Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method

Applied Numerical Mathematics 59 (2009) 1884–1893

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Applied Numerical Mathematics www.elsevier.com/locate/apnum

Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method ✩ Qun Lin, Hehu Xie ∗ LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 17 April 2007 Received in revised form 17 January 2009 Accepted 26 January 2009 Available online 11 February 2009

The paper provides a general procedure or method to produce asymptotic error expansion for the eigenvalue approximations of second order elliptic problems by the mixed finite element method. We obtain a transform lemma for the error of the eigenvalue approximations. As an application of the transform lemma, the asymptotic error expansion of the eigenvalue approximations for the second order elliptic problem by the lowest order Raviart–Thomas mixed finite element method is given by means of integral identity technique. Based on such an error expansion, Richardson extrapolation technique is applied to improve the accuracy of the eigenvalue approximations. © 2009 IMACS. Published by Elsevier B.V. All rights reserved.

MSC: 65N30 65N25 65L15 65B99 Keywords: Second order elliptic eigenvalue problem Mixed finite element method Asymptotic error expansion Richardson extrapolation Raviart–Thomas element

1. Introduction In this paper, we are concerned with the extrapolation method for the eigenvalue problems by the mixed finite element method. As we know, the mixed finite element method is a very important method in computing the numerical solutions of the partial differential equations. So far, there have been many works about this method. We mention the references [5,6,11,13,26–28] and [29]. Eigenvalue problems are very important in engineering society. In [1,2], Babuška and Osborn discuss many eigenvalue problems and the corresponding numerical methods (see also [4,7,30]). Here we are concerned with the following eigenvalue problem: Find λ, p such that

−∇ · ( A ∇ p ) + ϕ p = λρ p , A ∇ p · n = 0,

in Ω,

on ∂Ω,

(1.1) (1.2)

✩ This project was supported in part by the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300). Corresponding author. E-mail addresses: [email protected] (Q. Lin), [email protected] (H. Xie).

*

0168-9274/$30.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2009.01.011

Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

1885



ρ p 2 dΩ = 1,

(1.3)

Ω

¯ , C  ρ > c 0 > 0 on Ω¯ , ∇ and ∇· are gradient and where A = (ai j )2×2 is a symmetric positive definite matrix, ϕ > 0 in Ω divergence operator and Ω is a polygonal domain in R2 . The condition (1.3) just normalize the eigenfunction. In this paper, C , c and c 0 denote constants independent of the mesh size. The mixed finite element method [6,26,29,31] is employed to discretize the eigenvalue problem (1.1)–(1.3). There are many works for studying the numerical solution of eigenvalue problems by the mixed finite element method, see [4,14,16,17] and [31], and references cited therein. Our object of this paper is to present how to produce the asymptotic error expansion of eigenvalue approximations by the mixed finite element method. Based on the expansion, the extrapolation method can be applied to improve the accuracy of eigenvalue approximations. It is well known that the extrapolation method is an efficient procedure to increase the accuracy of approximations for many problems in numerical analysis [3,24,25]. If the asymptotic expansion has been obtained, the extrapolation method can be used to improve the accuracy of the numerical approximations. This technique has been well demonstrated in its application to the finite element methods [23,27]. The application of the extrapolation method to the eigenvalue problems was first proposed by Q. Lin and T. Lü [22], and was analyzed in [20,21,23] and [27]. So far, there are many results about the expansions and extrapolation for the eigenvalue problems by Galerkin finite element methods. To the best of our knowledge, this is the first analysis for the expansion and extrapolation for second order elliptic problems by the mixed finite element method. In this paper, we employ the sharp integral estimates [15,21,23] to derive asymptotic error expansion of the eigenvalue approximations by the mixed finite element method. Based on the expansion, we adopt extrapolation method to increase their accuracy. For simplicity, we only consider simple eigenvalues in this paper. For the extrapolation of multiple eigenvalues, we refer readers to [27]. The rest of this paper is organized in the following way. In Section 2, we introduce the mixed finite element method for the second order elliptic eigenvalue problem and derive the corresponding transform lemma for the eigenvalue approximations. In Section 3, as an application of the transform lemma, an asymptotic error expansion of the eigenvalue approximations by the lowest order Raviart–Thomas mixed finite element method is given. Based on the error expansion, extrapolation method is applied to improve the accuracy of the eigenvalue approximations. Some numerical results are given in Section 4 and Section 5 gives some concluding remarks. Throughout this article we use the standard notations as in Ciarlet [12], for example, the notations of the Sobolev space, inner product, norms, seminorms and discrete norms. 2. The mixed finite element method and the transform lemma In this section, we introduce the mixed finite element method to obtain the approximations of the second order elliptic eigenvalue problem (1.1)–(1.3) and give the corresponding transform lemma for the error of the eigenvalue approximations. For the sake of simplicity, we assume throughout this paper that the domain Ω ⊂ R2 is a rectangle. Nevertheless, we can find the transform lemma can be applied to general domain even with curved boundaries. We define a new vector-valued function u as follows u = − A∇ p. Then (1.1)–(1.3) can be transformed into the following equivalent formulation A −1 u + ∇ p = 0,

in Ω,

∇ · u + ϕ p = λρ p , u · n = 0,



(2.1)

in Ω,

(2.2)

on ∂Ω,

(2.3)

ρ p 2 dΩ = 1.

(2.4)

Ω

Let W := L 2 (Ω)

and





2



V := H(div, Ω) = v ∈ L 2 (Ω) : ∇ · v ∈ L 2 (Ω)

be the standard L 2 space on Ω with norm  · 0 and the Hilbert space equipped with the norm

 1/2 vV := v20 + ∇ · v20 , respectively. In addition, we set





V0 := v ∈ V: v · n = 0 on ∂Ω . The corresponding weak formulation for the problem (1.1)–(1.3) seeks (λ, u, p ) ∈ R × V0 × W such that p = 0 and

1886

Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

a(u, v) − (∇ · v, p ) = 0,

∀v ∈ V0 ,

(2.5)

(∇ · u, w ) + (ϕ p , w ) = λ(ρ p , w ),  ρ p 2 dΩ = 1,

∀w ∈ W ,

(2.6) (2.7)

Ω

where a(·,·) is a bilinear form defined by



a(u, v) =

A −1 u · v dΩ

Ω

and (·,·) denotes the standard L 2 -inner product. Let Th1 ,h2 be a finite element partition of Ω into uniform rectangles, where h1 and h2 are the mesh sizes in x- and y-axis, respectively. And in this paper we set h := max{h1 , h2 }. Let Vh1 ,h2 × W h1 ,h2 ⊂ V × W denote a pair of finite element spaces satisfying the Babuška–Brezzi condition. In addition, we set V0,h1 ,h2 = {v ∈ Vh1 ,h2 : v · n = 0 on ∂Ω}. Hence, the corresponding discrete mixed finite element version of (2.5)–(2.7) seeks a pair (λh1 ,h2 , uh1 ,h2 , p h1 ,h2 ) ∈ R × V0,h1 ,h2 × W h1 ,h2 such that p h1 ,h2 = 0 and a(uh1 ,h2 , v) − (∇ · v, p h1 ,h2 ) = 0,

∀v ∈ V0,h1 ,h2 ,

(∇ · uh1 ,h2 , w ) + (ϕ ph1 ,h2 , w ) = λh1 ,h2 (ρ ph1 ,h2 , w ),  ρ ph21 ,h2 dΩ = 1.

(2.8)

∀ w ∈ W h1 ,h2 ,

(2.9) (2.10)

Ω

˜ h1 ,h2 , p˜ h1 ,h2 ) ∈ V0,h1 ,h2 × W h1 ,h2 such that Moreover in this paper, we need to define the mixed finite element projection (u ˜ h1 ,h2 , v) − (∇ · v, p˜ h1 ,h2 ) = a(u, v) − (∇ · v, p ), a(u

∀v ∈ V0,h1 ,h2 ,

(∇ · u˜ h1 ,h2 , w ) + (ϕ p˜ h1 ,h2 , w ) = (∇ · u, w ) + (ϕ p , w ),

∀ w ∈ W h1 ,h2 .

(2.11) (2.12)

From (2.11), (2.12), we can obtain the following mixed finite element error equations:

˜ h1 ,h2 , v) − (∇ · v, p − p˜ h1 ,h2 ) = 0, a(u − u

∀v ∈ V0,h1 ,h2 ,     ˜ ˜ ∇ · (u − uh1 ,h2 ), w + ϕ ( p − ph1 ,h2 ), w = 0, ∀ w ∈ W h1 ,h2 .

(2.13) (2.14)

Let (Ih1 ,h2 , J h1 ,h2 ) denote an interpolation operator corresponding to the mixed finite element space Ih1 ,h2 × J h1 ,h2 : V × W −→ Vh1 ,h2 × W h1 ,h2 and we assume the interpolation satisfies the following assumptions [13] in this paper: (i) J h1 ,h2 p is the local L 2 (Ω) projection of p; (ii) (Ih1 ,h2 u, J h1 ,h2 p ) and (u, p ) satisfy



 ∇ · (u − Ih1 ,h2 u), w = 0,

(∇ · v, p − J h1 ,h2 p ) = 0,

∀ w ∈ W h1 ,h2 , ∀v ∈ Vh1 ,h2 .

(2.15)

Here, we just give the assumptions for the interpolation operator that are needed for deriving the following transform lemma for the eigenvalue approximations. Of course, there exists some kinds of interpolations (e.g. Raviart–Thomas and BDFM) that satisfy the assumptions (i) and (ii). Since the orthogonality (2.13) and (2.14) does not hold for eigenfunction (uh1 ,h2 , p h1 ,h2 ), we need a transform formula to decentralize the eigenvalue error into the weak form of interpolation error. In [20] and [21], there are transform lemmas for the error of the eigenvalue approximations by Galerkin finite element methods. Here, we give the transform lemma for the error of eigenvalue approximations by the mixed finite element method. Lemma 2.1. For the eigenvalue problems (2.5)–(2.7) and (2.8)–(2.10), there exists the following error formula

    λh1 ,h2 − λ = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p¯ h1 ,h2 − ϕ ( p − J h1 ,h2 p ), p¯ h1 ,h2 + a(u − Ih1 ,h2 u, u¯ h1 ,h2 ),

(2.16)

where (u, p ) and (Ih1 ,h2 u, J h1 ,h2 p ) are the exact solution and the mixed finite element interpolant of (u, p ), which satisfies the prop¯ h1 ,h2 , p¯ h1 ,h2 ) are defined as follows erties (i) and (ii), and (u

Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

¯ h1 ,h2 = u

uh1 ,h2

(ρ p , ph1 ,h2 )

,

p¯ h1 ,h2 =

p h1 ,h2

(ρ p , ph1 ,h2 )

1887

(2.17)

.

Proof. First we have

(ρ p , p¯ h1 ,h2 ) = 1. So

  λh1 ,h2 = λh1 ,h2 (ρ p , p¯ h1 ,h2 ) = λh1 ,h2 (ρ p˜ h1 ,h2 , p¯ h1 ,h2 ) + λh1 ,h2 ρ ( p − p˜ h1 ,h2 ), p¯ h1 ,h2 .

(2.18)

From (2.5)–(2.7), (2.8)–(2.10), (2.11), (2.12) and the symmetry of A, we have

λh1 ,h2 (ρ p˜ h1 ,h2 , p¯ h1 ,h2 ) = (∇ · u¯ h1 ,h2 , p˜ h1 ,h2 ) + (ϕ p¯ h1 ,h2 , p˜ h1 ,h2 ) = a(u˜ h1 ,h2 , u¯ h1 ,h2 ) + (ϕ p¯ h1 ,h2 , p˜ h1 ,h2 ) = (∇ · u˜ h1 ,h2 , p¯ h1 ,h2 ) + (ϕ p˜ h1 ,h2 , p¯ h1 ,h2 ) = λ(ρ p , p¯ h1 ,h2 ) = λ.

(2.19)

Then let’s consider λh1 ,h2 (ρ ( p − p˜ h1 ,h2 ), p¯ h1 ,h2 ). From (2.8)–(2.15) and the symmetry of A, we have

      λh1 ,h2 ρ ( p − p˜ h1 ,h2 ), p¯ h1 ,h2 = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p¯ h1 ,h2 + λh1 ,h2 ρ ( J h1 ,h2 p − p˜ h1 ,h2 ), p h1 ,h2   = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p h1 ,h2 + (∇ · u¯ h1 ,h2 , J h1 ,h2 p − p˜ h1 ,h2 ) + (ϕ p¯ h1 ,h2 , J h1 ,h2 p − p˜ h1 ,h2 )   = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p¯ h1 ,h2 + (∇ · u¯ h1 ,h2 , p − p˜ h1 ,h2 ) + (ϕ p¯ h1 ,h2 , J h1 ,h2 p − p ) + (ϕ p¯ h1 ,h2 , p − p˜ h1 ,h2 )     = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p¯ h1 ,h2 − ϕ ( p − J h1 ,h2 p ), p¯ h1 ,h2   + a(u − u˜ h1 ,h2 , u¯ h1 ,h2 ) − ∇ · (u − u˜ h1 ,h2 ), p h1 ,h2     = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p h1 ,h2 − ϕ ( p − J h1 ,h2 p ), p¯ h1 ,h2 + a(u − Ih1 ,h2 u, u¯ h1 ,h2 ) + a(Ih1 ,h2 u − u˜ h1 ,h2 , u¯ h1 ,h2 )     − ∇ · (u − Ih1 ,h2 u), p¯ h1 ,h2 − ∇ · (Ih1 ,h2 u − u˜ h1 ,h2 ), p¯ h1 ,h2     = λh1 ,h2 ρ ( p − J h1 ,h2 p ), p h1 ,h2 − ϕ ( p − J h1 ,h2 p ), p¯ h1 ,h2 + a(u − Ih1 ,h2 u, u¯ h1 ,h2 ). From (2.18), (2.19) and (2.20), we can obtain (2.16) and the proof is completed.

(2.20)

2

Lemma 2.1 (called transform lemma) decentralize the eigenvalue error into three interpolation errors. Combining the results of interpolation expansion [15,21,23], we can obtain the asymptotic error expansions of the eigenvalue approximations by the mixed finite element method. 3. The asymptotic error expansion for the lowest order Raviart–Thomas mixed finite element and eigenvalue extrapolation As an application of the transform lemma, the aim of this section is to get an asymptotic error expansion of the eigenvalue approximations by the lowest order Raviart–Thomas mixed finite element method on the rectangular mesh. Nevertheless, the transform lemma can be applied to other mixed finite element methods. In order to obtain the asymptotic error expansion of the eigenvalue approximations, we use the interpolation expansions which come from [15] (Theorem 3.1 and 3.4). Here we consider the Raviart–Thomas space of the lowest order, i.e.,





Vh1 ,h2 := v ∈ V: v|e ∈ Q 1,0 (e ) × Q 0,1 (e ), e ∈ Th1 ,h2 ,





W h1 ,h2 := w ∈ W : w |e ∈ Q 0,0 (e ), e ∈ Th1 ,h2 , where Q m,n (e ) indicates the space of polynomials of degree no more than m and n in x and y on e, respectively. Let us define Raviart–Thomas interpolation operator (Ih1 ,h2 , J h1 ,h2 ) Ih1 ,h2 × J h1 ,h2 : V × W −→ Vh1 ,h2 × W h1 ,h2 by the following conditions:

(3.1)

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Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

 (u − Ih1 ,h2 u) · ni ds = 0,

i = 1, 2, 3, 4,

si

 ( p − J h1 ,h2 p ) de = 0,

(3.2)

e

where si (i = 1, 2, 3, 4) are the four edges of the rectangle e and ni is the outward normal direction on si . ˜ h1 ,h2 , p˜ h1 ,h2 ) [1,4,13]: It is known that we have the following convergence rate for (λh1 ,h2 , uh1 ,h2 , p h1 ,h2 ) and (u

|λh1 ,h2 − λ|  ch2 ,

(3.3)

u˜ h1 ,h2 − uV +  p˜ h1 ,h2 − p 0 + uh1 ,h2 − uV +  ph1 ,h2 − p 0  ch p 3 .

(3.4)

As we know, the Raviart–Thomas interpolation defined by (3.2) satisfies the assumptions (i) and (ii). Then we can use the result of Lemma 2.1. Theorem 3.1. Let us define

M 2 (α , u, v) =



1

M 1 (α , u, v) = −



3



1



α11 (u 1 )xx + α12 (u 2 )xx v 1 dΩ +

1 3

Ω

3

N 1 (ϕ , p , w ) =

  (α22 )x (u 2 )x − α21 (u 1 )xx v 2 dΩ,

Ω





(α11 ) y (u 1 ) y − α12 (u 2 ) y y v 1 dΩ −

1



3

Ω

1







α22 (u 2 ) y y + α21 (u 1 ) y y v 2 dΩ,

Ω



ϕx p x w dΩ,

3 Ω

N 2 (ϕ , p , w ) =

1



ϕ y p y w dΩ.

3 Ω

Assume u ∈ V ∩ ( H (Ω))2 , p ∈ H 2 (Ω), A −1 = (αi j )2×2 and αi j ∈ W 2,∞ (Ω) (1  i , j  2), ρ , ϕ ∈ W 2,∞ (Ω), we have the following error expansion 3

λh1 ,h2 − λ = λh21 N 1 (ρ , p , p ) + λh22 N 2 (ρ , p , p ) − h21 N 1 (ϕ , p , p )

  − h22 N 2 (ϕ , p , p ) + h21 M 1 (α , u, u) + h22 M 2 (α , u, u) + O h3 .

(3.5)

Proof. First from (3.4) 1 − (ρ p , p h1 ,h2 ) =

=

 (ρ p , p ) − 2(ρ p , ph1 ,h2 ) + (ρ ph1 ,h2 , ph1 ,h2 )

1 2 1 2

ρ ( p − ph1 ,h2 ), p − ph1 ,h2



  = O h2 . Then we have

   ph1 ,h2 − p¯ h1 ,h2 0 = O h2 .

(3.6)

Moreover, from (2.16), (3.3) and the following interpolation error expansions which come from [15] (Theorem 3.1 and 3.4),









ρ ( p − J h1 ,h2 p ), p¯ h1 ,h2 = h21 N 1 (ρ , p , p ) + h22 N 2 (ρ , p , p ), ϕ ( p − J h1 ,h2 p ), p¯ h1 ,h2 = h21 N 1 (ϕ , p , p ) + h22 N 2 (ϕ , p , p ),

¯ h1 ,h2 ) = h21 M 1 (α , u, u) + h22 M 2 (α , u, u), a(u − Ih1 ,h2 u, u we can easily obtain (3.5).

2

Remark 3.1. Noting that we just need remainder term O(h3 ) (not O(h4 )), we can relax the regularity requirement for the eigenfunction and the coefficient matrix A.

Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

1889

After obtaining the asymptotic error expansion, we can use the extrapolation method to improve the accuracy of the eigenvalue approximations. Because eigenvalue is just a number, the extrapolation for the eigenvalue does not need interpolation postprocessing and this is different with the solution extrapolation [8,15,18,21,23]. So, the eigenvalue extrapolation is easier for implementation. Here, we use two types of extrapolation methods: global extrapolation and one direction extrapolation (splitting extrapolation) [15,18]. In order to use the global extrapolation method, we need construct the refined mesh Th1 /2,h2 /2 by dividing each element e i ∈ Th1 ,h2 into four small congruent elements e i , j ( j = 1, 2, 3, 4). The corresponding eigenpair approximation is denoted by (λh1 /2,h2 /2 , uh1 /2,h2 /2 , ph1 /2,h2 /2 ) ∈ R × V0,h1 /2,h2 /2 × W h1 /2,h2 /2 . Theorem 3.2. Assume u ∈ V ∩ ( H 3 (Ω))2 , p ∈ H 2 (Ω), A −1 = (αi j )2×2 and have 4λh1 /2,h2 /2 − λh1 ,h2 3

αi j ∈ W 2,∞ (Ω) (1  i , j  2), ρ , ϕ ∈ W 2,∞ (Ω), we

  = λ + O h3 .

(3.7)

Then

λ − λh1 /2,h2 /2 =

λh1 /2,h2 /2 − λh1 ,h2 3

  + O h3

(3.8)

provides an a posteriori error estimate (λh1 /2,h2 /2 − λh1 ,h2 )/3 for λ − λh1 /2,h2 /2 . Proof. First, from (3.5), we can obtain the extrapolation scheme (3.7). (3.8) is a direct consequence of (3.7).

2

As in [15] and [18], we can also use the one direction extrapolation (splitting extrapolation) method with partial refined meshes. Thus, this extrapolation method is more efficient and more suitable for parallel computations. Let (λh1 /2,h2 , uh1 /2,h2 , p h1 /2,h2 ) ∈ R × V0,h1 /2,h2 × W h1 /2,h2 and (λh1 ,h2 /2 , uh1 ,h2 /2 , p h1 ,h2 /2 ) ∈ R × V0,h1 ,h2 /2 × W h1 ,h2 /2 be the eigenpair approximations corresponding to meshes Th1 /2,h2 and Th1 ,h2 /2 , respectively. Theorem 3.3. Under the condition of Theorem 3.2, we have 4(λh1 /2,h2 + λh1 ,h2 /2 ) − 5λh1 ,h2 3

  = λ + O h3 .

(3.9)

Then

λ − λh1 /2,h2 = λ − λh1 ,h2 /2 =

λh1 /2,h2 + 4λh1 ,h2 /2 − 5λh1 ,h2 3 4λh1 /2,h2 + λh1 ,h2 /2 − 5λh1 ,h2 3

  + O h3 ,

(3.10)

  + O h3

(3.11)

provide a posteriori error estimates (λh1 /2,h2 + 4λh1 ,h2 /2 − 5λh1 ,h2 )/3 and (4λh1 /2,h2 + λh1 ,h2 /2 − 5λh1 ,h2 )/3 for λ − λh1 /2,h2 and λ − λh1 ,h2 /2 , respectively. Generally speaking, the cost of computing λh1 /2,h2 and λh1 ,h2 /2 is cheaper than that of computing λh1 /2,h2 /2 . Hence, the computation and storage can be saved. In addition, compared with (3.7), (3.9) is easy to compute in a parallel manner, so that this method is more efficient than the global method described in (3.7), especially for high dimensional problems. Remark 3.2. Our numerical results indicate that 4λh1 /2,h2 /2 − λh1 ,h2 3

  = λ + O h4 .

(3.12)

We conjecture that there exists the following eigenvalue error expansion

λh1 ,h2 − λ = λh21 N 1 (ρ , p , p ) + λh22 N 2 (ρ , p , p ) − h21 N 1 (ϕ , p , p )

  − h22 N 2 (ϕ , p , p ) + h21 M 1 (α , u, u) + h22 M 2 (α , u, u) + O h4 .

And based on the expansion, we can obtain the extrapolation formula (3.12).

(3.13)

1890

Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

Table 1 Approximation for the fourth eigenvalue of the model problem. M×N

2×2

4×4

λh1 ,h2

23.999999999

20.773284010

19.994161313

19.802707357

19.755068235

λhextra 1 ,h 2

–

19.697712014

19.734453747

19.738889372

19.739188528

4.2608e+000

errh errhextra

–

Rh

–

R hextra

–

8×8

16 × 16

32 × 32

1.0341e+000

2.5495e−001

6.3499e−002

1.5859e−002

−4.1497e−002

−4.7551e−003

−3.1943e−004

−2.0274e−005

2.042780237

2.020040668

2.005432885

2.001382570

3.125465545

3.895887743

3.977774816

–

4. Numerical results In this section we give some numerical examples to test the theoretical analysis. In order to illustrate the convergence rate, we should define some notations

λhextra = 1 ,h 2

4λh1 /2,h2 /2 − λh1 ,h2 3

,

errh = |λh1 ,h2 − λ|, errhextra = λhextra − λ, 1 ,h 2 Rh =

log(errh /errh/2 ) log(2)

R hextra =

,

log(errextra /errextra ) h h/ 2 log(2)

.

We know that R hextra indicates the convergence rate for the eigenvalue extrapolation method. 4.1. Model problem In this subsection, we consider the model problem

− p = λ p , in Ω, ∂p = 0, on ∂Ω, ∂n  p 2 dΩ = 1,

(4.1) (4.2) (4.3)

Ω

where Ω = [0, 1] × [0, 1]. It is known that the exact solution is pkl = cos(kπ x) cos(lπ y ),

  λ = k2 + l2 π 2 , 0  k, l, and k, l are integers.

(4.4)

The numerical results for the approximations of the fourth eigenvalue 2π 2 are listed in Table 1. 4.2. More general problem In this subsection, we are concerned with a more general eigenvalue problem (1.1)–(1.3), where

A= 1

1 + (x1 − 12 )2

(x1 − 12 )(x2 − 12 )

(x1 − 12 )(x2 − 12 )

1 + (x2 − 12 )2

,

1

ϕ = e (x1 − 2 )(x2 − 2 ) , ρ = 1 + (x1 − 12 )(x2 − 12 ) and Ω = [0, 1] × [0, 1].

Since the exact solution is not known, we choose an adequately accurate approximation λ = 21.73456901306545 as the exact fourth eigenvalue (has 8 significant digits at least) [20]. The numerical results are listed in Table 2. 4.3. Model problem on L-shaped domain Here, we consider the following eigenvalue problem

Q. Lin, H. Xie / Applied Numerical Mathematics 59 (2009) 1884–1893

1891

Table 2 Approximation for the fourth eigenvalue of more general problem. M×N

2×2

4×4

8×8

λh1 ,h2

26.100974025

22.835822336

22.008137479

21.802838676

21.751629100

λhextra 1 ,h 2

–

21.747438439

21.732242526

21.734405741

21.734559242

4.3664e+000

errh errhextra

–

Rh

–

R hextra

–

16 × 16

32 × 32

1.1013e+000

2.7357e−001

6.8270e−002

1.7060e−002

1.2869e−002

−2.3265e−003

−1.6327e−004

−9.7713e−006

1.9872995835 –

2.0091725281

2.0025854141

2.0006196044

2.4677227587

3.8328052687

4.0625787487

Table 3 Approximation for the fourth eigenvalue of more general problem. M×N

2×2

4×4

8×8

16 × 16

32 × 32

λh1 ,h2

10.821064737

10.651207652

9.9632204762

9.7236819414

9.6608851787

λhextra 1 ,h 2

–

10.594588623

9.7338914177

9.6438357631

9.6399529245

1.0114838075

0.3234966322

0.083958097

0.021161335

9.5486e−001

9.4168e−002

4.1119e−003

2.2908e−004

errh

1.1813408933

errhextra

–

Rh

–

R hextra

–

0.2239521112 –

1.6446506267

1.9460093143

1.9882388498

3.3419941807

4.5173465640

4.1658856613

Fig. 1. The construction method for graded mesh.

− p = λ p , p = 0,



in Ω,

on ∂Ω,

p 2 dΩ = 1,

(4.5) (4.6) (4.7)

Ω

on the L-shaped domain. In order to dealt with the singularity near the reentrant corner, we use the graded mesh with the construction method described in Fig. 1. A suit graded rectangular mesh with graded parameter q = 3 (Fig. 2) is applied to solve the eigenvalue problem (4.5)–(4.7) [19]. Here, we adopt λ = 9.6397238440219 as the smallest accurate eigenvalue and the numerical results are listed in Table 3. 4.4. Numerical conclusion From Table 1 and 2, with the extrapolation method, the approximation accuracy can be improved from O(h 2 ) to O(h4 ) and the asymptotic expansion derived is reasonable. Even on the L-shaped domain, the eigenfunctions lack regularity, we can find that the extrapolation method can also improve the accuracy of the eigenvalue approximations (see Table 3).

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Fig. 2. Graded mesh 16 × 16 with graded parameter q = 3.

5. Concluding remarks Our main object of this paper is to derive the transform lemma for the error of the eigenvalue approximations by the mixed finite element method. Combining the interpolation expansions [15,21,23], we obtained the asymptotic error expansion and extrapolation for the eigenvalue approximations. As a by-product, we can use the approximation of higher accuracy to form a class of a posterior error estimators [9,10,14,16,20,21] for the eigenvalues. Of course, based on the transform lemma and combining the interpolation expansions, we can obtain asymptotic error expansions and extrapolation for higher order Raviart–Thomas elements and BDFM elements on the rectangular meshes [6,21,23]. Of course the transform lemma can be applied to the triangular mixed finite element methods and this is our future work. Practically the extrapolation method may give “good" results even though the exact solution does not satisfy the regularity assumptions. References [1] I. Babuška, J.E. Osborn, Estimate for the errors in eigenvalue and eigenvector approximation by Galerkin methods with particular attention to the case of multiple eigenvalue, SIAM J. Numer. Anal. 24 (1987) 1249–1276. [2] I. Babuška, J.E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989) 275–297. [3] H. Blum, R. Rannacher, Finite element eigenvalue computation on domains with reentrant corners using Richardson’s extrapolation, J. Comp. Math. 8 (1990) 321–332. [4] D. Boffi, F. Brezzi, L. Gastaldi, On the convergence of eigenvalue for mixed formulations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) 131–154. [5] J. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math. 68 (1994) 311–324. [6] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. [7] F. Chatelin, Spectral Approximation of Linear Operators, Academic Press Inc., New York, 1983. [8] C. Chen, Y. Huang, High Accuracy Theory for Finite Element Methods, Hunan Sci. Tech. Press, China, 1995. [9] W. Chen, Higher order approximation of eigenvalue for the biharmonic eigenvalue problem by Ciarlet–Raviart scheme, Numer. Methods Partial Differential Equations 21 (3) (2004) 512–520. [10] W. Chen, Q. Lin, Approximation of an eigenvalue problem associated with the Stokes problems by the stream function–vorticity–pressure method, Appl. Math. 1 (2006) 73–88. [11] Z. Chen, Finite Element Methods and Their Applications, Springer-Verlag, Berlin, Heidelberg, 2005. [12] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [13] J. Douglas, J.E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985) 39–52. [14] R.G. Durán, L. Gastaldi, C. Padra, A posteriori error estimators for mixed approximations of eigenvalue problems, Math. Models Methods Appl. Sci. 9 (1999) 1165–1178. [15] G. Fairweather, Q. Lin, Y. Lin, J. Wang, S. Zhang, Asymptotic expansion and Richardson extrapolation of approximate solutions for second order elliptic problems by mixed finite element methods, SIAM. J. Numer. Anal. 44 (3) (2006) 1122–1149. [16] F. Gardini, A posteriori error estimates for an eigenvalue problem arising from fluid-structure interaction, Istituto Lombardo (Rend. Sc.), 138, 2004. [17] F. Gardini, Mixed approximation of eigenvalue problems: a superconvergence result, I.M.A.T.I.-C.N.R. Publ. N.9-PV, 2006. [18] C.B. Liem, T.M. Shih, T. Lü, The Splitting Extrapolation Method, World Scientific Press, Singapore, 1995. [19] Q. Lin, Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners, Numer. Math. 58 (1991) 631–640. [20] Q. Lin, H. Huang, Z. Li, New expansion of numerical eigenvalue for −u = λρ u by nonconforming elements, Math. Comp. 77 (2008) 2061–2084. [21] Q. Lin, J. Lin, Finite Element Methods: Accuracy and Improvement, China Sci. Press, Beijing, 2006. [22] Q. Lin, T. Lü, Asymptotic expansions for finite element eigenvalues and finite element solution, Math. Schrift., Bonn, 1984. [23] Q. Lin, N. Yan, The Construction and Analysis of High Efficiency Finite Element Methods, HeBei University Publishers, Baoding, 1995. [24] G.I. Marchuk, V.V. Shaidurov, Difference Methods and Their Extrapolations, Springer-Verlag, New York, 1983.

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