Verified Computations of Eigenvalue Exclosures for

Downloaded 02/26/16 to 129.13.197.229. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

SIAM J. NUMER. ANAL. Vol. 52, No. 2, pp. 975–992

c 2014 Society for Industrial and Applied Mathematics

VERIFIED COMPUTATIONS OF EIGENVALUE EXCLOSURES FOR EIGENVALUE PROBLEMS IN HILBERT SPACES∗ YOSHITAKA WATANABE† , KAORI NAGATOU‡ , MICHAEL PLUM§ , AND MITSUHIRO T. NAKAO¶ Abstract. This paper presents eigenvalue excluding methods for self-adjoint or non-self-adjoint eigenvalue problems in Hilbert spaces, including problems with partial differential operators. Eigenvalue exclosure means the determination of subsets of the complex field which do not contain eigenvalues of the given problem. Several verified eigenvalue excluding results for ordinary and partial differential operators are reported on. Key words. eigenvalue excluding, eigenvalue problems, differential operators, computer-assisted proof AMS subject classifications. 35P15, 65G20, 65N25 DOI. 10.1137/120894683

1. Introduction. Let X, Y be complex Hilbert spaces endowed withthe inner products ( u, v )X , ( u, v )Y and the norms uX = ( u, u )X , uY = ( u, u )Y , respectively, and let D(A) be a complex Banach space. Assume that D(A) ⊂ X ⊂ Y and the embedding D(A) → X is compact. Let a linear operator A : D(A) → Y and bounded linear operators Q : X → Y , B : X → Y be given, and consider the eigenvalue problem (1.1)

(A + Q)u = λBu

for eigenpairs [u, λ]t ∈ (D(A)\{0}) × C. Here the letter t stands for transposition. Concerning the eigenvalue problem (1.1), we have already proposed several numerical verification procedures for enclosing eigenpairs [6, 9, 7, 21]. They are based on a fixed-point formulation with a Newton-like operator. The purpose of this paper is to propose verified computations of eigenvalue exclosures for the eigenvalue problem (1.1) which should provide us with important information about eigenvalue distribution. It is an improvement of a previous eigenvalue excluding method for second-order elliptic PDE eigenvalue problems introduced by Nagatou [7]. The methods use numerical means, but all numerical errors are taken into account, and hence they imply rigorous proofs of all statements made. ∗ Received by the editors October 10, 2012; accepted for publication (in revised form) February 14, 2014; published electronically April 24, 2014. This work was supported by a grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (20224001, 21540134, 24340018). http://www.siam.org/journals/sinum/52-2/89468.html † Research Institute for Information Technology, Kyushu University, Higashi-ku, Fukuoka 8128518, Japan ([email protected], http://www.cc.kyushu-u.ac.jp/RD/watanabe/). ‡ Institut f¨ ur Analysis, Karlsruhe Institut f¨ ur Technologie, 76133 Karlsruhe, Germany, and Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan ([email protected]). § Institut f¨ ur Analysis, Karlsruhe Institut f¨ ur Technologie, 76133 Karlsruhe, Germany (michael. [email protected]). ¶ Sasebo National College of Technology, Sasebo, Nagasaki 857-1193, Japan (mtnakao@ sasebo.ac.jp).

975

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.


976

Y. WATANABE, K. NAGATOU, M. PLUM, AND M. T. NAKAO

Recently, Brown et al. proposed a method of eigenvalue enclosures and exclosures for non-self-adjoint ODE eigenvalue problems [1]. Their algorithm combines an interval arithmetic–based code for solving ODEs with the argument principle for counting zeros of analytic functions. We note that our proposed excluding procedures can be applied to partial differential operators. For self-adjoint eigenvalue problems, variational methods are a powerful tool for obtaining index-controlled eigenvalue enclosures, and thus automatically also exclosurers. These methods, however, totally break down for non-self-adjoint problems, while the approach proposed here is still applicable in the non-self-adjoint case. The paper is organized as follows. Section 2 describes two eigenvalue excluding theorems for the eigenvalue problem (1.1). By these theorems, eigenvalue exclosures are reduced to the question of the invertibility of some linear operators and norm bounds for their inverses. Section 3 is concerned with a computer-checkable criterion to verify the invertibility of these infinite dimensional linear operators by using a finite dimensional subspace and some constructive a priori error bounds for a projection onto it. In sections 4 and 5, computations of norm bounds for the inverse operators will be given. The paper concludes with several eigenvalue excluding results in the last section. For readers interested in details of our program code, the code is available for downloading from the first author’s web page. 2. Eigenvalue exclosures. This section describes two eigenvalue excluding theorems for problem (1.1). Assume that the operators A and B have the following properties. Here, A can be imagined as some leading principal part of a differential equation (e.g., the Laplacian) and B as a lower-order nonlinear term. (Concrete examples will be shown in section 6.) A1. A : D(A) → Y is bijective with bounded inverse. The operator A−1 : Y → X is then compact due to the compactness of the embedding D(A) → X. A2. There exists a constant Cp > 0 such that BuY ≤ Cp uX

(2.1)

∀u ∈ X.

A3. There exists a constant Cb > 0 such that A−1 BuX ≤ Cb uX

(2.2)

∀u ∈ X.

In our actual verified computations, explicit values of Cp and Cb will be needed. The concept of the first excluding theorem is due to the idea introduced in [6, p. 192]. Let μ ∈ C be a candidate point for which we believe (and want to prove) that no eigenvalue is close to μ, including μ itself. Defining (2.3)

L : D(A) → Y,

L u := Au − (μB − Q)u

we shift problem (1.1), (2.4)

L u = (λ − μ)Bu,

and obtain the following excluding result. Theorem 2.1. If the operator L has an inverse L −1 : Y → D(A), and if there exists a constant M > 0 such that (2.5)

L −1 φX ≤ M φY

∀φ ∈ Y,



EIGENVALUE EXCLOSURES IN HILBERT SPACES

977

then there is no eigenvalue of problem (1.1) in the open disk 1 (2.6) z ∈ C |z − μ| < . Cp M ˜ t ∈ (D(A)\{0}) × C satisfying (2.4), substituting Proof. For any eigenpair [˜ u, λ] L u˜ ∈ Y into condition (2.5) as φ and using A2, we have ˜ − μ| B˜ ˜ − μ| ˜ ˜Y = M |λ uY ≤ M Cp |λ uX ; ˜ uX ≤ M L u therefore ˜ − μ| ≥ |λ

1 . Cp M

The second eigenvalue excluding theorem is based on an operator on X. By using A−1 : Y → X, problem (1.1) is rewritten equivalently in the form u = A−1 (λB − Q)u on X. Therefore, using again a candidate point μ ∈ C and defining (2.7)

: X → X, L

u := u − A−1 (μB − Q)u, L

we obtain a shifted eigenvalue problem for [u, λ]t ∈ (D(A)\{0}) × C, (2.8)

u = (λ − μ)A−1 Bu, L

and the following excluding theorem. has an inverse (L )−1 : X → X, and if there Theorem 2.2. If the operator L > 0 such that exists a constant M (2.9)

)−1 φX ≤ M φX (L

∀φ ∈ X,

then there is no eigenvalue of problem (1.1) in the open disk 1 . (2.10) z ∈ C |z − μ| < Cb M ˜ t ∈ (D(A)\{0}) × C satisfying (2.8), taking φ = Proof. For any eigenpair [˜ u, λ] u˜ ∈ X in (2.9) and using A3, we have L u ˜ − μ| M ˜ − μ| Cb M ˜ L A−1 B˜ uX , ˜ uX ≤ M ˜ X = |λ uX ≤ |λ which implies the conclusion. with the point μ ∈ Remark 1. If we can verify the invertibility of L or L C replaced by a set Λ ⊂ C, it is also ensured that no λ ∈ Λ is an eigenvalue of problem (1.1) [7]. Then Theorems 2.1 and 2.2 are not needed if we are satisfied with Λ as the resulting eigenvalue-free set (cf. section 6.1.1). However, the confirmation )−1 usually requires a narrow subset Λ because of the of the existence of L −1 or (L properties of interval computations used in the verifying linear algebra for an interval matrix G, described in the following section. We can say that taking a point μ (and using Theorem 2.1 or Theorem 2.2) has a great advantage from the point of view of interval arithmetic. Moreover, we may happen to face a trial-and-error choice for sizes of Λ depending on the eigenvalue distribution. In section 6.1.1 this approach is compared with our new results in an example.



978


. This section describes a computer3. Invertibility condition for L and L checkable condition ensuring the invertibility of the linear operator L in (2.3) and in (2.7). Basically, the verification method is an extension of that for second-order L elliptic boundary value problems introduced by a part of the authors [10, 11]. From now on, the identity map on X is denoted by the symbol I. . We start with a relation between the invertibility of L and L Lemma 3.1. If L is bijective, so is L . : X → X is bijective. Since L = AL |D(A) , L is oneProof. Assume that L −1 to-one. For any φ ∈ Y we can define v = A φ ∈ X by A1, and there exists u ∈ X u = v, i.e., u = A−1 ((μB − Q)u + φ). Thus, u ∈ D(A) and L u = φ such that L holds. Let Xh be a finite dimensional approximation subspace of X dependent on the parameter h > 0. For example, in the case of a PDE eigenvalue problem, Xh is taken to be a finite element subspace with mesh size h. The orthogonal projection Ph : X → Xh is defined by (3.1)

( v − Ph v, vh )X = 0

∀vh ∈ Xh

∀v ∈ X.

Thus, since Xh is a closed subspace of X, any element u ∈ X can be uniquely decomposed into u h ∈ Xh , u ∗ ∈ X∗ ,

u = uh + u∗ , where

X∗ := (I − Ph )X. Defining (3.2)

f : X → Y,

f (u) := −(Q − μB)u,

we assume that A, Ph , and f have the following properties. For example, if A is the Laplacian, A4 is just based on partial integration, and A5 is the usual finite element approximation error estimate, while A6 and A7 incorporate appropriate bounds for the nonlinear terms. (Concrete examples will also be shown in section 6.) A4. The operator A satisfies (3.3)

( u, v )X = ( Au, v )Y

∀u ∈ D(A) ∀v ∈ X.

A5. There exists C(h) > 0 such that (3.4)

(I − Ph )uX ≤ C(h)AuY

∀u ∈ D(A).

A6. There exists ν1 > 0 such that (3.5)

Ph A−1 f (u∗ )X ≤ ν1 u∗ X

∀u∗ ∈ X∗ .

A7. There exist ν2 > 0 and ν3 > 0 such that (3.6)

f (u)Y ≤ ν2 Ph uX + ν3 (I − Ph )uX

∀u ∈ X.

Note that the concrete values of the constants C(h) and νi (i = 1, 2, 3) have to be known and must be evaluated in the rigorous mathematical sense, and C(h) must




979

have the property that C(h) → 0 as h → 0. Also note that the constants νi (i = 1, 2, 3) depend on the given candidate point μ ∈ C. We emphasize that especially the estimate (3.4) is indispensable in our argument and the compactness of the embedding D(A) → X is essential in getting the constant C(h) with the desired properties. We will show concrete examples of C(h) and νi (i = 1, 2, 3) in section 6. −I is a compact operator on X, the Fredholm alternative implies that L Since L is bijective if it is one-to-one, i.e., if the problem L u = 0 has only the trivial solution u = 0. We aim at sufficient conditions for this one-to-one property now. By defining the compact operator F : X → X by F u = A−1 f (u),

(3.7)

u = 0 can be rewritten in the fixed-point form the problem L (3.8)

u = F u.

This fixed-point equation on X is equivalently rewritten as Ph u = Ph F u, (3.9) (I − Ph )u = (I − Ph )F u. Now, let us define the Newton-like operator Nh : X → Xh by (3.10)

Nh u := Ph u − [I − F ]−1 h Ph (I − F )u.

Here [I − F ]−1 : Xh → Xh means the inverse of the restriction of the operator h Ph (I − F ) : X → Xh to Xh . Note that the existence of [I − F ]−1 h is equivalent to the invertibility of a matrix G defined in the next section, which is numerically checked in the actual verified computations. Since Ph u = Nh u and Ph u = Ph F u are equivalent, using a map T on X defined by T u := Nh u + (I − Ph )F u,

(3.11)

we find that the two fixed-point problems u = F u and u = T u are also equivalent. Next, for positive parameters γˆ and α ˆ , set Uh := {uh ∈ Xh | uh X ≤ γˆ } ⊂ Xh , ˆ } ⊂ X∗ , U∗ := {u∗ ∈ X∗ | u∗ X ≤ α and define a set U ⊂ X by (3.12)

U := Uh + U∗ .

Then a sufficient condition for the invertibility result is derived as follows [9]. Lemma 3.2. When an inclusion (3.13)

T U ⊂ int(U )

is invertible. holds, then L u = 0 and u = 0, u also satisfies u = T u. Proof. If there exists u ∈ X such that L Since T is a linear operator, for any t ∈ C, we have T (tu) = tT u = tu. Then, we can choose tˆ ∈ C satisfying tû ∈ ∂U. However, this contradicts T U ⊂ int(U ) and u = 0. T (tû) = tû. Therefore u = 0. That is, u = 0 is the unique solution of L



980


We describe a procedure to construct a candidate set U of X which is expected to satisfy the inclusion (3.13). From the unique decomposition of u ∈ U , we will check the condition for finite and infinite dimensional parts separately. Lemma 3.3. If one can check the conditions (3.14)

sup Nh uX < γˆ ,

u∈U

C(h) sup f (u)Y < α ˆ, u∈U

is invertible. then L Proof. We will check the inclusions (3.15)

Nh U ⊂ int(Uh ),

(I − Ph )F U ⊂ int(U∗ )

in the respective subspaces Xh and X∗ of X with “int” referring to the relative topologies in these subspaces. From the definition of Uh , the finite dimensional part of the inclusions (3.15) is implied by supu∈U Nh uX < γˆ . On the other hand, the infinite dimensional part of the inclusions (3.15) means (I − Ph )A−1 f (u) ∈ int(U∗ )

∀u ∈ U.

Therefore, from A5, the condition C(h) supu∈U f (u)Y < α ˆ is sufficient. In order to obtain the conditions of Lemma 3.3, for given positive parameters α ˆ and γˆ , we have to compute γ := sup Nh uX , u∈U

α := C(h) sup f (u)Y u∈U

and confirm γ < γˆ ,

α 0 denote an upper bound of a matrix 2-norm: (3.18)

−1 L1 2 ≤ ρ. LH 1 G

Evaluation of ρ, including a proof of the invertibility of G, can be reduced to the verified computation of the maximum singular value of a matrix [17]. Then we obtain the following result.


981



Lemma 3.4. The operator [I − F ]−1 h exists and satisfies [I − F ]−1 h sh X ≤ ρsh X

(3.19)

∀sh ∈ Xh .

Proof. Suppose that th , sh ∈ Xh are related by Ph (I − F )th = sh or, equvalently, by ( (I − F )th , vh )X = ( sh , vh )X

(3.20)

∀vh ∈ Xh .

By setting sh =:

N

sh,n φn ,

s := [sh,n ] ∈ CN ,

th,n φn ,

t := [th,n ] ∈ CN ,

n=1

th =:

N n=1

and by A4, (3.20) is equivalent to (3.21)

N

th,n (( φn , φm )X − ( f (φn ), φm )Y ) =

n=1

N

sh,n ( φn , φm )X ,

1 ≤ m ≤ N,

n=1

i.e., to the linear system Gt = A1 s. Since G is invertible by assumption, s = 0 implies t = 0, i.e., sh = 0 implies th = 0. Hence [I − F ]−1 h exists, and [I − F ]−1 sh X = th X = tH A1 t = LH 1 t2 −1 = LH A1 s2 1 G −1 ≤ LH L1 2 LH 1 G 1 s2 ≤ ρ sh X .

Now, for any u ∈ U such that u = uh + u∗ , uh ∈ Uh , u∗ ∈ U∗ , we obtain Nh u = Ph u − [I − F ]−1 h Ph (I − F )u = [I − F ]−1 h Ph (F u − F uh ) −1 f (u∗ ) = [I − F ]−1 h Ph A

from the linearity of f . Hence Lemma 3.4 and A6 imply (3.22)

Nh uX ≤ ρν1 α ˆ

∀u ∈ U.

Moreover, from A7, we have (3.23)

f (u)Y ≤ ν2 uh X + ν3 u∗ X ≤ ν2 γˆ + ν3 α ˆ

for each u ∈ U . Therefore, the following criterion for invertibility holds. Theorem 3.5. If (3.24)

κ := C(h)(ρν1 ν2 + ν3 ) < 1

is one-to-one, and therefore, bijective. holds, then the operator L



982


Proof. From (3.22), (3.23), in order to get the conditions γ < γˆ and α < α ˆ , we shall check the inequalities (3.25)

ˆ < γˆ ρν1 α

and

C(h)(ν2 γˆ + ν3 α ˆ) < α ˆ

for some α ˆ , γˆ > 0. By assumption (3.24), we have 1 − C(h)(ρν1 ν2 + ν3 ) > 0; hence for any fixed δ > 0, α ˆ > 0 can be taken satisfying α ˆ [1 − C(h)(ρν1 ν2 + ν3 )] > C(h)ν2 δ.

(3.26) Therefore, by setting (3.27)

γˆ = ρν1 α ˆ + δ,

the first inequality in (3.25) holds, and substituting δ of (3.27) into (3.26), we obtain α ˆ > C(h)(ˆ αν3 + ν2 γˆ ), which is the second inequality in (3.25). . This section is devoted to a computable upper bound 4. Computation of M . of M in Theorem 2.2 under the invertibility criterion (3.24) for L Theorem 4.1. Under assumption (3.24) of Theorem 3.5, M > 0 for (2.9) can be taken as (4.1) where

(4.2)

= W 2 , M

⎡ ν1 C(h)ν2 ρ ρ 1+ , ⎢ 1−κ W := ⎣ C(h)ν2 ρ , 1−κ

⎤ ρν1 2×2 1 − κ⎥ ⎦∈R . 1 1−κ

= I − F , under condition (3.24), for any φ ∈ X, there exists a Proof. Since L unique ψ ∈ X such that (4.3)

(I − F )ψ = φ.

Equation (4.3) can be regarded as the fixed-point problem (4.4)

ψ = Fψ + φ

for ψ. Using the projection Ph : X → Xh , (4.4) is decomposed into Ph ψ = Ph (F ψ + φ), (4.5) (I − Ph )ψ = (I − Ph )(F ψ + φ). Therefore by introducing the Newton-like operator Nˆh ψ := Ph ψ − [I − F ]−1 h Ph ((I − F )ψ − φ) and ˆh ψ + (I − Ph )(F ψ + φ), Tˆψ := N



983


the fixed-point problem ψ = Tˆ ψ is equivalent to (4.4). Now, decomposing ψ = ψh + ψ∗ ,

φ = φh + φ∗ ,

ψh , φh ∈ Xh , ψ∗ , φ∗ ∈ X∗ ,

we will estimate ψh X and ψ∗ X using φh X and φ∗ X . Since ψ = Tˆ ψ implies ˆh ψ = Ph ψ − [I − F ]−1 Ph ((I − F )ψ − φ) ψh = N h = [I − F ]−1 h (Ph F ψ∗ + φh ),

we have, from Lemma 3.4 and A6, ψh X = [I − F ]−1 h (Ph F ψ∗ + φh )X

(4.6)

≤ ρPh F ψ∗ + φh X ≤ ρ(ν1 ψ∗ X + φh X ). On the other hand, since ψ∗ = (I − Ph )(F ψ + φ) = (I − Ph )F (ψh + ψ∗ ) + φ∗ , A5 and A7 imply ψ∗ X = (I − Ph )F (ψh + ψ∗ ) + φ∗ X ≤ C(h)f (ψh + ψ∗ )Y + φ∗ X

(4.7)

≤ C(h)ν2 ψh X + C(h)ν3 ψ∗ X + φ∗ X . Substituting (4.6) into (4.7) shows ψ∗ X ≤ C(h)ν2 ρν1 ψ∗ X + C(h)ν2 ρφh X + C(h)ν3 ψ∗ X + φ∗ X = κψ∗ X + C(h)ν2 ρφh X + φ∗ X . Thus we have (4.8)

ψ∗ X ≤

C(h)ν2 ρ 1 φh X + φ∗ X . 1−κ 1−κ

Also substituting (4.8) into (4.6) implies C(h)ν2 ρ 1 φh X + ρν1 φ∗ X + ρφh X ψh X ≤ ρν1 1−κ 1−κ

ν1 C(h)ν2 ρ ρν1 φ∗ X . =ρ 1+ φh X + 1−κ 1−κ Consequently, we have (with componentwise inequality)

⎤ ⎡ ν1 C(h)ν2 ρ ρν1 ρ 1 + , ψh X ⎥ φh X ⎢ 1 − κ 1 − κ ≤⎣ ⎦ φ ψ∗ X 1 C(h)ν2 ρ ∗ X , 1−κ 1−κ and thus the desired result. Remark 2. In Theorem 4.1, if κ and C(h) converge to 0 as h → 0, we obtain W 2 → [ ρ0 ρν11 ]2 , which is always larger than 1. Consequently, if the true norm )−1 happens to be less than 1, Theorem 4.1 will overestimate it as h → 0. of (L



984


5. Computation of M . This section is devoted to a computable upper bound of M in Theorem 2.1. For the matrix (5.1)

[A2 ]mn := ( φn , φm )Y ,

1 ≤ m, n ≤ N,

ˆ > 0 be an upper bound satisfying let L2 be such that A2 = L2 LH 2 , and let ρ −1 LH L2 2 ≤ ρˆ. 1 G

(5.2) Then the following is shown. Theorem 5.1. If (5.3)

κ ˆ := C(h)ν3 (1 + ρˆν2 ) < 1,

then L is bijective, and M > 0 for (2.5) can be taken as ρˆ2 + C(h)2 (1 + ν2 ρˆ)2 . (5.4) M= 1−κ ˆ Proof. Suppose that ψ ∈ D(A) and φ ∈ Y are given which satisfy the equation L ψ = Aψ − f (ψ) = φ, i.e., Aψ = f (ψ) + φ.

(5.5)

Equation (5.5) is equivalent to the fixed-point form ψ = A−1 (f (ψ) + φ)

(5.6)

for ψ, and using the projection Ph : X → Xh , problem (5.6) can be decomposed into Ph ψ = Ph A−1 (f (ψ) + φ), (5.7) (I − Ph )ψ = (I − Ph )A−1 (f (ψ) + φ). Now, decomposing ψ into ψ = ψh + ψ∗ ,

ψh ∈ Xh ,

ψ∗ ∈ X∗ ,

we will estimate ψh X and ψ∗ X using φY . Since ψh = Ph A−1 (f (ψ) + φ), A4 shows that ( ψh , vh )X = ( A−1 (f (ψ) + φ), vh )X ,

(5.8)

= ( f (ψ) + φ, vh )Y , = ( f (ψh ) + f (ψ∗ ) + φ, vh )Y

(5.9)

∀vh ∈ Xh .

By setting ψh =:

N

tˆh,n φn ∈ Xh ,

ˆt := [tˆh,n ] ∈ CN ,

fˆ := [( f (ψ∗ ) + φ, φn )Y ] ∈ CN ,

n=1

(5.9) is equivalent to (5.10)

Gˆt = fˆ



985


with G given by (3.17). Defining the projection P0 : Y → Xh by ( v − P0 v, vh )Y = 0

(5.11)

it can be checked that P0 (f (ψ∗ ) + φ)Y =

(5.12)

∀vh ∈ Xh ,

v ∈ Y,

H −1 ˆ ˆ fˆ A−1 2 f = L2 f 2

because A2 is invertible. Therefore we have, recalling our assumption that G is invertible, H −1 ˆ ˆ f 2 ψh X = LH 1 t2 = L1 G −1 ˆ H −1 ≤ L G L2 2 L f 2 1

2

≤ ρˆP0 (f (ψ∗ ) + φ)Y ≤ ρˆ (f (ψ∗ )Y + φY ), and, from A7, ψh X ≤ ρˆν3 ψ∗ X + ρˆφY .

(5.13)

On the other hand, because ψ∗ = (I − Ph )A−1 (f (ψ) + φ), using A5 and A7 we obtain ψ∗ X ≤ C(h)f (ψh + ψ∗ ) + φY ≤ C(h)ν2 ψh X + C(h)ν3 ψ∗ X + C(h)φY .

(5.14)

Substituting (5.13) into (5.14) shows ψ∗ X ≤ C(h)ν2 ρˆν3 ψ∗ X + C(h)ν2 ρˆφY + C(h)ν3 ψ∗ X + C(h)φY =κ ˆ ψ∗ X + C(h)(ν2 ρˆ + 1)φY . Thus we have ψ∗ X ≤ C(h)

(5.15)

1 + ν2 ρˆ φY . 1−κ ˆ

Now substituting (5.15) into (5.13) implies ψh X ≤ C(h) =

ρν3 (1 + ν2 ρˆ)ˆ φY + ρˆφY 1−κ ˆ

ρˆ φY . 1−κ ˆ

Consequently, we have (5.16)

ψ2X = ψh 2X + ψ∗ 2X

2

2 ρˆ 1 + ν2 ρˆ 2 ≤ φY + C(h) φ2Y 1−κ ˆ 1−κ ˆ = M 2 φ2Y .

In particular, (5.16) shows that φ = 0 implies ψ = 0, i.e., that L is one-to-one. Furthermore, for any given φ ∈ Y , the equation (5.17)

ψ ∈ D(A),

L ψ = φ,


986



is equivalent to (5.18)

ψ ∈ X,

(I − A−1 f )ψ = A−1 φ.

Since A−1 f : X → X is compact, the Fredholm alternative holds for (5.18), whence L being one-to-one implies that (5.18), and hence (5.17), is uniquely solvable. Hence L is bijective, and (5.16) gives (2.5) with M from (5.4). Remark 3. κ ˆ → 0 and C(h) → 0 as h → 0 in (5.4) implies M/ρˆ → 1. Comparing with Theorem 4.1, the bound (5.4) is expected to converge to the exact operator norm of L −1 (cf. Tables 1 and 2 in the next section), i.e., the possible disadvantage of Theorem 4.1 mentioned in Remark 2 does not occur in Theorem 5.1. However, in order to obtain ρˆ in (5.2), an additional computation is needed. The question if M or gives the best overall result (i.e., if Theorems 2.1 and 5.1 or Theorems 2.2 and 4.1 M should be used) depends on the given problem and the choice of the approximation subspace Xh , especially the size of h > 0. Remark 4. In [6] the auxiliary self-adjoint problem (5.19)

u ∈ D(A),

( L u, L v )Y = κ( u, v )X

∀v ∈ D(A)

is considered, and a procedure to compute a positive lower bound for its bottom eigenvalue κ1 by a variational method is proposed in order to obtain an upper bound for L −1 and thus, by Theorem 2.1, eigenvalue exclosures for the non-self-adjoint problem (1.1). Also, approximate excluding computations for the Orr–Sommerfeld equation with Blasius profile are carried out in [6]. It is easily checked that M of (2.5) is √ an upper bound of 1/ κ1 . The variational approach used in [6], which is based on the Rayleigh–Ritz and Lehmann–Goerisch methods, is a powerful computer-assisted approach and has a significant advantage for problems on unbounded domains [15]. However, this approach requires a finite dimensional approximation subspace belong used in ing to D(A) for (5.19). The verified computation for obtaining M and M the present paper does not need such higher-order finite dimensional spaces since it is based on the weak formulation. 6. Exclosure results. Here, we report on several verified computations of eigenvalue exclosures by Theorems 2.1 and 2.2. We use the interval arithmetic toolbox INTLAB [16], version 6, with MATLAB 7.12.0.635 (R2011a) on an Intel Core 2 Duo 2.6 GHz computer (Mac OS X 10.6.8). 6.1. Second-order elliptic problems. Let Ω be a bounded convex domain in Rn (n = 1, 2, 3) with piecewise smooth boundary ∂Ω, and let H m (Ω) denote the L2 -Sobolev space of order m on Ω. We define H01 (Ω) := {u ∈ H 1 (Ω) | u = 0 on ∂Ω (in the trace sense)} with the inner product ( ∇u, ∇v )L2 (Ω) and the norm uH01 (Ω) := ∇uL2 (Ω) , where ( u, v )L2 (Ω) := Ω u¯ v dx and uL2 (Ω) denote the L2 -inner product and the L2 -norm on Ω, respectively. Consider the following second-order elliptic eigenvalue problem, where b ∈ L∞ (Ω)n and c ∈ L∞ (Ω): −Δu + b · ∇u + cu = λu in Ω, (6.1) u = 0 on ∂Ω.


987



We define D(A) = H 2 (Ω) ∩ H01 (Ω), Q = b · ∇ + c,

B = IH01 →L2 ,

X = H01 (Ω),

Y = L2 (Ω),

( u, v )X = (∇u, ∇v)L2 (Ω) ,

A = −Δ,

( u, v )Y = (u, v)L2 (Ω) .

It is well known that A1 holds [2] and A4 is an immediate consequence of partial integration. For A2, the constant Cp satisfying uL2(Ω) ≤ Cp ∇uL2 (Ω)

(6.2)

∀u ∈ H01 (Ω)

√ is the Poincaré or Rayleigh–Ritz constant. It is given by Cp = 1/ λ1 with λ1 denoting the smallest eigenvalue of −Δ√under Dirichlet boundary conditions. For example, if Ω = (0, 1) × (0, 1), Cp = 1/(π 2). For calculating a constant Cb satisfying ∇(−Δ)−1 uL2 (Ω) ≤ Cb ∇uL2 (Ω)

∀u ∈ H01 (Ω)

for A3, we use the Poincaré constant (6.2), ∇(−Δ)−1 u2L2 (Ω) = (∇(−Δ)−1 u, ∇(−Δ)−1 u)L2 (Ω) = (u, (−Δ)−1 u)L2 (Ω) ≤ uL2(Ω) (−Δ)−1 uL2 (Ω) ≤ Cp2 ∇uL2 (Ω) ∇(−Δ)−1 uL2 (Ω) , to obtain Cb = Cp2 . For A5, we note that Ph is now the usual H01 -projection, and (3.4) holds with C(h) = h/π and h/(2π) for bilinear and biquadratic elements, respectively, for rectangular meshes on square domains [8]. Moreover, C(h) = 0.493h for linear and uniform triangular meshes on convex polygonal domains [5]. Here, h > 0 stands for the side length for a given finite element mesh. Concerning A6 and A7, we can take ν1 = Cp (bL∞ (Ω)n + C(h)c − μL∞ (Ω) ), ν2 = bL∞(Ω)n + Cp c − μL∞ (Ω) , n

ν3 = bL∞(Ω)n + C(h)c − μL∞ (Ω)

for b2L∞ (Ω)n := i=1 bi 2L∞ (Ω) [14]. Note that if Ph and (−Δ)−1 commute [20], or b is differentiable [11], we can derive more accurate estimates for νi (i = 1, 2, 3). Also Kinoshita and others showed that it is possible to obtain a similar kind of accurate estimate for νi , even when b is not differentiable [3, 12, 13]. 6.1.1. Linearization of the Allen–Cahn equation. Our first example for the second-order elliptic problem (6.1) is the two-dimensional self-adjoint eigenvalue problem −Δu + ν(3u2h − 2(a + 1)uh + a)u = λu in Ω, (6.3) u = 0 on ∂Ω, where Ω = (0, 1) × (0, 1), ν and a are positive constants, and uh is an approximate solution of the Allen–Cahn equation −Δu = νu(u − a)(1 − u) in Ω, (6.4) u = 0 on ∂Ω.



988


1

0.35 0.3

0.8 0.25 0.6

0.2 0.15

0.4

0.1 0.2 0.05 0 1

0 1 0.8

0.8

1 0.6 0.6

0.4

0.4

0.2

0.2 0

0.8 0.6

0.4

0.4

0.2

1 0.6

0.8

0.2 0

0

0

Fig. 1. Shape of approximate solutions on the upper branch (left) and lower branch (right).

Obviously, (6.3) is the linearization of (6.4) at uh . It is shown in [15] that such linearizations (at approximate solutions) play an important role in some computerassisted existence and multiplicity proofs for nonlinear boundary value problems. It is known that (6.4) has at least two solution branches with respect to the parameter ν > 0 [19]. We consider finite element approximations uh on both the lower and the upper branch for ν = 150 and a = 0.01 with linear and uniform triangular meshes on Ω (Figure 1). We can take C(h) = 0.493h for the uniform partition size h > 0. Tables 1 and 2 show eigenvalue excluding results with h = 1/50 for the lower approximate and R solution branch and the upper approximate solution branch, respectively. R ) and 1/(Cp M ), respectively. Figure 2 also stand for the excluding radius 1/(Cb M shows an excluding area around zero. The dots indicate approximate eigenvalues obtained by the eig function of MATLAB. In this example, Theorem 2.1 gives wider excluding areas than Theorem 2.2. For the approximate solution on the lower branch, Theorem 2.1 succeeds to exclude the area [−16.5044, 18.7917] by using 10 candidate points μ (cf. Table 1). We also mention that when we replace μ by an interval Λ (cf. Remark 1), we needed to choose 2,000 subintervals (of equal width) and to verify for each subinterval until we could exclude just the small area the invertibility of L [−16.5044, −16.4], amounting to an approximately 200 times higher computational cost compared with excluding the whole area [−16.5044, 18.7917] by Theorem 2.1. If we move away from −16.5 (which is actually close to an eigenvalue), larger diameters for the intervals Λ can be chosen, e.g., Λ = [−1, 1]. Nevertheless, this example clearly demonstrates the advantages of our new approach. Table 1 Eigenvalue excluding result for the approximate solution on the lower branch. μ -16.4 -16 -14 -11 0 11 15 17 18 18.5

ρ 95.2682 33.4660 7.8622 3.6476 2.5819 5.8072 10.7335 18.6783 29.6737 42.0606

ρˆ 20.9648 7.3701 1.7375 0.8098 0.3604 0.8158 1.5105 2.6305 4.1804 5.9265

κ 0.5175 0.1884 0.0523 0.0304 0.0360 0.1106 0.2263 0.4134 0.6726 0.9647

M 197.8657 41.3290 8.3259 3.7853 2.7169 6.6304 14.0928 32.3534 92.0954 1209.0305

R 0.0997 0.4776 2.3709 5.2148 7.2656 2.9771 1.4007 0.6101 0.2143 0.0163

κ ˆ 0.5060 0.1844 0.0514 0.0301 0.0242 0.0713 0.1438 0.2611 0.4235 0.6064

M 42.5532 9.0611 1.8381 0.8388 0.3728 0.8878 1.7838 3.6002 7.3336 15.2295

R 0.1044 0.4903 2.4172 5.2972 11.9204 5.0048 2.4908 1.2341 0.6058 0.2917


989



Table 2 Eigenvalue excluding result for the approximate solution on the upper branch. μ -58 -24 0 24 35 40 43.5 44.5

ρ 0.9999 1.1270 1.4709 2.5661 4.5318 7.4412 14.2803 19.6070

ρˆ 0.0763 0.1074 0.1568 0.3107 0.5874 0.9978 1.9634 2.7156

κ 0.1884 0.1430 0.1311 0.1468 0.2020 0.3024 0.6175 0.8626

M 1.7296 1.6793 1.9581 3.2129 5.9213 11.0398 38.6360 147.6034

R 11.4133 11.7550 10.0815 6.1439 3.3337 1.7880 0.5109 0.1337

κ ˆ 0.0758 0.0691 0.0687 0.0836 0.1202 0.1837 0.3807 0.5343

M 0.0939 0.1255 0.1782 0.3505 0.6840 1.2486 3.2401 5.9601

R 47.3554 35.4209 24.9387 12.6774 6.4955 3.5587 1.3713 0.7454

0 -16.6168

19.6970

19.7422

0 47.1079

55.8075 55.9023

Fig. 2. Eigenvalue excluding area.

6.1.2. Two-dimensional convection-diffusion equation. Our next example for the second-order elliptic problem (6.1) is the two-dimensional non-self-adjoint eigenvalue problem on Ω = (0, 1) × (0, 1), −Δu + ν b · ∇u = λu in Ω, (6.5) u = 0 on ∂Ω, where ν a is positive constant, and b = [−y + 1/2, x − 1/2]t . Equation (6.5) originates from a stationary convection-diffusion equation, more precisely from the rotating Gaussian hill problem [18]. Figure 3 shows approximate eigenvalues computed by MATLAB (left) and excluding circles by Theorem 2.1 for ν = 5 (right). Xh is the same as in the previous example, and h = 1/100. 6.2. Orr–Sommerfeld problem. The Orr–Sommerfeld equation (6.6) (−D2 + a2 )2 u + iaRe[V (−D2 + a2 ) + V ]u = λ(−D2 + a2 )u u(x1 ) = u(x2 ) = u (x1 ) = u (x2 ) = 0,

in

(x1 , x2 ),

is one of the central equations governing the linearized stability theory of incompressible flows. Here, D = d/dx stands for the derivative, i the imaginary unit, a > 0 the wave number of the perturbation, and Re > 0 the Reynolds number of an underlying fluid which moves in a stationary flow with given real-valued flow profile V ∈ C 2 (x1 , x2 ). The Orr–Sommerfeld equation (6.6) is a non-self-adjoint eigenvalue problem for the eigenpair [u, λ]t . We focus on the case of the plane Poiseuille flow (6.7)

V = 1 − x2 ,

x1 = −1,

x2 = 1.



990

Y. WATANABE, K. NAGATOU, M. PLUM, AND M. T. NAKAO 40

25

30

20 15

20

10

10

5

0

0 ï5

−10

ï10

−20

ï15

−30 ï20

−40 1 10

2

10

3

4

10

10

5

10

10

20

30

40

50

Fig. 3. Approximate eigenvalues (left) and eigenvalue excluding areas (right).

Then for the subspace H02 (Ω) := v ∈ H 2 (Ω) | v(−1) = v (−1) = v(1) = v (1) = 0 for Ω = (−1, 1), we can take D(A) = H 4 (Ω) ∩ H02 (Ω), X = H02 (Ω), Y = L2 (Ω), with inner products ( u, v )X = ((−D2 + a2 )u, (−D2 + a2 )v)L2 (Ω) ,

( u, v )Y = (u, v)L2 (Ω) ,

and A = (−D2 + a2 )2 , Q = iaRe[V (−D2 + a2 ) + V ], B = −D2 + a2 . Also, we can take Cp = 1 and Cb = 1/(π 2 /4 + a2 ) [21]. For the case of (6.6), when we introduce a finite dimensional approximation subspace Xh ⊂ H02 (Ω), using base functions constructed from piecewise cubic Hermite interpolating polynomials with uniform partition size h, we can take √

3 2 a2 2 h 1 + h , ν1 = C(h)τ1 , ν2 = τ2 + τ3 Cb , ν3 = τ2 + C(h)τ3 , C(h) = p p where τ1 :=2τ3 Cb + τ2 + ReV ∞ , τ2 := − iaReV + μ∞ , τ3 := aReV ∞ , and √ √ p = 6 70/ 4 + 5 [21, 22]. Especially, for C(h), we use interpolation error estimates ) = 1/(Cb M in H 4 (Ω) ∩ H02 (Ω) [4]. Table 3 shows the eigenvalue excluding radii R of Theorem 2.2 and R = 1/(Cp M ) of Theorem 2.1 for Re = 5776, a = 1.019, μ1 = −100 + 1552.59i, μ2 = −200 + 1552.59i, and μ3 = −500 + 1552.59i. The sign “—” indicates that the assumptions of Theorem 2.2 or Theorem 2.1 did not hold. When the candidate excluding point μ is taken far from the eigenvalues of problem (1.1), Table 3 Eigenvalue excluding radii for (6.6). μ1 1/h 100 200 300 400 500

R — 0.5116 0.5362 0.4158 0.2167

μ2 R — — — — —

R 0.3143 1.2195 1.2333 0.9462 0.4881

R — — — — 21.2792

μ3 R 1.5169 2.9809 3.2040 2.6540 1.4648

R — — 31.7138 89.2753 115.7774


991



10000 8000

1650

6000 1600

4000 2000

1550

0 1500

-2000 -4000

1450 -6000 -8000 -6000 -4000 -2000

0

2000

4000

6000

8000

-150

-100

-50

0

50

Fig. 4. Local circles excluding eigenvalues (left) and its zoom (right).

Theorem 2.1 gives a larger excluding radius than Theorem 2.2. When μ is closer to an eigenvalue, the criterion κ ˆ < 1 could not be satisfied, while Theorem 2.2 works. We tends to point out that, especially for h smaller than 1/200, the round-off error for R be a serious problem; note that it is expected to be larger for a larger partition number. The left side of Figure 4 indicates excluding circles for Re = 5776 and a = 1.019 with h = 1/250. Each dot stands for an approximate eigenvalue obtained by MATLAB. The right side of Figure 4 shows a zoom near an approximate eigenvalue with the smallest real part. Concerning a stability proof based on eigenvalue exclosures for the Orr–Sommerfeld equation (6.6), see [22]. Acknowledgment. The authors heartily thank the two anonymous referees for their thorough reading and valuable comments. REFERENCES [1] M. Brown, M. Langer, M. Marletta, C. Tretter, and M. Wagenhofer, Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics, LMS J. Comput. Math., 13 (2010), pp. 65–81. [2] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. [3] T. Kinoshita, K. Hashimoto, and M. T. Nakao, On the L2 a priori error estimates to the finite element solution of elliptic problems with singular adjoint operator, Numer. Funct. Anal. Optim., 30 (2009), pp. 289–305. [4] T. Kinoshita and M. T. Nakao, On very accurate enclosure of the optimal constant in the a priori error estimates for H02 -projection, J. Comput. Appl. Math., 234 (2010), pp. 526–537. [5] F. Kikuchi and X. Liu, Determination of the Babuska-Aziz constant for the linear triangular finite element, Jpn. J. Ind. Appl. Math., 23 (2006), pp. 75–82. [6] J.-R. Lahmann and M. Plum, A computer-assisted instability proof for the Orr-Sommerfeld equation with Blasius profile, Z. Angew. Math. Mech., 84 (2004), pp. 188–204. [7] K. Nagatou, Numerical verification method for infinite dimensional eigenvalue problems, Jpn. J. Ind. Appl. Math., 26 (2009), pp. 477–491. [8] M. T. Nakao, N. Yamamoto, and S. Kimura, On best constant in the error bound for the H01 -projection into piecewise polynomial spaces, J. Approx. Theory, 93 (1998), pp. 491–500. [9] M. T. Nakao, N. Yamamoto, and K. Nagatou, Numerical verifications for eigenvalues of second-order elliptic operators, Jpn. J. Ind. Appl. Math., 16 (1999), pp. 307–320. [10] M. T. Nakao and Y. Watanabe, An efficient approach to the numerical verification for solutions of elliptic differential equations, Numer. Algorithms, 37 (2004), pp. 311–323. [11] M. T. Nakao, K. Hashimoto, and Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing, 75 (2005), pp. 1–14.



992


[12] M. T. Nakao and K. Hashimoto, Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications, J. Comput. Appl. Math., 218 (2008), pp. 106–115. [13] M. T. Nakao and T. Kinoshita, Some remarks on the behaviour of the finite element solution in nonsmooth domains, Appl. Math. Lett., 21 (2008), pp. 1310–1314. [14] M. T. Nakao and Y. Watanabe, Numerical verification methods for solutions of semilinear elliptic boundary value problems, NOLTA, 2 (2011), pp. 2–31. [15] M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresber. Dtsch. Math.-Ver., 110 (2008), pp. 19–54. [16] S. M. Rump, INTLAB—INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, ed., Kluwer Academic Publishers, Dordrecht, the Netherlands, 1999, pp. 77–104; also available online from http://www.ti3.tu-harburg.de/rump/ [17] S. M. Rump, Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse, BIT, 51 (2011), pp. 367–384. [18] M. Tabata, Discrepancy between theory and real computation on the stability of some finite element schemes, J. Comput. Appl. Math., 199 (2007), pp. 424–431. [19] Y. Watanabe and M. T. Nakao, Numerical verifications of solutions for nonlinear elliptic equations, Jpn. J. Ind. Appl. Math., 10 (1993), pp. 165–178. [20] Y. Watanabe, N. Yamamoto, M. T. Nakao, and T. Nishida, A numerical verification of nontrivial solutions for the heat convection problem, J. Math. Fluid Mech., 6 (2004), pp. 1–20. [21] Y. Watanabe, M. Plum, and M. T. Nakao, A Computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow, Z. Angew. Math. Mech., 89 (2009), pp. 5–18. [22] Y. Watanabe, K. Nagatou, M. Plum, and M. T. Nakao, A computer-assisted stability proof for the Orr-Sommerfeld problem with Poiseuille flow, NOLTA, 2 (2011), pp. 123–127.


Verified Computations of Eigenvalue Exclosures for

Verified Computations of Eigenvalue Exclosures for

Suggest Documents

VERIFIED COMPUTATIONS FOR HYPERBOLIC 3-MANIFOLDS

Verified Compilation of Floating-Point Computations - LRI

Verified Eigenvalue Evaluation for the Laplacian over Polygonal

Verified Computations of Laminar Premixed Flames - University of ...

Eigenvalue extraction from time domain computations - Advances in ...

A verified runtime for a verified theorem prover - Cambridge Computer ...

Approximation and eigenvalue extrapolation of Stokes eigenvalue ...

ModelPlex: Verified Runtime Validation of Verified ... - AndrÃ© Platzer

Eigenvalue location for chain graphs

Eigenvalue Solvers for Atomistic Simulations of ...

THE EIGENVALUE PERTURBATION BOUND FOR

Eigenvalue spacings for regular graphs

S3 Computations for ELFP - PLOS

Response of Predators to Western Sandpiper Nest Exclosures

Stem growth of woody species at the Nkuhlu exclosures, Kruger ...

Stem growth of woody species at the Nkuhlu exclosures ... - Koedoe

Lazy Codes For Scientific Computations

Incoop: MapReduce for Incremental Computations

First-Principles Computations for KTN

LIMIT THEOREMS FOR LINEAR EIGENVALUE STATISTICS OF

universality of local eigenvalue statistics for random

“Application of MATLAB for Engineering Computations”

Verified Validation of Program Slicing

Error Detection for Polynomial Computations