Norm bound computation for inverses of linear operators in Hilbert spaces

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Norm bound computation for inverses of linear operators in Hilbert spaces Yoshitaka Watanabe a,∗ , Kaori Nagatou b , Michael Plum b , Mitsuhiro T. Nakao c a Research Institute for Information Technology, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8518,

Japan b Institut für Analysis, Karlsruher Institut für Technologie, Englerstraße 2, 76131 Karlsruhe, Germany c National Institute of Technology, Sasebo College, 1-1 Okishin-cho, Sasebo, Nagasaki, 857-1193, Japan

Received 25 July 2014; revised 24 September 2015

Abstract This paper presents a computer-assisted procedure to prove the invertibility of a linear operator which is the sum of an unbounded bijective and a bounded operator in a Hilbert space, and to compute a bound for the norm of its inverse. By using some projection and constructive a priori error estimates, the invertibility condition together with the norm computation is formulated as an inequality based upon a method originally developed by the authors for obtaining existence and enclosure results for nonlinear partial differential equations. Several examples which confirm the actual effectiveness of the procedure are reported. © 2015 Elsevier Inc. All rights reserved. MSC: 65G20; 47F05; 35P15 Keywords: Numerical verification; Solvability of linear problem; Differential operators

* Corresponding author.

E-mail addresses: [email protected] (Y. Watanabe), [email protected] (K. Nagatou), [email protected] (M. Plum), [email protected] (M.T. Nakao). URL: http://www.cc.kyushu-u.ac.jp/RD/watanabe/ (Y. Watanabe). http://dx.doi.org/10.1016/j.jde.2015.12.041 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction Let X, Y be complex Hilbert spaces√endowed with the 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 that the embedding D(A) → X is compact. Let linear operators A : D(A) → Y and Q : X → Y be given. This paper will consider a linear operator defined by L := A + Q :

D(A) → Y,

(1)

and propose a procedure for proving invertibility of L , and for computing a constant M > 0 satisfying L −1 φX ≤ MφY ,

∀φ ∈ Y,

(2)

i.e. a bound for the operator norm of L −1 : Y → X. In the context of computer-assisted proofs for nonlinear equations, the operator L stands for the linearization of a given nonlinear problem, and the verification of the invertibility of L and the computation of a norm bound for L −1 play an essential role in, for example, Newton-type or Newton–Cantorovich-type arguments which aim at proving the existence of a solution of the nonlinear problem with a mathematical rigorous error bound [4,11,15,16,18,19,22]. Our proposed approach is in fact an extension of the methods presented in [10,11,14,27] which are based on finite dimensional spectral norm estimation for Galerkin approximations to L −1 . Our bounds are expected to converge, as the Galerkin space increases, to the exact operator norm of L −1 and to provide accurate and efficient enclosure results for the solution of nonlinear problems. The procedure uses numerical means, but all numerical errors are taken into account, and hence it implies rigorous proofs of all statements made. We also note that our proposed method for invertibility and norm bounds has applications to eigenvalue exclosures in Hilbert spaces [28]. Other computational approaches to bounds for L −1 have already been proposed by one of the authors [16,18,19] and Oishi [15,22], for example. The method described in [16,18,19] is based on eigenvalue bounds, which are obtained by the Rayleigh–Ritz and the Lehmann–Goerisch method with additional base functions and some homotopic steps, together with verified computations for rather small matrix eigenvalue problems. It does not need any infinite dimensional projection error estimates and is applicable to differential equation problems on unbounded domains. However when L is non-self-adjoint, higher-order base functions are needed since then eigenvalue bounds for L ∗ L are required. In contrast, the verified computation for obtaining M used in the present paper does not need higher-order finite dimensional spaces since it is based on the weak formulation. Oishi’s method bounds the operator norm of L −1 by estimations based on numerical computation of the matrix norm of its Galerkin approximation, together with error bounds for the Galerkin projection, similar to the approach used in [8] already. This procedure, in principle, could be applicable to general Banach spaces and operators, and has also connections with the ideas of the present paper. The paper is organized as follows. Section 2 describes assumptions on the given linear operator and introduces some finite dimensional approximation subspaces. Section 3 is concerned with a criterion to verify the invertibility of L and to compute norm bounds for L −1 by using

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a finite dimensional approximation subspace and some constructive a priori error estimates for a projection onto it. The paper concludes with several computer-assisted results in the last section. 2. Function spaces and finite dimensional approximation This section describes assumptions on the linear operator L and introduces some finite dimensional approximation subspace. Assume that the operator A has the following properties. A1. A : D(A) → Y is bijective with bounded inverse A −1 . The operator A−1 := ID(A)→X ◦ A −1 : Y → X is then compact due to the compactness of the embedding D(A) → X. A2. The operator A satisfies ( u, v )X = ( Au, v )Y ,

∀u ∈ D(A),

∀v ∈ X.

(3)

Let Xh be a finite dimensional approximation subspace of X dependent on the parameter h > 0. For example, in case of a partial differential equation problem, Xh can be taken to be a finite element subspace with mesh size h. The orthogonal projection Ph : X → Xh is defined by ( v − Ph v, vh )X = 0,

∀vh ∈ Xh ,

∀v ∈ X.

(4)

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

uh ∈ Xh , u∗ ∈ X∗ ,

where X∗ := (I − Ph )X. We assume that Ph and Q have the following properties. A3. There exists C(h) > 0 such that (I − Ph )uX ≤ C(h)AuY ,

∀u ∈ D(A).

(5)

A4. Q is bounded and there exist τ1 > 0 and τ2 > 0 such that QuY ≤ τ1 Ph uX + τ2 (I − Ph )uX ,

∀u ∈ X.

(6)

Assumption A4 indicates some more detailed information about boundedness of the operator Q : X → Y . For the success of our approach, concrete values of the constants C(h) and τi (i = 1, 2) have to be known and must be evaluated in the rigorous mathematical sense, and C(h) must have the property that C(h) → 0 as h → 0. Also note that the constants τi depend on Q and Ph . We emphasize that especially the estimate (5) is indispensable in our argument, and the

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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 τi in Section 4. 3. The invertibility of L and computation of M This section is devoted to a computable bound M in (2), together with a proof of the invertibility of the linear operator L . Let {φn }N n=1 be a basis of Xh with N := dim Xh , and let A1 , A2 , G, L1 , and L2 be N × N matrices defined by [A1 ]mn :=( φn , φm )X ,

(7)

[A2 ]mn :=( φn , φm )Y ,

(8)

Gmn :=( φn , φm )X + ( Qφn , φm )Y

(9)

for 1 ≤ m, n ≤ N , and Li (i = 1, 2) such that Ai = Li LH i . Usually (but not necessarily) Li is taken to be the Cholesky factor of Ai (note that Ai is positive definite), i.e. Li is a lower triangular matrix. Here the letter H stands for conjugate transposition. Now let ρ > 0 be an upper bound satisfying −1 LH 1 G L2 2 ≤ ρ,

(10)

which of course in particular requires invertibility of the matrix G (see also Remark 2 below). Roughly speaking, G is a finite dimensional projection of L , and (10) reflects the invertibility of this projection. We may then hope for invertibility of the full operator L , if the projection error is not too large. The precise condition connecting between ρ and the projection error is condition (11) in the following theorem; its proof can be found in [28, Theorem 5.1]. Theorem 1. If κ := C(h)τ2 (1 + ρτ1 ) < 1,

(11)

then L is invertible, and M > 0 in (2) can be taken as  ρ 2 + C(h)2 (1 + ρτ1 )2 M= . 1−κ

(12)

Remark 1. In the examples treated in the present paper (Section 4) we will focus on linear operators only. We emphasize again that also in the context of computer-assisted proofs for nonlinear equations including ODEs/PDEs, the verification of the invertibility of L and the computation of a norm bound for L −1 plays an essential role, and Theorem 1 is expected to provide accurate and efficient enclosure results for solutions of nonlinear problems. We will report on concrete computer-assisted proofs of nonlinear equations by using Theorem 1 in our upcoming papers. Remark 2. In (12), κ → 0 and C(h) → 0 as h → 0 implies M/ρ → 1, therefore M is expected to converge to the exact operator norm of L −1 , as h → 0.

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Remark 3. 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 [21]. Remark 4. A part of the authors has considered a verification procedure of the invertibility of L and a norm bound for L −1 for second-order linear elliptic operators (14) (in the next section); [27,17]. Theorem 1 is an extension to more general operators in Hilbert spaces, and an improvement of Theorem 4.2 in [27] because here the invertibility of L is derived from the condition (10) by using the Fredholm alternative. Remark 5. In [7] the auxiliary self-adjoint eigenvalue problem u ∈ D(A),

( L u, L v )Y = κ( u, v )X ,

∀v ∈ D(A),

(13)

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 a bound for L −1 in the context of a computer-assisted instability proof for the Orr–Sommerfeld equation with Blasius profile. It √ is easily checked that M in (2) is an upper bound of 1/ κ1 . The variational approach used in [7], which is based on the Rayleigh–Ritz and the Lehmann–Goerisch method, is a powerful computer-assisted approach and has a significant advantage for problems on unbounded domains [19]. However, this approach requires a finite dimensional approximation subspace belonging to D(A) for (13). 4. Examples In this section, we report on several verified norm computations by Theorem 1. We use the interval arithmetic toolbox INTLAB [20] Version 6 with MATLAB 7.14.0.739 (R2012a) on Fujitsu PRIMERGY TX300 S5 (CPU: Intel Xeon E5520 2.27 GHz, OS: Red Hat Enterprise Linux Server release 5.6), for subsection 4.1 and subsection 4.3, and Sun Fortran 95 Ver. 8.3 (supporting interval arithmetic) on Sun SPARC Enterprise M3000 (CPU: SPARC64VII 2.52 GHz, OS: Solaris 10) for subsection 4.2. 4.1. Second-order elliptic operators Let  be a bounded convex domain in Rn (n = 1, 2, 3) with piecewise smooth boundary ∂, and let H m () denote the L2 -Sobolev space of order m on . We define H01 () := {u ∈ H 1 () | u = 0 on ∂} with the inner product ( ∇u, ∇v )L2 () and the norm uH 1 () := ∇uL2 () , where 0  2 ( u, v )L2 () := uv¯ dx and uL2 () denote the L -inner product and L2 -norm on , respec

tively. Consider the following second-order elliptic operator L u = − u + b · ∇u + cu

(14)

with b ∈ L∞ ()n and c ∈ L∞ () under the Dirichlet boundary condition u |∂ = 0. We define

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D(A) = H 2 () ∩ H01 (), X = H01 (), Y = L2 (), A = − , Q = b · ∇ + c, ( u, v )X = (∇u, ∇v)L2 () , ( u, v )Y = (u, v)L2 () .

(15)

It is well known that A1 holds [3], and A2 is an immediate consequence of partial integration. For A3, we note that Ph is now the usual H01 -projection, and (5) holds with C(h) = h/π and h/(2π) for bilinear and biquadratic elements, respectively, for rectangular meshes on square domains [9]. Moreover, C(h) = 0.493h for linear elements and uniform triangular meshes on convex polygonal domains [6]. Here, h > 0 stands for the element side length for a given finite element mesh. Concerning A4, we can take τ1 = bL∞ ()n + Cp cL∞ () ,

τ2 = bL∞ ()n + C(h)cL∞ () ,

 for b2L∞ ()n := ni=1 bi 2L∞ () [14]. Here, the constant Cp stands for the Poincaré or Rayleigh–Ritz constant satisfying uL2 () ≤ Cp ∇uL2 () ,

∀u ∈ H01 ().

(16)

√ For example, if  = (0, 1) × (0, 1), Cp = 1/(π 2). Note that if Ph and (− )−1 commute [23], or b is differentiable [11], we can derive more accurate estimates for τi (i = 1, 2). Kinoshita, et al. also showed that it is possible to obtain a similar kind of accurate estimate for τi , even though b is not differentiable [5,12,13]. 4.1.1. Linearization of a reaction–diffusion equation Our first example for the second-order elliptic operator (14) is a one-dimensional self-adjoint operator L u = −u +

1 (3u2h − 2(a + 1)uh + a + δPh A−1 )u, ε2

(17)

where  = (−1, 1), ε = 0, δ ∈ R, a > 0, A is as in (15), and uh is an approximate solution of the two-coupled system ⎧ 2  ⎨ −ε u −v  ⎩ u=v

= u(1 − u)(u − a) − δv = u = 0

in in on

, , ∂

(18)

obtained by the Newton–Raphson method with floating-point computation. By solving the problem for v in (18) and inserting Ph v into the first equation, the system (18) is reduced to a single (but non-local) nonlinear equation, and (up to a factor ε 2 ) L in (17) is its linearization [25]. A motivation for studying such kinds of linearizations is, for example, the following. In order to prove the existence of a solution of (18) by Newton–Cantorovich-type arguments [19] or by an approach based on Schauder’s fixed-point theorem [11], the linearization at an approximate solution such as (17) is required and the verification of the invertibility of L and the computation of a norm bound for L −1 play a central role in these arguments. Numerical evidence tells us that equation (18) apparently has various non-trivial solutions depending on the parameters ε, δ, and a [25]. We divide the interval (−1, 1) into K equal parts and take Xh as the set of continuous piecewise linear functions, linear on each subinterval and 0

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Fig. 1. Shape of approximate solutions to problem (18); upper: uh , lower: vh (ε = 0.08, δ = 0.2, a = 0.25).

on ∂. Then we can choose N = dim Xh = K − 1, h = 2/K, Cp = 2/π , and C(h) = h/π . Fig. 1 shows the shapes of approximate solutions uh (upper) and vh (lower), all for ε = 0.08, δ = 0.2 and a = 0.25. Table 1 shows results for M in (12) by Theorem 1 for each of these approximate solutions. The sign “–” indicates that the assumption (11) of the theorem did not hold. Table 2 shows values of M, ρ, κ and uh L∞ () for the approximate solution of type S2.

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Table 1 Values of M for the operator (17). K

S1

S2

S3

S4

AS1

AS2

AS3

30 40 50 100 200 500 1000 2000

1.41229 0.85657 0.70424 0.55151 0.55177 0.54238 0.54105 0.54073

– – 41.12986 3.80938 3.05137 2.88635 2.86418 2.85942

2.90467 1.34176 1.04250 0.77931 0.72844 0.71500 0.71311 0.71267

27.55243 2.88221 1.95094 1.31132 1.20264 1.17464 1.17072 1.23919

– – 3.34208 1.26220 1.06545 1.01866 1.01223 1.01072

12.53017 2.02820 1.39069 0.93513 0.85595 0.83544 0.83256 0.83185

– 2.44820 1.54232 0.97586 0.88360 0.85998 0.85669 0.85593

Table 2 Verification results of S2 for the operator (17). K

M

κ

ρ

uh L∞ ()

20 30 40 50 100 200 500 1000 2000

– – – 41.12986 3.80938 3.05137 2.88635 2.86418 2.85942

5.53932 2.53056 1.43760 0.92434 0.23253 0.05825 0.00933 0.00234 0.00059

2.69341 2.78157 2.81406 2.82926 2.84980 2.85498 2.85645 2.85675 2.85757

0.65495 0.65236 0.65145 0.65103 0.65046 0.65032 0.65029 0.65028 0.65028

Table 3 Invertibility results and bound of M for the operator described in Section 4.1.2. 1/ h 10 20 50 100 130

c=0

c = −20 − 10i

M

κ

ρ

M

κ

ρ

2.46871 0.42433 0.27608 0.24758 0.24190

0.89637 0.45066 0.18054 0.09029 0.06946

0.22223 0.22423 0.22479 0.22487 0.22489

– – 1.88921 0.72031 0.63005

3.84695 1.90955 0.76213 0.38094 0.29302

0.45100 0.44615 0.44498 0.44481 0.44478

4.1.2. Linearization of a two-dimensional convection–diffusion equation Our next example is a two-dimensional non-self-adjoint operator (14) on  = (0, 1) × (0, 1) with b = 10[−y + 1/2, x − 1/2]T and c ∈ C which originates from a stationary convection– diffusion equation. Here the letter T stands for transposition. Note that b is not a gradient field. Table 3 shows values of M, ρ and κ for c = 0 and c = −20 − 10i, respectively. We take linear and uniform triangular meshes on  with the uniform partition size h > 0. Some a posteriori estimates of L −1 are also described in [27]. 4.2. Linearization of the Orr–Sommerfeld problem Consider a one-dimensional non-self-adjoint operator on  = (x1 , x2 )   2 ˜ + V  )u − λh u ˜ − λs u ˜ h ˜ u + iaR(V u L = λ ( u, u0 )L2 ()

(19)

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together with Dirichlet boundary condition, u(x1 ) = u(x2 ) = u (x1 ) = u (x2 ) = 0, which is the linearization, re-scaled by a parameter s > 0, of the normalized Orr–Sommerfeld equation ⎧ 2 ˜ + V  )u ˜ u + iaR(V ⎨ ( u, u0 )L2 () ⎩ u(x1 ) = u(x2 )

˜ = λ u in , = ξ, = u (x1 ) = u (x2 ) = 0

(20)

at an approximate solution [uh , λh ]T . We obtained [uh , λh ]T by a Galerkin method with Leg˜ := −D 2 + a 2 , where D = d/dx stands for the derivative, i the endre polynomials. Here, imaginary unit, a > 0 the wave number of a single-mode perturbation, R > 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 ]. u0 denotes a fixed normalizing function, e.g. some rough eigenfunction approximation of (20), and ξ > 0 is a scaling parameter. The equation (20) is an eigenvalue problem for the eigenpair [u, λ]T and one of the central equations governing the linearized stability theory of incompressible flows. We note again that to apply some Newton-type fixed-point argument to (20), the invertibility of L together with a bound of the operator norm of L −1 is indispensable. We will report on details in a forthcoming article. We focus on the case of the plane Poiseuille flow V = 1 − x2,

x1 = −1,

x2 = 1.

(21)

Denote

H02 () := v ∈ H 2 () | v(−1) = v  (−1) = v(1) = v  (1) = 0 , ˜ L2 () is a norm equivalent to vH 2 and ( v, ˜ w ˜ )L2 () can be chosen as the inner v 2 2 product on H0 (). We define a Hilbert space X := H0 () × C with the inner product ˜ 1 , u ˜ 2 )L2 () + λ1 λ¯2 ([u1 , λ1 ]T , [u2 , λ2 ]T )X := ( u and norm  · X . We take D(A) = (H 4 () ∩ H02 ()) × C endowed with the norm [u, λ]D(A) := uH 4 () + |λ|, and Y = L2 () × C with the inner product ([u1 , λ1 ]T , [u2 , λ2 ]T )Y = ( u1 , u2 )L2 () + λ1 λ¯2 , and

 ˜2 A= , I

Q

  ˜ + V  )u − λh u ˜ − λs u ˜ h u iaR(V . = λ ( u, u0 )L2 () − λ

For the operator (19), when we introduce a finite dimensional approximation subspace Xh ⊂ H02 (), using base functions constructed from piecewise cubic Hermite interpolating polynomials with uniform partition number K, we can take √ C(h) =

3

t1

 2

h

  a2 2 ˜ h 2 2 + t 2 u0 2 2 + 1, 1 + h , τ1 = (t2 + t3 t4 )2 + s 2  u 4 L () L () t1

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Table 4 Invertibility results and bound of M for the operator (19). (a, R) = (1.019, 2500)

K 100 200 500 1000 1500

τ2 =

(a, R) = (1.020551, 5772.221818)

M

κ

ρ

M

κ

ρ

– 0.04908 0.01611 0.01469 0.01445

2.82873 0.70512 0.11273 0.02818 0.01253

0.01426 0.01426 0.01426 0.01426 0.01426

– – 0.03153 0.01508 0.01375

14.82084 3.69384 0.59050 0.14761 0.06561

0.01284 0.01284 0.01284 0.01284 0.01284

 √  √ (t2 + C(h)t3 )2 + C(h)2 u0 2L2 () , where h = 2/K, t1 := 6 70/ 4 + 5, t2 :=

iaRV − λh ∞ , t3 := aRV  ∞ , and t4 := 1/(π 2 /4 + a 2 ) [24,26]. Table 4 shows values of M, ρ and κ for (a, R) = (1.019, 2500) (uh L2 () ∼ 94.45875) and (a, R) = (1.020551, 5772.221818) (uh L2 () ∼ 158.42133). In the better parameter constellation, problem (20), (21) is known to have an eigenvalue in the unstable region {Re(λ) < 0} [2]. The scaling parameters are s = 20 and ξ = 1000. 4.3. Fourth-order elliptic operator Our final example is a fourth-order differential operator L u = 2 u −

1 (1 − 3u2h )u ε2

(22)

in  = (0, 1) × (0, 1), with Navier boundary condition u |∂ = u |∂ = 0, for ε ∈ R\{0} and a given function uh . We choose D(A) = {v ∈ H 4 () | v = v = 0 on ∂} endowed with the H 4 -norm, X = H 2 () ∩ H01 (), Y = L2 (), ( u, v )X = ( − u + u, − v + v )L2 () so that uX is equivalent to the usual H 2 -norm, ( u, v )Y = (u, v)L2 () , and A = (− + I )2 ,

Q = 2 + ((3u2h − 1)/ε 2 − 1)I.

We choose Xh as the linear hull of all functions sin(mπx) sin(nπy) (m, n ∈ {1, . . . , K}). Using the orthogonality of the base functions, we can take C(h) = 1/(π 2 (K 2 + 1) + 1), τ1 = 2 + s/(2π 2 + 1), and τ2 = 2 + sC(h) where s =  − 1 + (3u2h − 1)/ε 2 L∞ () . We take uh ∈ Xh as an approximation of the nonlinear equation 

2 u = u =

1 (u − u3 ) ε2 u = 0

in

,

on

∂

(23)

obtained by a Newton–Raphson method using floating-point computation. Problem (23) occurs e.g. as the result of a reduction process applied to some reaction–diffusion system [1], and again, in order to prove the existence of a solution of (23) by Newton-type arguments, the linearization at an approximate solution, i.e. (22), is required. Fig. 2 shows the shape of uh for ε = 0.01 (left)

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Fig. 2. Approximate solutions of (23) for ε = 0.01 (left) and ε = 0.005 (right). Table 5 Invertibility results and bounds of M for the operator (22). K 10 15 20 25 30

ε = 0.01

ε = 0.005

M

κ

ρ

M

κ

ρ

– 0.02083 0.00987 0.00759 0.00678

1.97785 0.40738 0.13476 0.05811 0.02971

0.00600 0.00600 0.00600 0.00600 0.00600

– – – 0.02130 0.00945

20.86776 4.20289 1.34763 0.55948 0.27388

0.00298 0.00298 0.00298 0.00298 0.00298

and ε = 0.005 (right). Table 5 presents values of M, ρ and κ for ε = 0.01 and ε = 0.005, respectively. Acknowledgments The authors heartily thank the anonymous referee for his/her thorough reading and valuable comments. This work was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science, and Technology of Japan (No. 24340018). The computation was mainly carried out using the computer facilities at Research Institute for Information Technology, Kyushu University, Japan. References [1] E. Berchio, F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electron. J. Differential Equations 2005 (2005) 1–20. [2] M. Brown, M. Langer, M. Marletta, C. Tretter, M. Wagenhofer, Eigenvalue enclosures and exclosures for non-selfadjoint problems in hydrodynamics, LMS J. Comput. Math. 13 (2010) 65–81. [3] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. [4] J.G. Heywood, W. Nagata, W. Xie, A numerically based existence theorem for the Navier–Stokes equations, J. Math. Fluid Mech. 1 (1999) 5–23. [5] T. Kinoshita, K. Hashimoto, 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) 289–305. [6] F. Kikuchi, X. Liu, Determination of the Babuska–Aziz constant for the linear triangular finite element, Jpn. J. Ind. Appl. Math. 23 (2006) 75–82.

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