数理解析研究所講究録
第 1719 巻 2010 年 118-129118

無限次元固有値問題に対する固有値の非存在証明

Some eigenvalue excluding methods for infinite dimensional operators
渡部 善隆 (Yoshitaka Watanabe) \dag er

Michael Plum \dager

長藤 かおり (Kaori Nagatou) *中尾 充宏 (Mitsuhiro T. Nakao) ’\dag er

九州大学情報基盤研究開発センター (Research Institute for Information Technology, Kyushu University)*

九州大学大学院数理学研究院 (Faculty of Mathematics, Kyushu University)\d ag er Faculty

1of Mathematics, Karlsruhe University

Introduction
Consider the following eigenvalue problem$(A+Q)u=\lambda Bu$ ,

(1)

where
: $D(A)arrow Y,$: $Xarrow Y$ and: $Xarrow Y$ are linear operators for the complex Hilbert
spaces $D(A)\subset X\subset Y$ . The inner products and norms of $X$ andare denoted $(u, v)_{X},$,$(u, v)_{Y}$ and, respectively. Here, note that it is possible that we$(u,v)_{X}$
use two inner products
.
and
$A$

$B$

$Q$

$Y$

$\langle u,$$v\rangle x$$\Vert u\Vert_{Y}=\sqrt{(u,u)_{Y}}$$\Vert u\Vert_{X}=\sqrt{\langle u,u\rangle_{X}},$$\langle u,v)x$Assumption 1
AlA2

For all $\phi\in Y,$ $A\psi=\phi$ has the unique solution $\psi\in D(A)\subset X$ . Denote this mapping by$A^{-1}:Yarrow X$.

The operator$A$

satisfies$(u,v)_{X}=(Au,v)_{Y}$ ,A3$\forall u\in D(A)$$\forall v\in X$.

(2)

There exists a constant $C_{p}>0$ such that$\Vert Bu\Vert_{Y}\leq C_{p}\Vert u\Vert_{X}$

A4

,

,

$\forall u\in X$.

(3)

There exists a constant $C_{b}>0$ such that$\Vert A^{-1}Bu\Vert_{X}\leq C_{b}\Vert u\Vert_{X}$In actual validated computation, explicit values of$D(A)arrow X$ is compact, $A^{-1}$is also compact.

$C_{p}$,

and$\forall u\in X$$C_{b}$.

(4)

have to be evaluated. If the imbedding

119
1.1

Example 1

In the case of linear elliptic problem for a bounded domain$c\in L^{\infty}(\Omega)$

in $\mathbb{R}^{n}(n=1,2,3)$ , where$\Omega$$b\in L^{\infty}(\Omega)^{n}$,$\}(5)$$\{\begin{array}{ll}-\triangle u+b\cdot\nabla u+cu =\lambda u in \Omega,u =0 on \partial\Omega,\end{array}$$A=-\Delta$,$Q=b\cdot\nabla+c$$D(A)=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$,$(u, v)_{X}=\langle u,$ $v\rangle_{X}=(\nabla u, \nabla v)_{L^{2}(\Omega)}$

The constant$1/(\pi\sqrt{2})$

,

and

,

$B=I_{Xarrow Y}$$X=H_{0}^{1}(\zeta))$

,

,

,

$Y=L^{2}(\zeta\})$$(u, v)_{Y}=(u, v)_{L^{2}(\Omega)}=:,\int_{\Omega}uvdx$.

is the Poincar\’e or Rayleigh-Ritz constant. For example, if $\Omega=(0,1)\cross(0,1),$ $C_{p}=$.

$C_{p}$$C_{b}=C_{p}^{2}$

1.2 Example2
In the case of Orr-Sommerfeld equation(6)$\{\begin{array}{l}(-D^{2}+a^{2})^{2}u+iaR[V(-D^{2}+a^{2})+V’’]u=\lambda(-D^{2}+a^{2})u,u(x_{1})=u(x_{2})=u’(x_{1})=u’(x_{2})=0,\end{array}$$A=(-D^{2}+a^{2})^{2}$$\Omega=[x_{1}, x_{2}]$,,$Q=iaR[V(-D^{2}+a^{2})+V’’]$ ,$D(A)=H^{4}(\Omega)\cap H_{0}^{2}(\Omega)$,$X=H_{0}^{2}(\Omega)$$(u, v)_{X}=\langle u,$ $v\rangle_{X}=((-D^{2}+a^{2})u, (-D^{2}+a^{2})v)_{L^{2}(\Omega)}$

If $\Omega=(-1,1),$ $C_{p}=1$ , and$C_{b}$$B=-D^{2}+a^{2}$ ,,,$Y=L^{2}(\Omega)$,$(u, v)_{Y}=(u, v)_{L^{2}(\Omega)}=:\int_{\Omega}uvdx$.

can be taken as $C_{b}=1/(\pi^{2}/4+a^{2})[9]$ .

1.3 Example3
In the case of a linearized problem of the Kolmogorov equation [5]$\triangle^{2}\psi+R(J(\phi_{N}, \Delta\psi)+J(\psi, \triangle\phi_{N}))=\lambda R\Delta\psi$in$\Omega$,(7)

where $\Omega=(-\pi/\alpha, \pi/\alpha)\cross(-\pi, \pi),$ $J(u, v)=u_{x}v_{y}-u_{y}v_{x},$ $R>0,0<\alpha<1$ andare in function spaces such that$\psi\in X^{3}$

and$X^{k}:=X^{k}0X_{1}^{k}\cdots$ ,$X_{0}^{k}$$X_{m}^{k}$$:= \{\sum_{n=1}^{\infty}a_{n}\cos(ny)$$a_{n} \in \mathbb{C},\sum_{n=1}^{\infty}n^{2k}a_{n}^{2}<\infty\}$$:= \simeq\{\sum_{n=-\infty}^{\infty}a_{n}\cos(m\alpha x+ny)$,$a_{n} \in \mathbb{C},\sum_{n=-\infty}^{\infty}((\alpha m)^{2k}+n^{2k})a_{n}^{2}<$We can set$A=\Delta^{2}$,$Q=R(J(\phi_{N}, \Delta\cdot)+J(\cdot, \triangle\phi_{N}))$,$B=R\triangle$

,

oo

$\},$

$m\geq 1$.

$\phi_{N}\in X^{4}$

120$D(A)=X^{4}$ ,$(u, v)_{X}=(\Delta u, \Delta v)_{L^{2}(\Omega)}$$X=X^{3}$,,$Y=X^{0}$,$(u, v)_{Y}=(u, v)_{L^{2}(\Omega)}=:\int_{\Omega}uvdx$.

Here, for this problem (7), we introduce one more inner product$\langle u,$$v\rangle x=(u_{xxx}+3u_{xxy}+3u_{xyy}+u_{yyy}, v_{xxx}+3v_{xxy}+3v_{xyy}+v_{yyy})_{L^{2}(\Omega)}$

has the third order differential term. Under thesebecausewhich implies-seminorm$C_{b}=R\alpha^{-2}$$C_{p}=R\alpha^{-1}$.
definitions, we can takeand$H^{3}$$Q$$|u|_{H^{3}(\Omega)}$

2 Eigenvalue excludings
Our concept of excluding method is due to the idea by Lahmann-Plum [2](pp.l92). Let
be acandidate point which is suspected that no eigenvalue is close to . We consider equivalently shifted$\mu\in \mathbb{C}$$\mu$eigenvalue problem of Eq.(l) by$\hat{L}u=(\lambda-\mu)Bu$

,

(8)

where
$\wedge u:=Au-f(u)$ :$D(A)arrow Y$,

(9)

.

(10)

and
$f(u):=-(Q-\mu B)u$ :$Xarrow Y$

Then the following excluding result is obtained.

Lemma 1 If the operator$\hat{L}$has the inverse$\hat{L}^{-1}$: $Yarrow D(A)$ , and there exists $\hat{M}>0$ such that$\Vert\hat{L}^{-1}\phi\Vert_{X}\leq\hat{M}\Vert\phi\Vert_{Y}$then there is $no$ eigenvalue$\tilde{\lambda}$,,$\forall\phi\in Y$(11)

of Eq.(l) in the area such that$| \tilde{\lambda}-\mu|<\frac{1}{C_{p}\hat{M}}$

.

(12)

Proof.
For any eigenpair $(A. u)\in \mathbb{C}\cross D(A)$ such that$L\overline{u}=(\tilde{\lambda}-\mu)B\tilde{u}$since$\hat{L}\tilde{u}\in Y$, substituting$\hat{L}\tilde{u}$,into the condition (11) as$\overline{u}\neq 0$$\phi$,

and using A3, we have$\Vert\tilde{u}\Vert_{X}\leq\hat{M}\Vert\hat{L}\tilde{u}\Vert_{Y}$$\leq\hat{M}C_{p}|\tilde{\lambda}-\mu|\Vert\tilde{u}\Vert_{X}$,therefore$| \tilde{\lambda}-\mu|\geq\frac{1}{C_{p}\hat{M}}$then the result is derived.

,
ロ

121
Next, we show an another excluding criterion using an operator on $X$ .
From the assumption A2, the weak problem of Eq.(l) is$(u, v)x=((\lambda B-Q)u, v)_{Y}$Using$A^{-1}$.

$\forall v\in X$

.

(13)

: $Yarrow X$ , the weak problem (13) can be rewritten equivalently in the form
$u=A^{-1}(\lambda B-Q)u$

on

$X$

, hence$u+A^{-1}Qu=\lambda A^{-1}Bu$Then we have shifted eigenvalue problem forsuch that$(u, \lambda)\in X\cross \mathbb{C}$$Lu=(\lambda-\mu)A^{-1}Bu$

where$Lu:=u-A^{-1}f(u)$ :

.

(14)

,

$Xarrow X$

.

(15)

Then, an another excluding lemma is obtained as follows.

Proof.
For any eigenpair$(A. \tilde{u})\in \mathbb{C}\cross X$

of Eq.(13) which satisfies,$L\tilde{u}=(\overline{\lambda}-\mu)A^{-1}B\tilde{u}$taking $\phi\in X$ as$L\tilde{u}$$\tilde{u}\neq 0$,

in (16), we have$\Vert\tilde{u}\Vert_{X}\leq M\Vert L\overline{u}\Vert_{X}$$=|\overline{\lambda}-\mu|M\Vert A^{-1}B\tilde{u}\Vert_{X}$$\leq|\overline{\lambda}-\mu|C_{b}M\Vert u\Vert_{X}$,by A4, therefore$| \tilde{\lambda}-\mu|\geq\frac{1}{C_{b}M}.\square$

Now we will show the relation between the invertibility of

$L$

and

$\hat{L}$.

Proof. Assume $L:Xarrow X$ has the inverse. For any $\phi\in Y$ , there exists $v\in X$ such that $v=A^{-1}\phi$by Al, and there exists $u\in X$ such that $Lu=v$ , namely,$u-A^{-1}f(u)=A^{-1}\phi$$\Rightarrow$$u=A^{-1}(f(u)+\phi)$

.

122

Then by the definition of$A^{-1},$$u\in D(A)$ and$Au=f(u)+\phi$$\hat{L}u=\phi$$\Rightarrow$.
口Combining Lemmata, the following excluding theorem is obtained.
Theorem 1 Assume the operator
that

$L$

has the inverse: $Xarrow X$ , and there exists $M>0$ such$L^{-1}$$\Vert L^{-1}\phi\Vert_{X}\leq M\Vert\phi\Vert_{X}$,$\forall\phi\in X$

,

(18)

then there is $no$ eigenvalue A of Eq.(13) in the area such that$| \tilde{\lambda}-\mu|<\frac{1}{C_{b}M}$

.

(19)

Moreover if there exists $\hat{M}>0$ such that$\Vert\hat{L}^{-1}\phi\Vert_{X}\leq\hat{M}\Vert\phi\Vert_{Y}$then also there is $no$ eigenvalue$\tilde{\lambda}$

,

,

(20)

$\forall\phi\in Y$

of Eq.(l) in the area such that$| \tilde{\lambda}-\mu|<\frac{1}{C_{p}\hat{M}}$

3 lnvertibility condition of

.

(21)

$L$

This section describes a computable condition assuring the invertibility of the linear operator
that

$L$

such

$Lu=u-A^{-1}f(u)$ .

Basically, this verification method is an extension of the one for solutions of second-order elliptic boundary value problems introduced by a part of the authors [6, 7]. From now on, the identity maps on $X$
are denoted by the symbol .
$I$

3.1 Finite dimensional subspaoe and projection error
First we introduce a finite dimensional approximation subspaceorthogonal projection defined by$(v-P_{h}v, v_{h})_{X}=0$

Since$S_{h}$,$S_{h}\subset X$$\forall v_{h}\in S_{h}$, and let

$P_{h}$: $Xarrow S_{h}$ be the

.

(22)

is the closed subspace of $X$ , any element $u\in X$ can be uniquely decomposed into
$u=u_{h}+u_{*}$,$u_{h}\in S_{h},$$u_{*}\in S.$

,

where
$S_{*};=\{u_{*}\in X|u_{*}=(I-P_{h})u, u\in X\}$

We assume$P_{h}$

has the following properties.

.

123

Assumption 2
A5

There exists $C(h)>0$ such that$\Vert(I-P_{h})u\Vert_{X}\leq C(h)\Vert Au\Vert_{Y}$

A6

,

(23)

$\forall u\in D(A)_{;}$

There exists $\nu_{1}>0$ such that$\Vert P_{h}A^{-1}f(u_{*})\Vert x\leq\nu_{1}\Vert u_{*}\Vert_{X}$,$\forall u_{*}\in S_{*}$

.

(24)

There exist $\nu_{2}>0$ and $\nu_{3}>0$ such that

A7$\Vert f(u)\Vert_{Y}\leq\nu_{2}\Vert P_{h}u\Vert_{X}+\nu_{3}\Vert(I-P_{h})u\Vert_{X}$,$\forall u\in X$

.

(25)

For the case of Example 1 (5),-projection, and it can be taken as $C(h)=h/\pi$
is the usual$h/(2\pi)$andfor bilinear and biquadratic element, respectively, for the rectangular mesh on the squaredomain [3]. And $C(h)=0.493h$ for the linear and uniform triangular mesh of the convex polygonal
domain [1]. Here, $h>0$ stands for the maximum mesh size for given finite elements.
, since $P_{h}(-\Delta)^{-1}(b\cdot Vu. +cu_{*}-\mu u_{*})\in S_{h}$ ,For$P_{h}$$H_{0}^{1}$$u_{*}\in S_{*}$$\Vert\nabla P_{h}(-\Delta)^{-1}(b\cdot\nabla u_{*}+cu_{*}-\mu u_{*})\Vert_{L^{2}(\Omega)}^{2}$$=(\nabla P_{h}(-\triangle)^{-1}(b\cdot\nabla u_{*}+cu_{*}-\mu u_{*}), \nabla P_{h}(-\Delta)^{-1}(b\cdot\nabla u_{*}+cu_{*}-\mu u_{*}))_{L^{2}(\Omega)}$$=(b\cdot\nabla u_{\star}+cu_{*}-\mu u_{*}, P_{h}(-\Delta)^{-1}(b\cdot\nabla u_{*}+cu_{*}-\mu u_{\star}))_{L^{2}(\Omega)}$$\leq\Vert b\cdot\nabla u_{*}+cu_{*}-\mu u_{\star}\Vert_{L^{2}(\Omega)}C_{p}\Vert P_{h}\nabla(-\Delta)^{-1}(b\cdot\nabla u_{*}+cu_{*}-\mu u_{*})\Vert_{L^{2}(\Omega)}$,

we have$\Vert P_{h}\nabla(-\Delta)^{-1}(b\cdot\nabla u_{*}+cu_{*}-\mu u_{*})\Vert_{L^{2}(\Omega)}\leq C_{p}\Vert b\cdot\nabla u_{*}+cu_{*}-\mu u_{*}\Vert_{L^{2}(\Omega)}$$\leq C_{p}(\Vert b\cdot\nabla u_{*}\Vert_{L^{2}(\Omega)}+\Vert(c-\mu)u_{*}\Vert_{L^{2}(\Omega)})$$=C_{p}(\Vert|b|_{E}\Vert_{L\propto(\Omega)}+C_{p}\Vert c-\mu\Vert_{L\approx(\Omega)})\Vert\nabla u_{*}\Vert_{L^{2}(\Omega)}$

Therefore we can take$\nu_{1}=C_{p}(\Vert|b|_{E}$.

I $L\propto(\Omega)p(\Omega)$ ,$\nu_{2}=\nu_{3}=\Vert|b|_{E}\Vert_{L^{\infty}(\Omega)}+C_{p}\Vert c-\mu\Vert_{L^{\infty}(\Omega)}$.

3.2 Newton-like method
We will show that the problem $Lu=0$ has only unique solution $u=0$ . Defining $F:Xarrow X$ by$Fu=A^{-1}f(u)$ ,(26)

the problem $Lu=0$ can be rewritten equivalently in the fixed-point form
$u=Fu$ .

In order to prove the uniqueness $(u=0)$ of the fixed-point of $F$ on $X$ , for a nonempty, bounded, convex
and closed set $U\subset X$ centered zero, we will check$\overline{FU}\subset$

int $(U)$ .

124

From uniqueness of decomposition for
rewritten as$S_{h}$

and$S_{*}$, the fixed-point equation $u=Fu$ on $X$ is equivalently$\{\begin{array}{l}P_{h}u = P_{h}Fu,(27)(I-P_{h})u = (I-P_{h})Fu.\end{array}$Now, let us define the Newton-like operator$\mathcal{N}_{h}$: $Xarrow S_{h}$ by$\mathcal{N}_{h}u:=P_{h}u-[I-F]_{h}^{-1}P_{h}(I-F)u$.

means the inverse of the restriction of the operator $P_{h}(I-F):Xarrow S_{h}$ toHere $[I-F]_{h}^{-1}$ :. Note that the existence of $[I-F]_{h}^{-1}$ is equivalent to the invertibility of a matrix, which is numerically, using a map on $X$
checked in the actual verified computations. Since
defined by$S_{h}arrow S_{h}$$S_{h}$$T$$P_{h}u=P_{h}\mathcal{N}_{h}u\Leftrightarrow P_{h}u=P_{h}Fu$$Tu=\mathcal{N}_{h}u+(I-P_{h})Fu$,

we find that the two fixed-point problems: $u=Fu$ and $u=Tu$ are equivalent.
Next, for positive constants$\hat{\gamma}$

and$\hat{\alpha}$, set$U_{h}:=\{u_{h}\in S_{h}|\Vert u_{h}\Vert x\leq\hat{\gamma}, \}\subset S_{h}$.

$U$$:=\{u_{*}\in S_{*}|\Vert u_{*}\Vert x\leq\hat{\alpha}_{:}\}\subset S.$

and define a candidate set $U\subset X$ by$U:=U_{h}+U_{*}$

,

,

.

Then a sufficient condition for the invertibility result is as follows [4].

Lemma 4 When an inclusion$\overline{TU}\subset$

holds, then$L$

int $(U)$(28)

is invertible.

Proof. If there exists $u\in X$ such that $Lu=0$ and $u\neq 0,$, we haveoperator, for any$u$also satisfies $u=Tu$ . Since$T$

is linear$t\in \mathbb{R}$$T(tu)=tT(u)$$=tu.$

Then, we can choose$\hat{t}\in \mathbb{R}$satisping$\hat{t}u\in\partial U$

However, this contradicts withunique solution of $Lu=0$ .

$\overline{TU}\subset$.

int $(U)$ and $T(tu)=tu.$ Therefore $u=0$ . That is, $u=0$ is a$\square$

We now describe a procedure to construct the candidate setof $X$ which is expected to satisfy the
inclusion (28). From the unique decomposeness of $u\in U$ , we will check for finite and infinite partseparately.
, can be written as
int
The finite dimensional part of the inclusion,$\overline{\mathcal{N}_{h}U}\subset$$(U_{h})$$\sup_{u\in U}\Vert \mathcal{N}_{h}u\Vert x<\hat{\gamma}$

.

$U$

125
On the other hand, the infinite dimensional part of the inclusion,$\overline{(I-P_{h})FU}\subset$

int$(U_{*})$, means$(I-P_{h})A^{-1}f(u)\in$ int $(U_{*})$

for any $u\in U$ . Therefore, from the assumption A5 (23), the condition$C(h) \sup_{u\in U}\Vert f(u)\Vert_{Y}<\hat{\alpha}$

is sufficient. From this we can derive the following lemma.

3.3 Criterion for the invertibility of$L$

In order to confirm the verification Lemma 5, for given positive parameters

compute$\gamma:=\sup_{\hat{u}\in U}\Vert \mathcal{N}_{h}\hat{u}\Vert x$,$\alpha:=C(h)\sup_{\hat{u}\in U}\Vert f(u)\Vert_{Y}$$\hat{\alpha}$

and$\hat{\gamma}$, we have to,and confirm$\gamma<\hat{\gamma}$,$a<\hat{\alpha}$.

contains the infinite dimensional term . Moreover,In the actual computation, the candidate set
it is impossible to avoid the effect of rounding error of floating point arithmetic. However, by norm
estimates, and interval arithmetic software taking into account effects of rounding error, we can obtain$U$mathematically rigorous upper bounds forthese computations in more detail.
For any $u\in U$ such that $u=u_{h}+u_{*},$$\gamma$$U_{*}$

and a and with possible over-estimates. Let us describe$u_{h}\in U_{h},$$u_{*}\in U_{*}$, we obtain$\mathcal{N}_{h}u=P_{h}u-[I-F]_{h}^{-1}P_{h}(I-F)u$$=[I-F]_{h}^{-1}P_{h}A^{-1}f(u_{*})$

from the linearity of .
Here setting$f$$s_{h}:=P_{h}A^{-1}f(u_{*})= \sum_{n=1}^{N}s_{h,n}\acute{\phi}_{n}\in S_{h}$$\mathcal{N}_{h}u:=\sum_{n=1}^{N}t_{h,n}\hat{\phi}_{n}\in S_{h}$where$\{\hat{\phi}_{n}\}_{1}^{N}$is basis of$S_{h}$,,$t:=[t_{h,n}]\in \mathbb{C}^{N}$$s:=[s_{h,n}]\in \mathbb{C}^{N}$,,

with $N:=\dim S_{h}$ , the definition of $[I-F]_{h}^{-1}$ implies$((I-F)\mathcal{N}_{h}u, v_{h})x=(s_{h}, v_{h})x$,$\forall v_{h}\in S_{h}$

.

(29)

126

From A2 (2), the eq.(29) is equivalent as$\sum_{n=1}^{N}t_{h,n}((\hat{\phi}_{n},\hat{\phi}_{m})_{X}-(f(\phi_{n}),\hat{\phi}_{m})_{Y})=\sum_{n=1}^{N}s_{h,n}(\hat{\phi}_{n},\hat{\phi}_{m})_{X}$,$1\leq m\leq N$.

Therefore defining$[A_{1}]_{mn}:=(\hat{\phi}_{n},\hat{\phi}_{m})_{X}$,$[A_{2}]_{mn}:=-(f(\phi_{n}),\hat{\phi}_{m})_{Y}$,$G:=A_{1}+A_{2},\ovalbox{\tt\small REJECT}$$[A_{3}]_{mn}:=(\hat{\phi}_{n},\hat{\phi}_{m}\rangle x$and$L_{3}$

is the matrix decomposed factor of$A_{3}$

such that$Gt=A_{1}s$,$A_{3}=L_{3}L_{3}^{T}$

we have,and,$\Vert \mathcal{N}_{h}u\Vert_{X}=\Vert L_{3}^{T}t\Vert_{E}\leq\rho\Vert s_{h}\Vert_{X}$(30)

where $\rho>0$ is an upper bound satisfying(31)

for the matrix 2-norm. Evaluations of
singular value of a matrix.
Here note that when $(u, v)_{X}=\langle u,$$\Vert\cdot\Vert_{E}$$v\rangle_{X},$$\rho$

can be reduced to the computation of the maximum$A_{1}=L_{3}L_{3}^{T}$then$\rho$

is estimated by

From assumption A6$\Vert s_{h}\Vert_{X}\leq\nu_{1}\Vert u_{*}\Vert_{X}\leq\nu_{1}\hat{\alpha}$

,

(32)

hencefore$\Vert \mathcal{N}_{h}u\Vert x\leq\rho\nu_{1}\hat{\alpha}$.

,$\forall u\in U$

Moreover, from assumption A7,$\Vert f(u)\Vert_{Y}\leq\nu_{2}\Vert u_{h}\Vert x+\nu_{3}\Vert u_{*}\Vert_{X}$$\leq\nu_{2}\hat{\gamma}+\nu_{3}\hat{\alpha}$

.

(33)
(34)

Therefore, the following criterion for verification holds.
Theorem 2 If$\kappa:=C(h)(\rho\nu_{1}\nu_{2}+\nu_{3})<1$

holds, then the operator$L$

has the inverse.

(35)

127
4 Upper bound of $M$
We can get an upper bound $M>0$ satisfies$\Vert L^{-1}\phi\Vert x\leq M\Vert\phi\Vert_{X}$under the invertibility criterion for$L$,$\forall\phi\in X$.

by the following theorem.

Theorem 3 Under the same assumption (35), an upper bound $M>0$ for (16) can be taken as$M=\Vert \mathcal{M}\Vert_{E}$

,

where
$\mathcal{M}=[^{\rho}(1+\frac{\nu_{1}C(h)\nu_{2}\rho}{1-\kappa(h)\nu_{2}\rho 1-\kappa})\frac{C}{}$

5

Upper bound for$\frac\frac{1-\kappa\rho\nu_{1}1}{1-\kappa}]\in \mathbb{R}^{2\cross 2}$

.

(36)

$\hat{M}$We can obtain an upper bound $\hat{M}>0$ satisfies$\Vert\hat{L}^{-1}\phi\Vert x\leq\hat{M}\Vert\phi\Vert_{Y}$under the invertibility criterion for$\hat{L}$.

,$\forall\phi\in Y$

by the following theorem.

Defining$[A_{4}]_{nm};=(\hat{\phi}_{m},\hat{\phi}_{n})_{Y}$$L_{4}$

is the matrix decomposed factor of $A_{4}:A_{4}=L_{4}L_{4}^{T}$ , and

,

(37)

$\hat{\rho}>0$

is an upper bound satisfying(38)

Then the following can be shown.

6 Verified examples
Consider the two-dimensional self-adjoint eigenvalue problem:$-\triangle u+\nu(3u_{h}^{2}-2(a+1)u_{h}+a)u$$=$$\lambda u$

$u$

$=$

$0$

$\{$

in

on

$\Omega$$\partial\Omega$

,
,

(40)

128

and
so-called Allen-Cahn equation:
where } $=(0,1)\cross(0,1),$$\zeta$

$\nu$

$a$

are positive constants, and$-\Delta u$$=$$\nu u(u-a)(1-u)$

$u$

$=$

$0$

$u_{h}$(41)’$onin$$\{$

is an approximate solution of the$\partial\Omega\Omega.$

It is known that this equation has two solution branches with respect to the parameter $\nu>0[8]$ . We
considered both case in which
are lower and upper branch finite element solutions for $\nu=150$ and$a=0.01$ in linear and uniform triangular mesh of the . We can take $C(h)=0.493h$ for the uniform
partition size $h>0$ .
Setting $w_{h}=\nu(3u_{h}^{2}-2(a+1)u_{h}+a),$ $A=-\Delta,$ $Q=c=w_{h},$ $B=I_{Xarrow Y},$ $C_{p}=1/(\pi\sqrt{2}),$ $C_{b}=1/(2\pi)$ ,$u_{h}$$\Omega$$D(A)=H^{2}(\Omega)\cap H_{0}^{1}(\Omega),$$Y=L^{2}(\Omega)$$X=H_{0}^{1}(\Omega),$

, For

$u$

. S., since$E$$\Vert\nabla P_{h}(-\Delta)^{-1}(c-\mu)u_{*}\Vert_{L^{2}(\Omega)}^{2}\leq\Vert\nabla(-\Delta)^{-1}(c-\mu)u_{*}\Vert_{L^{2}(\Omega)}^{2}$$=((c-\mu)u_{*}, (-\Delta)^{-1}(c-\mu)u_{*})_{L^{2}(\Omega)}$$\leq\Vert(c-\mu)u_{*}\Vert_{L^{2}(\Omega)}C_{p}\Vert\nabla(-\Delta)^{-1}(c-\mu)u_{*}\Vert_{L^{2}(\Omega)}$,

we have$\Vert\nabla(-\Delta)^{-1}(cu-\mu)u_{*}\Vert_{L^{2}(\Omega)}\leq C_{p}\Vert c-\mu\Vert_{L(\Omega)}\propto\Vert u_{*}\Vert_{L^{2}(\Omega)}$

Therefore using$\Vert u_{*}\Vert_{L^{2}(\Omega)}\leq C(h)\Vert u_{*}\Vert_{H_{0}^{1}(\Omega)}$,

we can take$\nu_{1}=C_{p}C(h)\Vert c-\mu\Vert_{L^{\infty}(\Omega)}$

The norm$\Vert c-\mu\Vert_{L(\Omega)}\propto$,$\nu_{2}=C_{p}\Vert c-\mu\Vert_{L(\Omega)}\propto$,$\nu_{3}=C(h)\Vert c-\mu\Vert_{L(\Omega)}\propto$.

can be estimated as$\Vert c-\mu\Vert_{L^{\infty}(\Omega)}=\Vert 3\nu(u_{h}-\frac{a+1}{3})^{2}-\frac{\nu(a+1)^{2}}{3}+a\nu-\mu\Vert_{L\propto(\Omega)}$

Since$0\leq u_{h}(x)\leq\Vert u_{h}\Vert_{L^{\infty}(\Omega)}$$u_{h}(x)=\Vert u_{h}\Vert_{L^{r}(\Omega)}$orfor any $x\in\Omega,$$u_{h}(x)= \frac{a+1}{3}$$\Vert c-\mu\Vert_{L(\Omega)}\propto$

can be attained when $u_{h}(x)=0$ or.

6.1 Excluding result for $h=1/50$ (lower solution)
The approximate absolute minimum eigenvalue is 16.616847.
$\overline{\overline{-16499.917121.98780.5426229.58850.08590.530646.96180.0946}}\mu.\rho\hat{\rho}\kappa MR_{1}\hat{\kappa}\hat{M}R_{2}$

$-16$
$-14$
$-11$
$0$

11
15
17
18
18.3

350991
82357
38255
27078
60905
112572
19589731.121637800577296
18222
0.84930.37800.8556
15841
27588
43843
53258

01974
0.05470.03170.03750.11580.23700.43330.70510.8626469723
91809
41686
29916
73341
157184368329112462129310210.4293
21500
47352
65981
26915
12558
0.535901755
0.067301932
0.05380.03130.02510.074501506
0.27350.44380.542496071
19324
0.88080.39130.9342
18855
38403
7.97251177250.462422991
5.044411356747560
23564
1.15690.55720.2773

129$R_{1}$

and$R_{2}$

stand for the excluding radius $1/(C_{b}M)$ and $1/(Cp\hat{M})$ , respectively.

6.2 Excluding result for $h=1/50$ (upper solution)$\frac{Theapproximateabso1ute\min imumeigenva1ueis47.107986}{\mu\rho\hat{\rho}\kappa\Lambda IR_{1}\hat{\kappa}\hat{M}R_{2}}\overline{\overline{-231.19150.11400.14851.842110.71610.07160.133333.3452}}$

$0$

23
35
40
43
44
44.5

1.54262.59734.75287.804313.2038
17.320920.5637

9.1674 0.0714 0.1873
2.15320.1645 0.1369
5.7532 0.0856 0.3537
3.43110.3126 0.1517
2.9966 0.1256 0.7218
6.58810.6161 0.2114
1.5926 0.1922 1.3232
12.3947
1.0465 0.3167
0.5984 0.3477 2.8337
32.9839
1.8087 0.5655
76.7302 0.2572 0.4664 4.5781
2.3901 0.7550
0.0847 0.5599
0.9042 232.8762
090422328762008470559966147
6284812.8481$\cdot$23.7298
12.5644
6.1561
3.35791.56790.97040.6716

参考文献[1] Kikuchi, F. and Xuefeng, L., Determination of the Babuska-Aziz constant for the linear triangular
finite element, Japan J. $Ind$ . Appl. Math. 23, 75-82 (2006).
[2] Lahmann, J. and Plum, M., A computer-assisted instability proof for the Orr-Sommerfeld equation
with Blasius profile, J. A $ppl$ . Math. Mech. $(ZA$ MM 84(3), 188-204 (2004).
[3] Nakao, M. T., Yamamoto, N. and Kimura, S., On best constant in the optimal error estimates for-projection into piecewise polynomial spaces, Joumal of Approximation Theory 93, 491-500
the$)$$H_{0}^{1}$(1998).
[4] Nakao, M. T., Yamamoto, N. and Nagatou, K. Numerical verffications for eigenvalues of secondorder elliptic operators, Japan Joumal of Industmal and Applied Mathematics 16, 307-320 (1999).
[5] Nagatou, K. A computer-assisted proof on the stability of the Kolmogorov flows of incompressible

viscous fluid, Joumal of Computational and Applied Mathematics 169, 33-44 (2004).
[6] Nakao, M. T. and Watanabe, Y.. An efficient approach to the numerical verification for solutionsof elliptic differential equations, Numerical Algorithms, 37, 311-323 (2004).
[7] Nakao, M. T., Hashimoto, K. and Watanabe, Y., A numerical method to verify the invertibility oflinear elliptic operators with applications to nonlinear problems, Computing, 75, 1-14 (2005).
[8] Watanabe, Y., and Nakao, M. T., Numerical verifications of solutions for nonlinear elliptic equations,Japan Joumal of Industrial and Applied Mathematics 10, 165-178 (1993).
[9] Watanabe, Y., Plum, M. and Nakao, M. T., Computer-assisted instability proof for Poiseuille flow,

J. Appl. Math. Mech. (ZAMM) 89(1), 5-18 (2009).