆 𝑖 ) 𝑐1 𝑐1 𝑛

+ 𝑞𝐶5 ∑ (𝑞𝐶5

(81)

𝑚=1

(82)

𝑛−𝑚 𝐶𝑖 󵄨 󵄨 𝐶6 𝜉 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 𝜂∗ (𝐻)) 𝑐1

⋅ 𝛿ℎ𝑛 (𝜆 𝑖 ) }

∗

where 𝐶8 := (𝐶1 /𝑐1 ){1 + 𝑞𝐶5 /(1 − 𝑞𝐶5 𝐶6 𝜉|𝜆 𝑖 |𝜂 (𝐻)𝐶𝑖 /𝑐1 )} and 𝐶8∗ := (𝐶1 /𝑐1 ){1 + 𝑞𝐶5∗ /(1 − 𝑞𝐶5∗ 𝐶6∗ 𝜉|𝜆∗𝑖 |𝜂(𝐻)𝐶𝑖 /𝑐1 )}. Proof. At first, by Lemma 2, we have Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ1 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝛿ℎ1 (𝜆 𝑖 ) , Φ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ1 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝜂∗ (ℎ1 ) 𝛿ℎ1 (𝜆 𝑖 ) ≤ 𝐶𝑖 𝜂∗ (𝐻) 𝛿ℎ1 (𝜆 𝑖 ) , Θ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗1 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝛿ℎ∗1 (𝜆∗𝑖 ) , Φ (𝑀∗ (𝜆∗𝑖 ) , 𝑀ℎ∗1 (𝜆∗𝑖 )) ≤ 𝐶𝑖 𝜂 (ℎ1 ) 𝛿ℎ∗1 (𝜆∗𝑖 ) ≤ 𝐶𝑖 𝜂 (𝐻) 𝛿ℎ∗1 (𝜆∗𝑖 ) , 󵄨 󵄨󵄨 𝑐 󵄨󵄨𝜆 𝑗,ℎ − 𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶𝑖 𝛿ℎ1 (𝜆 𝑖 ) 𝛿ℎ∗ (𝜆∗𝑖 ) . 1 󵄨 󵄨 1

≤

𝑞𝐶5 𝐶1 {1 + } 𝛿 (𝜆 ) 󵄨 󵄨 󵄨 𝑐1 1 − 𝑞𝐶5 𝐶6 𝜉 󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 𝜂∗ (𝐻) 𝐶𝑖 /𝑐1 ℎ𝑛 𝑖

≤ 𝐶8 𝛿ℎ𝑛 (𝜆 𝑖 ) , (90)

(83) Θ (𝑀 (𝜆 𝑖 ) , 𝑀ℎ𝑛 (𝜆 𝑖 )) ≤ 𝐶𝑖 𝜀ℎ𝑛 (𝜆 𝑖 ) . (84) (85) (86) (87)

(91)

The 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 [14] and can prove that solving the eigenvalue problem needs nearly the same work as solving the corresponding boundary value problem for Algorithm 7.

8

Mathematical Problems in Engineering Table 1: Ω = (0, 1)2 , b = (1, 0)𝑇 .

ℎ √2/32 √2/64 √2/128 √2/256 √2/512

Dof 961 3969 16129 65025 261121

𝜆𝑐1,ℎ 20.034801 20.000601 19.992056 19.989921 19.989387

𝜆𝑐2,ℎ 49.798457 49.648112 49.610543 49.601152 49.598805

𝜆 1,ℎ 20.034800 20.000601 19.992056 19.989921 19.989387

Table 3: Ω = (0, 1)2 , b = (10, 0)𝑇 . 𝜆 2,ℎ 49.798418 49.648109 49.610543 49.601152 49.598805

𝜆𝑐2,ℎ 𝜆 1,ℎ 𝜆 2,ℎ Dof 𝜆𝑐1,ℎ ℎ √2/32 961 44.637275 74.100045 44.637249 74.097049 √2/64 3969 44.713643 74.284384 44.713642 74.284210 √2/128 16129 44.732812 74.332012 44.732812 74.332001 √2/256 65025 44.737609 74.344013 44.737609 74.344013 √2/512 261121 44.738809 74.347019 44.738809 74.347019

Table 2: Ω = (0, 1)2 , b = (3, 0)𝑇 . ℎ √2/32 √2/64 √2/128 √2/256 √2/512

Dof 961 3969 16129 65025 261121

𝜆𝑐1,ℎ

𝜆𝑐2,ℎ

22.019219 21.996691 21.991078 21.989676 21.989326

51.765675 51.639857 51.608476 51.600635 51.598675

𝜆 1,ℎ 22.019218 21.996691 21.991078 21.989676 21.989326

Table 4: Ω = (−1, 1)2 \ [0, 1]2 , b = (1, 0)𝑇 . 𝜆 2,ℎ 51.765637 51.639853 51.608475 51.600635 51.598675

ℎ Dof √2/32 2945 √2/64 12033 √2/128 48641 √2/256 195585 √2/512 784385

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: 𝜆 𝑖,ℎ : the 𝑖th finite element eigenvalue by solving the eigenvalue problem directly. 𝜆𝑐𝑖,ℎ : the 𝑖th finite element eigenvalue by multilevel correction method solving the eigenvalue problem. Example 10. Consider the convection-diffusion equation in Ω, 𝑢 = 0, on 𝜕Ω,

(92)

where Ω = (0, 1)2 and b = b(𝑥)=(𝑏1 , 𝑏2 )𝑇 . The eigenvalues of (92) are 𝜆 𝑖,𝑗 =

𝑏12 + 𝑏22 + 𝜋2 (𝑖2 + 𝑗2 ) , 4

𝜆𝑐6,ℎ 41.931339 41.782028 41.741203 41.729602 41.726147

𝜆 1,ℎ 9.921856 9.900929 9.893785 9.891239 —

𝜆 6,ℎ 41.931223 41.782007 41.741199 41.729601 —

Table 5: Ω = (−1, 1)2 \ [0, 1]2 , b = (3, 0)𝑇 .

5. Numerical Experiments

−Δ𝑢 + b ⋅ ∇𝑢 = 𝜆𝑢,

𝜆𝑐1,ℎ 9.921856 9.900929 9.893785 9.891239 9.890301

(93)

where 𝑖, 𝑗 ∈ N+ . The corresponding adjoint eigenvalue problem has eigenvalues 𝜆∗𝑖,𝑗 = 𝜆 𝑖,𝑗 (see [10]). We obtain that 𝜆 1 = (𝑏12 + 𝑏22 )/4 + 2𝜋2 and 𝜆 2 = 𝜆 3 = (𝑏12 + 𝑏22 )/4 + 5𝜋2 , so the multiplicity of 𝜆 2 is 2. For (92) with b = (1, 0)𝑇 , b = (3, 0)𝑇 , and b = (10, 0)𝑇 , 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 𝜆𝑐𝑖,ℎ and 𝜆 𝑖,ℎ is nearly the same. 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 (𝑏12 + 𝑏22 )/4 + 9.639724 and the sixth eigenvalue is (𝑏12 + 𝑏22 )/4 + 41.47451 (see [3]). From Tables 4–6 and Figure 2, we can get the same accurate

ℎ √2/32 √2/64 √2/128 √2/256 √2/512

Dof 2945 12033 48641 195585 784385

𝜆𝑐1,ℎ 11.913459 11.898825 11.893259 11.891108 11.890268

𝜆𝑐6,ℎ 43.897920 43.773604 43.739093 43.729074 43.726016

𝜆 1,ℎ 11.913459 11.898825 11.893259 11.891108 —

𝜆 6,ℎ 43.897831 43.773591 43.739091 43.729074 —

Table 6: Ω = (−1, 1)2 \ [0, 1]2 , b = (10, 0)𝑇 . ℎ √2/32 √2/64 √2/128 √2/256 √2/512

Dof 2945 12033 48641 195585 784385

𝜆𝑐1,ℎ 34.613575 34.636342 34.640137 34.640327 34.640073

𝜆𝑐6,ℎ 𝜆 1,ℎ 𝜆 6,ℎ 66.310273 34.613578 66.311082 66.439222 34.636341 66.439144 66.467962 34.640137 66.467962 66.473790 34.640327 66.473790 66.474694 — —

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)𝑇 ; 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 [21] and it

Mathematical Problems in Engineering

9

10−1 10−1

Error

Error

10−2

10−2

10−3

10−3 10−4 103

104 Number of degree of freedom

|𝜆c1,h − 𝜆1 | when b = [1; 0] |𝜆1,h − 𝜆1 | when b = [1; 0] |𝜆c1,h − 𝜆1 | when b = [3; 0] |𝜆1,h − 𝜆1 | when b = [3; 0]

105

103

|𝜆c1,h − 𝜆1 | when b = [10; 0] |𝜆1,h − 𝜆1 | when b = [10; 0] A slope with −1

104 Number of degree of freedom

|𝜆c2,h − 𝜆2 | when b = [10; 0] |𝜆2,h − 𝜆2 | when b = [10; 0] A slope with −0.97

|𝜆c2,h − 𝜆2 | when b = [1; 0] |𝜆2,h − 𝜆2 | when b = [1; 0] |𝜆c2,h − 𝜆2 | when b = [3; 0] |𝜆2,h − 𝜆2 | when b = [3; 0]

(a)

105

(b)

Figure 1: Ω = (0, 1) × (0, 1), error curves for the 1st (a) and 2nd (b) eigenvalues.

10−1

Error

Error

10−2

10−2

10−3

10−3

104

105 Number of degree of freedom

|𝜆c1,h − 𝜆1 | when b = [1; 0] |𝜆1,h − 𝜆1 | when b = [1; 0] |𝜆c1,h − 𝜆1 | when b = [3; 0] |𝜆1,h − 𝜆1 | when b = [3; 0] (a)

|𝜆c1,h − 𝜆1 | when b = [10; 0] |𝜆1,h − 𝜆1 | when b = [10; 0] A slope with −0.67

104

105 Number of degree of freedom

|𝜆c6,h − 𝜆6 | when b = [1; 0] |𝜆6,h − 𝜆6 | when b = [1; 0] |𝜆c6,h − 𝜆6 | when b = [3; 0] |𝜆6,h − 𝜆6 | when b = [3; 0]

|𝜆c6,h − 𝜆6 | when b = [10; 0] |𝜆6,h − 𝜆6 | when b = [10; 0] A slope with −0.84

(b)

Figure 2: Ω = (−1, 1)2 \ [0, 1]2 , error curves for the 1st (a) and 6th (b) eigenvalues.

10 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.

Acknowledgments 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. LH[2014]7061 and LKS[2013]06).

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Mathematical Problems in Engineering [10] A. Naga and Z. Zhang, “Function value recovery and its application in eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 50, no. 1, pp. 272–286, 2012. [11] T. L¨u and Y. Feng, “Splitting extrapolation based on domain decomposition for finite element approximations,” Science in China Series E: Technological Sciences, vol. 40, no. 2, pp. 144–155, 1997. [12] Y. Yang, H. Bi, and S. Li, “The extrapolation of numerical eigenvalues by finite elements for differential operators,” Applied Numerical Mathematics, vol. 69, pp. 59–72, 2013. [13] Q. Lin and H. Xie, “A multi-level correction scheme for eigenvalue problems,” Mathematics of Computation, vol. 84, no. 291, pp. 71–88, 2015. [14] H. Xie, “A multigrid method for eigenvalue problem,” Journal of Computational Physics, vol. 274, pp. 550–561, 2014. [15] H. Xie, “A type of multilevel method for the Steklov eigenvalue problem,” IMA Journal of Numerical Analysis, vol. 34, no. 2, pp. 592–608, 2014. [16] X. Ji, J. Sun, and H. Xie, “A multigrid method for Helmholtz transmission eigenvalue problems,” Journal of Scientific Computing, vol. 60, no. 2, pp. 276–294, 2014. [17] L. Chen, Programming of Finite Element Methods in MATLAB, 2010, http://www.math.uci.edu/∼chenlong/226/Ch3FEMCode .pdf. [18] P. G. Ciarlet, “Basic error estimates for elliptic proplems,” in Finite Element Methods (Part1), Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 21–343, Elsevier Science Publishers, North Holland, The Netherlands, 1991. [19] J. Xu and A. Zhou, “A two-grid discretization scheme for eigenvalue problems,” Mathematics of Computation, vol. 70, no. 233, pp. 17–25, 2001. [20] Q. Lin, H. Xie, and J. Xu, “Lower bounds of the discretization error for piecewise polynomials,” Mathematics of Computation, vol. 83, no. 285, pp. 1–13, 2014. [21] J. Xu and A. Zhou, “Local and parallel finite element algorithms based on two-grid discretizations,” Mathematics of Computation, vol. 69, no. 231, pp. 881–909, 2000. [22] X. Dai and A. Zhou, “Three-scale finite element discretizations for quantum eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 46, no. 1, pp. 295–324, 2007/08. [23] H. Bi, Y. Yang, and H. Li, “Local and parallel finite element discretizations for eigenvalue problems,” SIAM Journal on Scientific Computing, vol. 35, no. 6, pp. A2575–A2597, 2013. [24] Y. Yang and J. Han, “The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem,” Computer Methods in Applied Mechanics and Engineering, vol. 289, pp. 249–266, 2015.

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