数理解析研究所講究録 1441 巻 2005 年 61-74

6l

Numerical verification by infinite dimensional
Newton’s method for stationary solutions of theNavier-Stokes problems
九州大学大学院数理学研究院 中尾 充宏 (Mitsuhiro T. Nakao),
Faculty of Mathematics, Kyushu University
九州大学大学院数理学研究院 長藤 かおり (Kaori Nagatou),
Faculty of Mathematics, Kyushu University
九州大学大学院数理学府 橋本 弘治 (Kouji Hashimoto)
Graduate School of Mathematics, Kyushu University
概要

We present a numerical method to enclose stationary solutions of the NavierStokes equations, especially 2-D driven ca ity problem with regularized boundary
condition. Our method is based on the infinite dimensional Newton’s method by
estimating the inverse of the corresponding linearized operator. The method canbe applied to the case for high Reynolds numbers and we show some numericalexamples which confirm us the actual effectiveness.

Key words: Numerical enclosure method, driven cavity flows, infinite dimensional
Newton’s method.

1

Introduction
We consider the following Navier-Stokes equations$-\Delta u+R\cdot(u\cdot\nabla)u+\nabla p=$$f$$\mathrm{d}\mathrm{i}\mathrm{v}u=$

$0$

$\{$

in ,
in ,
$\Omega$(1.1)$\Omega$$u=g$ on on,
where , and $R$ are the velocity vector, pressure and the Reynolds number, respectively
and the flow region is a convex polygonal domain in R. In what follows, for eachdenote the -Sobolev space of order $m$ on . The function
rational number $m$ , let$g=(g_{1},g_{2})\in H^{1/2}(\partial\Omega)$ ,$f=(f_{1}, f_{2})$ means a density of body forces with $f\in(H^{1}(\Omega))$ andonsatisfying
where we assume that there exists a function
$u$

$p$

$\Omega$$H^{m}(\Omega)$$\Omega$$L^{2}$$\varphi$$\in H^{2}(\Omega)$$(\varphi_{y}, -\varphi_{x})=g$

an.

82
The above problem was disccused by Wieners [7] for low Reynolds numbers. The
method proposed in it is based on Newton-Kantorovich theorem but it would not beable to apply to high Reynolds numbers, because the estimation for the inverse of thelinearized operator directly depends on the Reynolds number. We also uses Newton
type verification condition, but the method which verifies the invertibility of linearized
operator is different from the Wieners’ formulation. Our method has an advantage whichenables us to verify the invertibility of the linearlized operator, even for high Reynolds
numbers, provided that the approximation space is sufficiently accurate and that theinverse operator acually exists in the rigorous sense.

In Section 2 we introduce a stream function formulation of our problem and consider
the linearized operator which is needed in the infinite dimensional Newton method. In
Section 3, we formulate the verification method by computer to verify the invertibilityof the linearized operator. We then derive the infinite dimensional Newton method to
enclose the solution in Section 4 and we show some enclosure results in Section 5.

Stream function and the linearized operator

2

We first introduce a stream function satisfying $u=(\psi_{y}, -\psi_{x})$ by the incompressibility
condition in (1.1), where subscripts and denote the partial derivative for andrespectively. Using this function we can rewrite the equations (1.1) as$\psi$$x$$\Delta^{2}\psi+R\cdot J(\psi, \triangle\psi)$$=$$\psi$

$=$

$\{$

$=$

$\frac{\partial\psi}{\partial n}$

$x$

$y$

$(f_{2})_{x}-(f_{1})_{y}\mathrm{i}\mathrm{n}\Omega$$\varphi$$\frac{\mathit{8}\varphi}{\partial n}$

on on,
on ,

,
(2.1)

$\partial\Omega$where is a bilinear form defined by $J(u, v)=u_{x}v_{y}-u_{y}v_{x}$ andwe havederivative. Newly denoting as
$J$

$u$

$\Delta^{2}u+\Delta^{2}\varphi+R$$\psi$$\frac{\partial}{\partial n}$

stands for the normal$-\varphi$. $J(u+\varphi, \Delta(u+\varphi))$$=$$(f_{2})_{x}-(fi)_{y}$

$=$

$0$

$\{$
$u= \frac{\partial u}{\partial n}$$\mathrm{i}\mathrm{n}\Omega$on,$\partial\Omega$.

of (2.2), where
Our aim is to verify the existence of a weak solutionon{} with the inner product $<u$ ,$u\in H_{0}^{2}(\Omega)$$v \in H^{2}(\Omega)|v=\frac{\mathit{8}v}{\partial n}=0$

for ,
$u$

$v\in H_{0}^{2}(\Omega)$.

$y$$\partial\Omega$(2.2)$H_{0}^{2}(\Omega)\equiv$$v>_{H_{0}^{2\equiv}}(\Delta u, \Delta v)_{L^{2}}$

63

Let
Usually$S_{h}$

that depends on $h(0 <h<1)$ .
be a finite dimensional subspace ofis taken to be a finite element subspace with mesh size .
$H_{0}^{2}(\Omega)$$h$$S_{h}$

We calculate an approximate solution
space, satisfying for all$u_{h}\in C^{1}(\Omega)$of (2.2) in the finite dimensional,, (2.3)$v_{h}\in S_{h}$$(\Delta u_{h}+\Delta\varphi, \Delta v_{h})_{L^{2}}+(R\cdot J(u_{h}+\varphi, \Delta(u_{h}+\varphi))$ $v_{h})_{L^{2}}=((f_{2})_{x}-(f_{1})_{y}, v_{h})_{L^{2}}$and calculate
represented asby smoothing of$u_{s}\in C^{2}(\Omega)$

Lu $\equiv\Delta^{2}u+R$ . { ($J$$u_{s}+\varphi$, $\Delta u)+J(u$ , A

and is considerd as the operator from:the existence of the inverse$\mathcal{L}$$\mathcal{L}^{-1}$$u_{h}$. Then the linearized operator at$(u_{s}+\varphi))$$H_{0}^{2}(\Omega)$to$H^{-2}(\Omega)arrow H_{0}^{2}(\Omega)$$u_{s}$

is},in weak sense. We will verify
and formulate the infinite dimensional$H^{-2}(\Omega)$

Newton’s method.

3

Invertibility of the linearized operator
By direct computations, we find that for any $q\in H^{-2}(\Omega)$ there exists a unique solution
satisfying$v\in H_{0}^{2}(\Omega)$$\mathrm{A}^{2}v=$

rn ,
.
on
$\Omega$

$q$

$\{$

$=$

$v= \frac{\partial v}{\partial n}$

$0$

(3.1)

$\partial\Omega$of the equation (3.1) then $K$
For $q\in H^{-2}(\Omega)$ , let $Kq$ be the unique solution. Using the following compact operator on
tois a compact operator from$v\in H_{0}^{2}(\Omega)$$H^{-1}(\Omega)$$H_{0}^{2}(\Omega)$$H_{0}^{2}(\Omega)$$F_{1}(u)\equiv-R\cdot K${$J(u_{s}+\varphi$, Au) $+J(u,$$\Delta(u_{s}+\varphi))$},the equation Cu $=0$ is equivalent to the fixed point equation$u=F_{1}(u)$

.

(3.2)

84

In order to show the invertibility of the linearized operator , by the Fredholm alternative,.
we only have to show the uniqueness of the solution of the equation$\mathcal{L}$$\mathcal{L}u=0$

Now let$P_{h}$:$H_{0}^{2}(\Omega)rightarrow S_{h}$denote the$H_{0}^{2}$for all$(\mathrm{A}(\mathrm{u}-P_{h}u), \Delta v_{h})_{L^{2}}=0$-projection defined by$v_{h}\in S_{h}$,and we derive some error estimations for P&. In what follows, we restrict ourselves to thatis the set of piecewise bicubic
the domain is a unit square $(0, 1)$ $(0,1)$ , and that
Hermite functions with uniform mesh on (e.g., [5]). However, our verification principle
can also be applied to more general domains and approximation subspaces, when theappropriate a priori error estimates are obtained.
$\Omega$$S_{h}$$\mathrm{x}$$\Omega$

At first we derive the following interpolation error estimation.
denote the cubic Hermite interpolation on $\Omega=(0,1)^{2}$ . For $u\in$Lemma 1. Let
we have$\mathrm{I}_{\Omega}$$H^{4}(\Omega)\cap H_{0}^{3}(\Omega)$$||u- \mathrm{I}_{\Omega}u||_{H_{0}^{2}}\leq\frac{h^{2}}{\pi^{2}}||\Delta^{2}u||_{L^{2}}$

.

(3.3)

Proof. At first, we have$|u|_{H^{4}(\Omega)}=||\Delta^{2}u||_{L^{2}}$where$|\cdot$$|_{H^{4}(\Omega)}$fordenotes the$u\in H^{4}(\Omega)\cap H_{0}^{3}(\Omega)$$H^{4}$

seminorm on 0 defined by$|u|_{H^{4}}^{2}= \sum_{n_{1}+n_{2}=4,n_{1},n_{2}\in \mathrm{N}\cup\{0\}}||\frac{\partial^{4}u}{\partial x^{n_{1}}\partial y^{n_{2}}}||_{L^{2}}^{2}$

Actually, by expanding as
Fourier series, we have

$u=\Sigma_{m,n=1}^{\infty}a_{mn}$ $\sin$mvr$\sin n\pi y$$|u|_{P_{l}^{4}}^{2}= \frac{1}{4}\sum_{m,n=1}^{\infty}\{(m\pi)^{2}+(n\pi)^{2}\}^{4}a_{mn}^{2}=||\Delta^{2}u||_{L^{2}}^{2}$

Denoting$\mathrm{I}_{H}$, noting the convergence of the.

the Hermite interpolation on $I=(0,1)$ , we have by [5]$||u-\mathrm{I}_{H}u||_{H_{0}^{2}}$$\leq$$||u-\mathrm{I}_{H}u||_{H_{0}^{2}}$$\leq$$||u^{\prime/}||_{L^{2}}$for$\frac{h^{2}}{\pi^{2}}||u^{(4)}||_{L^{2}}$$u\in H_{0}^{2}(I)$,for $u\in H^{4}(I)\cap H_{0}^{2}(I)$ .

65

Representing$\mathrm{I}_{H}$$||u-\mathrm{I}_{\Omega}u||_{H_{0}^{2}}$as$\mathrm{I}_{H_{\mathrm{z}}}$and$\mathrm{I}_{H_{y}}$

for

$x$

and

$y$

direction respectively, we have$=$$||u-\mathrm{I}_{I_{x}}u$$\leq$$||u-\mathrm{I}_{H_{x}}u||_{H_{0}^{2}}+||\mathrm{I}_{H_{x}}(u-\mathrm{I}_{H_{y}}u)||_{H_{0}^{2}}$$\leq$$||u-\mathrm{I}_{H_{x}}u||_{H_{0}^{2}}+||\mathrm{I}_{H_{x}}(u-\mathrm{I}_{H_{y}}u)-(u-\mathrm{I}_{H_{y}}u)||_{H_{0}^{2}}+||u-\mathrm{I}_{H_{y}}u||_{H_{0}^{2}}$$\leq$$\frac{h^{2}}{\pi^{2}}||\frac{\partial^{4}u}{\partial x^{4}}||L^{2} +|| \frac{\partial^{2}}{\partial x^{2}}(u-\mathrm{I}_{H_{y}}u)||_{L^{2}}+\frac{h^{2}}{\pi^{2}}||\frac{\partial^{4}u}{\partial y^{4}}||_{L^{2}}$$\leq$$\frac{h^{2}}{\pi^{2}}||\frac{\partial^{4}u}{\partial x^{4}}||_{L^{2}}+\frac{h^{2}}{\pi^{2}}||\frac{\partial^{4}}{\partial x^{2}\partial y^{2}}u||_{L^{2}}+\frac{h^{2}}{\pi^{2}}||\frac{\partial^{4}u}{\partial y^{4}}||_{L^{2}}$$\leq$$+\mathrm{I}_{I_{x}}u-\mathrm{I}_{H_{x}}\mathrm{I}_{H_{y}}u||_{H_{0}^{2}}$$\frac{h^{2}}{\pi^{2}}|u|_{H^{4}}=\frac{h^{2}}{\pi^{2}}||\Delta^{2}u||_{L^{2}}$1
Using the estimation (3.3) and the relation$||u-P_{h}u||_{H_{0}^{2}}= \inf_{\xi\in S_{h}}||u-\xi||_{H_{0}^{2}}\leq||u-\mathrm{I}_{\Omega}u||_{H_{0}^{2}}$we have the following error estimatinos for$P_{h}$,.

In what follows, we will discuss under the assumption that the error estimation (3.3) is. Of course, as this assumption is not yet assured, we havevalid for all
to validate it in another paper or to decide a correct constant, which might be greaterthan , for further study.
$u\in H^{4}(\Omega)\cap H_{0}^{2}(\Omega)$$\frac{h}{\pi}F2$

Lemma 2. For$u\in H^{4}(\Omega)\cap H_{0}^{2}(\Omega)$$||u-P_{h}u||_{H_{0}^{2}}$$\leq$$||u-P_{h}u||_{H_{0}^{1}}$$\leq$$||u-P_{h}u||_{L^{2}}$$\leq$

we have$\frac{h^{2}}{\pi^{2}}||\Delta^{2}u||_{L^{2}}$

,

(3.4)

$\frac{h^{3}}{\pi^{3}}||\Delta^{2}u||_{L^{2}}$

,

(3.5)

$\frac{h^{4}}{\pi^{4}}||\Delta^{2}u||_{L^{2}}$

.

(3.6)

Proof. The first estimation (3.4) is trivial by Lemma 1.

ee
For $e\equiv u-P_{h}u$ we denote$\triangle^{2}\phi$

$=$

$e$

$\phi=\frac{\partial\emptyset}{\partial n}$

$=$

$0$

$\{$

rn
on

$\phi\in H^{4}(\Omega)\cap H_{0}^{2}(\Omega)$as the solution of the follow ing equations:(3 . 7)$\Omega$an.

Then we have$||u-P_{h}u||_{L^{2}}^{2}$$=$$(u-P_{h}u, u-P_{h}u)_{L^{2}}$$=$$(e, \Delta^{2}\phi)_{L^{2}}$

$=$

$($

$=$

(Ae,

$\leq$$||\Delta e||_{L^{2}}||\Delta(\phi-p_{h}\phi)||_{L^{2}}$$\leq$$||\Delta e||_{L^{2}}$$\leq$

Ae,$\Delta\phi)_{L^{2}}$$\Delta(\phi-P_{h}\phi)_{L^{2}}$.

$\frac{h^{2}}{\pi^{2}}||\Delta^{2}\phi||_{L^{2}}$$|| \Delta e||_{L^{2}}\cdot\frac{h^{2}}{\pi^{2}}||e||_{L^{\mathit{2}}}$.

holds and the third estimation (3.6) is proved.
Therefore
Finally using (3.4) and (3.6) we can prove the second estimation (3.5) as follows:$||u-P_{h}u||_{L^{2}} \leq\frac{h^{2}}{\pi^{2}}||u-P_{h}u||_{H_{0}^{2}}$$||u-P_{h}u||_{H_{0}^{1}}^{2}$$=$$(\nabla(u-P_{h}u), \nabla(u-P_{h}u))_{L^{2}}$$=$$(-\Delta(u-P_{h}u), u-P_{h}u)_{L^{2}}$$\leq$$||\Delta(u-P_{h}u)||_{L^{2}}||u-P_{h}u||_{L^{2}}$$\leq$$\frac{h^{2}}{\pi^{2}}||\Delta^{2}u||_{L^{2}}$.

$\frac{h^{4}}{\pi^{4}}||\Delta^{2}u||_{L^{2}}$.

1Now, as in [1] or [3], we decompose (3.2) into the finite and infinite dimensional parts:$P_{h}u=$$P_{h}F_{1}(u)$

,

$\{$

$(I-P_{h})u=$$(I-P_{h})F_{1}(u)$

(3.8)

.

Since we apply a Newton-like method only for the former part of (3.8), we define thefollowing operator:$\Lambda_{h}^{\Gamma 1}(u)\equiv P_{h}u-[I-F_{1}]_{h}^{-1}(P_{h}u-P_{h}F_{1}(u))$,

67of the. And we assume that the restriction towhere I is the identity map on. The validity of this assumption
has the inverse
operator $P_{h}[I-F_{1}]$ :can be numerically confirmed in actual computations.
$S_{h}$$H_{0}^{2}(\Omega)$$[I-F_{1}]_{h}^{-1}$$S_{h}arrow S_{h}$We next define the operator$T_{1}$:$H_{0}^{2}(\Omega)arrow H_{0}^{2}(\Omega)$$T_{1}(u)\equiv N_{h}^{1}(u)+(I-P_{h})F_{1}(u)$Then$T_{1}$

becomes a compact map on$u=T_{1}(u)\Leftrightarrow u=F_{1}(u)$by.

$H_{0}^{2}(\Omega)$

and we have the following equivalence relation.

, which is
Our purpose is to find a unique fixed point of in a certain set
called a ‘candidate set’. Given positive real numbers and we define the corresponding
candidate set by$U\subset H_{0}^{2}(\Omega)$$T_{1}$$\gamma$$\alpha$$U$$U\equiv U_{h}\oplus[\alpha]$

(3.9)

,

where
orthogonal complement of$U_{h}\equiv\{\phi_{h}\in S_{h}|||\phi_{h}||_{H_{-0}^{2}}\leq\gamma\})[\alpha]\equiv\{\phi_{[perp]}\in S_{[perp]}|||\phi[perp]||_{H_{0}^{2}}\leq\alpha\}$$\overline{T_{1}(U)}\subset$$S_{h}$

in$H_{0}^{2}(\Omega)$

and Si means the. If the relation(3.10)

int(tf)holds, by Schauder’s fixed point theorem and the linearity of , there exists a fixed point
in and the fixed point is unique, i.e., $u=0$ , which implies that the operatorofis invertible. Decomposing (3.10) into finite and infinite dimensional parts we have asufficient condition for (3.10) as follows:$T_{1}$$u$$U$$T_{1}$$C$$\sup_{u\in U}||N_{h}^{1}(u)||_{H_{0}^{2}}<\gamma$(3.11).

$\{$$\sup_{u\in U}||(I-P_{h})F_{1}(u)||_{H_{0}^{2}}<\alpha$

We now derive the following theorem in which the verification condition (3.11) is numerically and simply described.
Theorem 1. Let$C_{0}$$=$$\frac{h}{\pi}$,$\{\phi_{i}\}$be the basis of$C_{1}^{s}=||\nabla(u_{s}+\varphi)||_{\infty}$,$S_{h}$and define the following constatns:$C_{2}^{s}=|| \nabla\frac{\partial(u_{s}+\varphi)}{\partial x}||\infty +|| \nabla\frac{\partial(u_{s}+\varphi)}{\partial y}||_{\infty}$

88,$C_{3}^{s}$$=$$||\nabla\Delta(u_{s}+\varphi)||_{\infty}$$K_{1}$$=$$C_{1}^{s}+C_{0}^{2}C_{3}^{s}$$K_{2}$$=$$C_{1}^{s}+C_{0}C_{3}^{s}C_{p}$$K_{3}$$=$$\sqrt{2}C_{1}^{s}+C_{p}(C_{2}^{s}+C_{0}C_{3}^{s})$$C_{p}= \frac{1}{\pi\sqrt{2}}$, $M_{1}=||L^{T}G^{-1}L||_{E}$ ,

,

where
the Euclidian vector norrn,by,,denotes the matrix norm corresponding tois the Poincare constant, the matrix $G=(G_{ij})$ is defined$||\nabla v||_{\infty}\equiv(||\nabla v_{x}||_{\infty}^{2}+||\nabla v_{y}||_{\infty}^{2})^{\frac{1}{2}},$$C_{p}$$||\cdot$$||_{E}$,$G_{ji}\equiv R(J(u_{s}+\varphi, \Delta\phi_{i})+J(\phi_{i}, \Delta\langle u_{s}+\varphi))$$\phi_{j})_{L^{2}}+(\Delta\phi_{i}, \Delta\phi_{j})_{L^{2}}$,and $D=LL^{T}$ is a Cholesky decomposition for the matrix $D=(D_{ij})$ defined by$D_{ij}\equiv(\Delta\phi_{\mathrm{i}}, \Delta\phi_{j})_{L^{2}}$.

For these constants, if the inequality(3.12)$RC_{0}(K_{1}+K_{2}K_{3}M_{1}RC_{0})<1$

holds then the operator$\mathcal{L}$is invertible.

Proof. We show sufficient conditions for (3.11). Denoting u $=u_{1}+u_{2}$ ,, and thus
[I
, by some simple calculations we have$u_{1}\in U_{h}$$N_{h}^{1}(u)$$[\alpha]$

$=$

, u2

$\in$

$-F_{1}]_{h}^{-1}P_{h}F_{1}(u_{2})$(3.13)$||N_{h}^{1}(u)||_{H_{0}^{2}}\leq M_{1}||P_{h}F_{1}(u_{2})||_{H_{0}^{2}}$holds. (See [1] or [3] for details to such estimation.) Using error estimation in Lemma 2,
we can estimate
as follows:$||P_{h}F_{1}(u_{2})||_{H_{0}^{2}}$$||P_{h}F_{1}(u_{2})||_{H_{8}^{2}}\leq RC_{0}K_{3}\alpha$.

Thus we derive a sufficient condition for the first inequality in ($M_{1}RC_{0}K_{3}\alpha<\gamma$.

(3.14)$3.11\rangle$

as(3.15)

$\epsilon \mathrm{a}$Now we estimate the left hand side of the second inequality in (3.11). Noting that$||(I-P_{h})F_{1}(u)||_{H_{0}^{2}}$$R||(I-P_{h})KJ(u_{s}\neq\varphi, \Delta u)||_{H_{0}^{2}}$$\leq$$+R||(I-P_{h})KJ(u, \Delta(u_{s}+\varphi))||_{H_{0}^{2}}$$RC_{0}K_{2}\gamma+RC_{0}K_{1}\alpha$$\leq$,

we obtain the sufficient condition for the second inequality in (3.11) as$RC_{0}(K_{1}\alpha+K_{2}\gamma)<\alpha$(3.16).

Combining the conditions (3.15) and (3.16) we finally obtain the sufficient condition for(3.11) as $RC_{0}(K_{1}+K_{2}K_{3}M_{1}RC_{0})<1$ .

I

4Verification procedure for nonlinear problem

In what follows we assume that the invertibility of the linearized operator is confirmedby the method described in the previous section. We will verify the existence of solutions
for allfor (2.2) in the neighborhood of$\mathcal{L}$$ux\in C^{1}(\Omega)\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}_{\acute{\mathrm{J}}^{\gamma}}\mathrm{i}\mathrm{n}\mathrm{g}$$(\Delta ux+\Delta\varphi, \Delta v_{h})_{L^{2}}+(R\cdot J(u_{s}+\varphi, \Delta(u_{s}+\varphi))$

Considering the function$\Delta^{2}u$$\overline{u}$,$v_{h}\in S_{h}$$v_{h})_{L^{2}}=((f_{2})_{x}-(f_{1})_{y},v_{h})_{L^{2}}$. (4.1)

satisfying$=$$-\Delta^{2}\varphi-R\cdot J(u_{s}+\varphi, \Delta(u_{s}+\varphi))+(f_{2})_{x}-(f_{1})_{y}$

$=$

$0$

$\{$

in 42,,
on(4.2)$\partial\Omega$$\overline{u}=\frac{\partial\overline{u}}{\partial n}$

and writing$w\equiv u-\overline{u}$Noting that
estimates for$v_{0}\equiv\overline{u}-ux$$u_{X}=P_{h}\overline{u}$$v_{0}$$||v_{0}||_{H_{0}^{2}}$,$\leq$, $u-u_{X}$ can be represented as $w+v_{0}$ .

, we see that$v_{0}\in S_{[perp]}$

and, by Lemma 2 and its proof; the error

can be derived:$\frac{h^{2}}{\pi^{2}}||-\Delta^{2}\varphi-R\cdot J(u_{s}+\varphi, \Delta(u_{s}+\varphi))+(f_{2})_{x}-(f_{1})_{y}||_{L^{2}}$,70$||v_{0}||_{H_{0}^{1}}$$||v_{0}||_{L^{2}}$$\frac{h}{\pi}||v_{0}||_{H_{0}^{2}}$$\leq$,$\frac{h^{2}}{\pi^{2}}||v_{0}||_{H_{0}^{2}}$$\leq$.

Now we can rewrite (2.2) as$\triangle^{2}w$$=$$-R\cdot J$

$+R$

$\{$

$=$

$w= \frac{\partial w}{\mathit{8}n}$

$0$

.

on

$J$

( $w+ux+v_{0}+\varphi$ , A $(w+u_{X}+v_{0}+\varphi)$ )) in 0,
,A
(

(4.3)

$(u_{s}+\varphi)$$u_{s}+\varphi$

an.

Thus defining the following compact map on$H_{0}^{2}(\Omega)$

F2(w) $\equiv RK$ { $J(u_{s}+\varphi$ , A $(u_{s}+\varphi))-J(w+u_{X}+v_{0}+\varphi$ , $\Delta(w+u_{X}+v_{0}+\varphi))$ }, (4.4)

we have the fixed point equation$w=F_{2}(w)$

.

(4.5)

Now we formulate the infinite dimensional Newton method for the equation (4.5). Note,that $w-[I-F_{2}’(-v_{0}-u_{X}+u_{s})]^{-1}$ $(I-F_{2})(w)$ can be equivalently represented as
at $-v_{0}-$ ug %andwhere $F_{2}’(-v_{0}-u_{X}+u_{s})$ stands for Fr\’echet derivative of$\mathcal{L}^{-1}q(w)$$F_{2}$$q(w)\equiv R\{J(us+\varphi, \Delta\langle u_{s}+\varphi))-J$$u_{s}$( $w+ux+v_{0}+\varphi$ , A $(w+u_{X}+v_{0}+\varphi)$ )$+J(u_{s}+\varphi, \Delta w)+J(w, \Delta\acute{(}u_{s}+\varphi))\}$.

Then we have the relation$w=F_{2}(w)\Leftarrow\Rightarrow w=T_{2}(w)$where$T_{2}(w)\equiv \mathcal{L}^{-1}q(w)$

,

(4.6)

is a compact map on

We intend to find a fixed point of$T_{2}$$W=\{w\in H_{0}^{2}(\Omega)|||w||_{H_{0}^{2}}\leq\alpha\}$,$H_{0}^{2}(\Omega)$.

in a set $W$ defined by
(4.7)

71

where

is a positive number. If the relation$\alpha$(4.8)$T_{2}(W)\subset W$

holds, by Schauder’s fixed point theorem there exists a fixed point ofsufficient condition for (4.8) is$\sup_{w\in W}||T_{2}(w)||_{H_{0}^{2}}\leq\alpha$$T_{2}$

in $W$ . Since a

(4.9)

,

by estimating the left hand side of (4.8), we can derive the following theorem.

Theorem 2. Assume that the invertibility condition (3.12) holds. Using the same
constants in Theorem 1, we define the following constants:$\kappa\equiv$$C_{0}R(K_{1}+K_{2}K_{3}M_{1}C_{0}R)$$\frac{C_{0}RM_{1}K_{2}}{1-\kappa}$,$\tau_{2}=\frac{1}{1-\kappa}$$\tau_{1}$$=$$\tau_{3}$$=M_{1}(C_{0}RK_{3}\tau_{1}+1)$$b=$$||v_{0}||_{H_{0}^{2}}$,$C_{4}= \frac{1}{\pi}$

,

,

,

$\tau_{4}=M_{1}C_{0}RK_{3}\tau_{2}$,,is an embedding constant satisfying
where
have used the optimal embedding estimates$C_{4}$$C_{4}= \frac{1}{\pi}$[6]. Moreover for a matrix

and wewhich can be derived by the result in$||\nabla u||_{L^{4}}\leq C_{4}||\Delta u||_{L^{2}}$$S=(\begin{array}{ll}\tau_{1}^{2}+\tau_{3}^{2} \tau_{1}\tau_{2}+\tau_{3}\tau_{4}\tau_{1}\tau_{2}+\tau_{3}\tau_{4} \tau_{2}^{2}+\tau_{4}^{2}\end{array})$

and

for

$u\in H_{0}^{2}(\Omega)$$M_{2}\equiv||S||_{;}^{\frac{1}{E2}}$

define thefollowing constants:,$C_{2}^{X}=|| \nabla\frac{\partial(u_{X}+\varphi)}{\partial x}||\infty +|| \nabla\frac{\partial(u_{X}+\varphi)}{\partial y}||_{\infty}$$C_{1}^{X}$$=$$||\nabla(ux+\varphi)||_{\infty}$$C_{3}^{X}$$=$$||\nabla\Delta(u_{X}+\varphi)||_{\infty}$$D_{2}^{\delta}$$=$$||J(u_{X} -u_{s}, \Delta(u_{s}+\varphi))||_{L^{2}}$If there exists a real number,$D_{1}^{\delta}=||\nabla(u_{X}-u_{s})||_{L^{2}}$$\alpha>0$

,

,

,

$D_{3}^{\delta}=||\Delta(u\mathrm{x}-u_{s})||_{L^{2}}$,.

satisfying$M_{2}R\{C_{4}^{2}(\alpha+b)^{2}+C_{4}^{2}\alpha D_{3}^{\delta}+C_{3}^{X}C_{p}C_{0}b+\alpha D_{1}^{\mathit{5}}C_{p}$$+C_{0}b(\sqrt{2}C_{1}^{X}+C_{p}C_{3}^{X})+C_{\mathrm{p}}^{2}D_{2}^{\delta}+C_{p}C_{1}^{X}D_{3}^{\delta}\}\leq\alpha$,(4.10)

72
then there exists a fixed point of

Proof. For$q(w)\in H^{-2}(\Omega\rangle$$\mathcal{L}\phi=q(w)$$\phi=\frac{\partial\phi}{\partial n}=0$

consider the solution$\phi\in H_{0}^{2}(\Omega)$of the problem

in ,
on

$\phi=\phi_{h}+\phi_{1}$$||\phi_{h}||_{H_{0}^{2}}$in $W$ .

$\Omega$$\{$

Then writing$T_{2}$(4.11)

an.

,$\phi_{h}\in S_{h}$,$\emptyset[perp]\in S_{[perp]}$, we have,$\leq$$M_{1}RC_{0}K_{3}||\phi[perp]||_{H_{\mathrm{o}}^{2}}+M_{1}||P_{h}Kq(w)||_{H_{0}^{2}}$$\leq$$RC_{0}(K_{1}||\phi_{[perp]}||_{H_{0}^{2}}+K_{2}||P_{h}Kq(w)||_{H_{0}^{2}})+||(I-P_{h})Kq(w)||_{H_{0}^{2}}$$\{$$||\phi_{[perp]}||_{H_{0}^{2}}$Noting that $\kappa<1$ holds because of the invertibility of$||\phi_{h}||_{H_{0}^{2}}$$\mathcal{L}$(4.12), we have$\leq$$\tau_{3}||P_{h}Kq(w)||_{H_{0}^{2}}+\tau_{4}||(I-P_{h})Kq(w)||_{H_{0}^{2}}$,$\leq$$\tau_{1}||P_{h}Kq(w)||_{H_{0}^{2}}+\tau_{2}||(I-P_{h})Kq(w)||_{H_{0}^{2}}$.

$\{$$||\phi_{[perp]}||_{H_{0}^{2}}$.

(4.13)

Therefore we obtain$||\phi||_{H_{0}^{2}}\leq M_{2}||Kq(w)||_{H_{0}^{2}}\leq M_{2}||q(w)||_{H^{-2}}$.

Furthermore, we have the estimations$||q(w)||_{H^{-2}}=\sup|<q(w),$$\theta>_{H^{-2},H_{0}^{21}}\theta\in H_{0}^{2},||\theta||_{H_{0}^{2}}=1$$\leq R\{C_{4}^{2}(\alpha+b)^{2}+C_{4}^{2}\alpha D_{3}^{\delta}+C_{3}^{X}C_{p}C_{0}b+\alpha D_{1}^{\delta}C_{p}$$+C_{0}b(\sqrt{2}C_{1}^{X}+C_{p}C_{3}^{X})$$+C_{p}^{2}D_{2}^{\delta}+C_{p}C_{1}^{X}D_{3}^{\delta}\}$Thus we obtain$||\mathcal{L}^{-1}q(w)||_{H_{0}^{2}}\leq$$M_{2}R\{||C_{4}^{2}(\alpha+b)^{2}+C_{4}^{2}\alpha D_{3}^{\delta}+C_{3}^{X}C_{p}C_{0}b+\alpha D_{1}^{\delta}C_{p}$(4.14)

73$+C_{0}b(\sqrt{2}C_{1}^{X}+C_{p}C_{3}^{X})+C_{p}^{2}D_{2}^{\delta}+C_{p}C_{1}^{X}D_{3}^{\delta}\}$.

This means that the inequality (4.10) is a sufficient condition for $T_{2}(W)\subseteq W$ and thedesired assertion is proved.

1

5

Numerical examples
Particularly, we consider the two dimensional driven cavity problem with $f=0$ andin (1.1), where $\varphi(x, y)=x^{2}(1-x)^{2}y^{2}(1-y)$ .

$g=(\varphi_{y}, -\varphi_{x})$In calculations, we used interval arithmetic in ord er to avoid the effects of rounding errors in the floating-point computations. The computations were carried out on the DELL
) using MATLAB (Ver. 6.5.1) and thePrecision Workstation 650 (Intel Xeon
interval arithmetic toolbox INTLAB (Ver, 4.2.1) coded by Prof. Rump in TU Hamburg’means
Harburg ([4]). The verification results are shown in Table 1, in which ’smallest
the smallest bound a satisfing the verification condition (4.9) and the solution in (2.2).
is enclosed as$3.2\mathrm{G}\mathrm{H}\mathrm{z}$$\alpha$$u$$||u-u_{X}||_{H_{0}^{2}(\Omega)}\leq||v_{0}||_{H_{0}^{2}(\Omega)}+\alpha$$F1$ :

Verification Results for Driven Cavity Problem $(h=1/22)$$\ovalbox{\tt\small REJECT} R$$M_{1}$

100$M_{2}$$||v_{0}||_{H^{2}}$1.4511 9.2855e-4
1.03552.4945 1.0094e-3

1.0183$D_{3}^{\delta}$$\alpha$1.2940e-4 1.1199e-3
1.3666e-4 7.5815e-3$\underline{200}$
It seems that Wieners’ method would not be able to apply to the Reynolds number
higher than 20 in [7]. On the other hand, we enclosed the stationary solution for the
Reynolds number over 100, and our method can be applied, in principle, to more higher
Reynolds numbers by using more accurate approximation subspaces, i.e., smaller mesh
sizes

74

参考文献
[1] Nakao, M. T. and Watanabe, Y., An efficient approach to the numerical verification
for solutions of the elliptic differential equations, Numerical Algorithms 37 (2004),

311-323.
[2] Nakao, M. T. and Yamamoto, N., Numerical verification of solutions for nonlinear
residual method, J. Math. Anal. AppL, 217 (1998)
elliptic problems using an$L^{\infty}$

246-262.
[3] Nakao, M. T., Hashimoto, K. and Watanabe, Y., A numerical method to verify the
invertibility of linear elliptic operators with applicaions to nonlinear problems, MHF
Preprint Series, Kyushu University MHF 2003-1, to appear in Computing.

[4] Rump, S. M., INTLAB-INTerval LABoratory, a Matlab toolbox for verified
computations, version 4.2.1. Inst. Informatik, TU Hamburg-Harburg (2002),.
.ti3.tu-harburg. de / “rumphttp:$//\mathrm{w}\mathrm{w}\mathrm{w}$$/\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{b}/$[5] Schultz, M. H., Spline Analysis, Prentice-Hall, London (1973).
[6] Talenti, G. , Best constant in Sobolev inequality, Ann. Math. Pura AppL, 110, (1976)

353-372.
[7] Wieners, , Numerical enclosures for solutions of the Navier-Stokes equations forsmall Reynolds numbers, in Alefeld, G.(ed.) et al.: Numerical methods and error
bounds, Proceedings of the IM ACS-GAMM international symposium, Oldenburg,(1996), 280-286.
Germany, July 9-12, 1995. Berlin: Akademie Verlag, Math.
$\mathrm{C}$${\rm Res}$$\mathrm{S}9$