数理解析研究所講究録
1169 巻 2000 年 15-26

15Harmonic Relations between
Functions and Green’s Matrices for Boundary
Value Problems$\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}^{)}\mathrm{S}$YAMAMOTO Tetsuro (山本哲朗)
Department of Mathematical Sciences
Ehime University
Matsuyama 790-8577, Japan

1

Introduction

A principal purpose of numerical analysis is to project continuous problems
onto finite dimensional spaces and attack the original problems by analyzing
the discretized systems. Although infinite dimensional spaces are quite different from finite dimensional spaces, there is a beautiful harmony between
a continuous problem and the discretized one.
The purpose of this paper is to show this through boundary value problems of Sturm-Liouville type and their discretizations by finite difference
methods (FDMs). It is also shown that the FDMs work well even for thecase where local truncation errors diverge as the mesh-size tends to zero.
First, in \S 2, we present some new results on inversion of tridiagonal matrices. Next, in \S 3, relations between Green’s functions and Green’s matrices
are described for continuous and discrete boundary value problems. Furthermore, in \S 4, an example is given such that even if the truncation error
does not converge to zero as approaches to zero, the centered finite difference method does converge to the true solution as. This suggests
us a robustness of FDM. Finally, on the basis of the result in \S 4, a meshrefinement technique for the Shortley-Weller (S-W) method is proposed in
obtained by the centered three-point formula.
\S 5, which improves the
In fact, an application of our technique to the example in \S 4 yields numerical results withaccuracy, while the usual centered formula gives
accuracy.
$h$$h$$harrow \mathrm{O}$$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\dot{\mathrm{t}}\mathrm{s}$$\mathrm{O}(h^{2}\log\frac{1}{h})$$\mathrm{O}(h^{\frac{1}{2}})$16

2Inversion of TRidiagonal Matrices
We consider an$n\cross n$

nonsingular tridiagonal matrix$A=$
Concerning the inverse of$A$,$a_{i},$of the form

$A$$c_{i}\neq 0$$\forall i$.

, the following is known:

Theorem 2.1 (Yamamoto-Ikebe [13]). Define the two sequence
as follows:$\{u_{m}\}$,$\{v_{m}\}$

,

(2.1)

$u_{0}=0,$$u_{1}=h_{1},$$u_{m}=- \frac{1}{c_{m-1}}(am-1um-2+b_{m-1}u_{m-1})$$(m\geq 2)$$v_{n+1}=0,$$v_{n}=h_{2},$$v_{m}=- \frac{1}{a_{m+1}}(b_{m}+1vm+1+cm+1vm+2)$$(m\leq n-1)$ ,(2.2)

and may be chosen arbitrarily, but may not be zero. Then
$A^{-1}=(\alpha_{ij})$ is given bywhere$h_{1},$$h_{2},$$a_{1}$$c_{n}$$\alpha_{ij}=$
where$\prod’$(2.3)$(i\leq j)(i\geq j)$,

is understood to be$\prod_{k=2}^{j}\frac{c_{k-1}}{a_{k}}$$(j\geq 2)$1$(j=1)$ .

$\prod_{k=2}^{j}\frac{c_{k-1}}{a_{k}}=J\{$In [13], this theorem has been proved as a corollary of the more general
inversion formula for a nonsingular $(q,p)$ -type band matrix $A=(a_{ij})$ with
if $i-j=q$ or $j-i=p$ .
$a_{ij}=0$ if $i>j+q$ or $j>i+p$ andWe can also prove Theorem 2.1 by directly verifying$a_{ij}\neq 0$$b_{1}\alpha_{1j}+C1\alpha 2j=\delta_{1j}$$a_{i}\alpha i-1j+b_{iij}\alpha+c_{i}\alpha i+1j=\delta ij$’,$2\leq i\leq n-1$$a_{n}\alpha_{n-}1j+b_{n}\alpha_{nj}=\delta_{nj}$.

Furthermore, Theorem 2.1 includes the following results as a special case:

17

Theorem 2.2 (Gantmakher-Krein [4], Bukhberger-Emel’yanenko [3]).
If is a nonsingular symmetric and irreducible tridiagonal matrix, then there
exist two column vectors $u=(u_{i})$ and $v=(v_{i})$ such that$A$

$A^{-1}==(u\mathrm{m}:\mathrm{n}(i,j)v_{\max}(i,j))$ .

(2.4)

Conversely, if $u=(u_{i})$ and $v=(v_{i})$ are given and a matrix $B=(u_{\min(i,j)\max}v(i,j))$is nonsingular, thenis a symmetric tridiagonal matrix with nonzero superdiagonal elements.
$B^{-1}$

Theorem 2.3 (Ikebe [5]). Under the assumptions of Theorem 2.1, fourcolumn vectors $u=(u_{i}),$ $v=(v_{i}),$ $x=(x_{i})$ and $y=(y_{i}),$ $i=1,2,$
exist such thatis given by$\ldots,$$n$$A^{-1}=(\alpha_{ij})$$u_{i}v_{j}$$(i\leq j)$$X_{jy_{i}}$$(i\geq j)$$\alpha_{ij}=\{$$u_{i}v_{i}=x_{iy_{i}}$’Proof. In Theorem 2.1, put$U_{i}=u_{i}$,$V_{j}=- \frac{1}{a_{1}h_{1}v_{0}}\prod_{k=2}\frac{c_{k-1}}{a_{k}}j’$$X_{j}=- \frac{u_{j}}{a_{1}h_{1}v_{0}}\prod_{k=2}\frac{c_{k-1}}{a_{k}}j’$Then $U=(U_{i}),$ $V=(V_{i}),$ $X=(X_{i})$ andand in Theorem 2.3.

$u,$ $v,$ $x$,$Y_{i}=v_{i}$$\mathrm{Y}=(Y_{i})$,.

satisfy the conditions for$\square$$y$

On the basis of Theorem 2.1, we now state the following result which appears
to be new.
Theorem 2.4. Under the assumptions of Theorem 2.1, there exist threecolumn vectors $u=(u_{i}),$ $v=(v_{i})$ and $d=(d_{i})$ such that

$A^{-1}==GD$

,

(2.5)

where
and $G=(g_{ij})$ with $g_{ij}=u_{\min(i,j)}v_{\max(i,j}$). If we
add a normalization condition $d_{1}=1$ , then and $D$ are uniquely determined
and is symmetric if and only if $D=I$ .
$D=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(d_{1,\ldots,n}d)$

$G$

$A$

18

Proof. A proof of this result will be given in [11].

$\square$Remark 2.1. A matrix $R=(r_{ij})$ is called a Green’s matrix (cf. [2]) if there
such that,exist number$a_{1},$$a_{n},$$\ldots$$b_{1},$$b_{n}$$\ldots,$$a_{i}b_{j}$$(i\leq j)$$b_{i}a_{j}$$(i\geq j)$$r_{ij}=a_{\min}(i,j)b\mathrm{m}\mathrm{a}\mathrm{x}(i,j)=\{$.

Hence the matrix in (2.5) is a Green’s matrix. We shall call the expression (2.5) a $GD$ decomposition of the inverse of a nonsingular irreducible
tridiagonal matrix . The expression (2.5) with $d_{1}=1$ may be called thenormalized $GD$ decomposition of the inverse of a nonsingular irreducible
tridiagonal matrix .
$G$

$A$

$A$

Now, for certain tridiagonal matrices, we can derive the explicit formula for. Only two results are presented here without proofs.
$A^{-1}$

Theorem 2.5. Let$a_{2}+.b_{2}-b_{1}$$-..b_{2}.$$A=(^{a_{1}+b_{1}}-a_{2}$.

$a_{n}+b_{n}-b_{n-1}1$,$a_{i},$$b_{i}\neq 0$$\forall i$

.

(2.6)

.

$-\cdot.a_{n}.$Then $A^{-1}=(\alpha_{ij})$ is given by$\frac{1}{w_{j}}\sum_{\lambda=}^{i}1c^{-}\sum_{\mu}^{n}\lambda 1=j+1\mu+1-1c$$(i\leq j)$$\frac{1}{w_{j}}\sum_{\lambda=}^{j}1c-1\sum\lambda\mu n=+1i+1\mu C^{-1}$$(i\geq j)$$\alpha_{ij}=\{$

(2.7)

,

where
$c_{1}=a_{1}$,$c_{\lambda}=a_{1} \square \frac{b_{i}}{a_{i}}=b_{\lambda-}1\prod_{ii=1=2}^{-}’\frac{b_{i-1}}{a_{i}}\lambda-1\lambda 1(\lambda\geq 2)$,$w_{j}= \frac{a_{j}}{c_{j}}\sum_{1\lambda=}^{n+}c_{\lambda}^{-}11$

$A=$

.

Theorem 2.6. Let,$a_{i},$$b_{i}\neq 0$$\forall i$

(2.8)

,

19

and

$c_{\lambda},$$w_{j}$

be defined as in Theorem 2.5. Then

$A^{-1}=$

,

(2.9)

where
$z_{i}= \sum_{\lambda}^{i}c_{\lambda}-1$,$i=1,2,$$\ldots,$$n+1$ .

The right-hand side of (2.9) gives a normalized $GD$ decomposition of$A^{-1}$.

Proofs of Theorems 2.5 and 2.6 will be given in [11].

3Green’s Functions and Green’s Matrices

Consider the Sturm-Liouville equation$a<x<b$ ,$L[u]=- \frac{d}{dx}(p(x)\frac{du}{dx})+q(x)u=f(x)$(3.1)

subject to the boundary conditions$B_{a}[u]=\alpha_{1}u(a)+\alpha_{2}u(/)a=0$$B_{b}[u]=\beta_{1}u(b)+\beta_{2}u’(b)=0$

,

(3.2)
(3.3)

,

where $p\in C^{1}[a, b],$ $p(x)>0,$ $f\in C[a, b],$ $q(x)\geq 0$ and,areconstants satisfying. Then, as is w.ell known, theGreen function $G(x, \xi)$ for the operator$q,$$\alpha_{1}^{2}+\alpha_{2}^{2}\neq 0,$$- \frac{d}{dx}(p(x)\frac{d}{dx})$$\alpha_{1},$$\alpha_{2},$$\beta_{1},$$\beta_{2}$$\beta_{1}^{2}+\beta_{2}^{2}\neq 0$: $\mathscr{D}=\{u\in c2[a, b]|B_{a}[u]=B_{b}[u]=0\}arrow C[a, b]$

is constructed as follows: Letvalue problems$L[u]=0$ ;$\overline{u}(x)$and$\overline{v}(x)$$u(a)=\alpha_{2},$be the solutions of the initial$u’(a)=-\alpha_{1}$

,

(3.4)

20

and
$L[v]=0$ ;respectively. Then$\overline{u}(x)$and$v(b)=\beta_{2},$$\overline{v}(x)$$v’(b)=-\beta_{1}$

(3.5)

,

are linearly independent over $[a, b]$ and$- \frac{\overline{u}(x)\overline{v}(\xi)}{p(\xi)W(\xi\cdot\overline{u},\overline{v})}$,$G(X, \xi)=\{$$- \frac{\overline{v}(x)\overline{u}(\xi)}{p(\xi)W(\xi\cdot\overline{u},\overline{v})}$,$(a\leq X\leq\xi)$$(\xi\leq X\leq b)$$(a\leq X\leq\xi)$$\frac{1}{c}\overline{u}(x)\overline{v}(\xi)$$=\{$$(\xi\leq x\leq b)$$\frac{1}{c}\overline{v}(x)\overline{u}(\xi)$denotes the Wronskian determinant of and and $c=$
. Observe that is a nonzero constant independent of . If
we discretize $(3.1)-(3.3)$ by the usual difference formula:where$\overline{v}$$\overline{u}$$W(\xi;\overline{u},\overline{v})$$-p(\xi)W(\xi;\overline{u},\overline{v})$

$\xi$

$c$

$a=x_{0}<X1<\cdots<x_{n}<x_{n}+1=b$ ,$x_{i-\frac{1}{2}}= \frac{1}{2}(_{X_{i1}+}-X_{i})$,$h_{i}=x_{i}-X_{i1}-$,$i=1,2,$$\ldots,$$n+1$ ,$- \frac{d}{dx}(p(x)\frac{du}{dx})x=xj.=.-[p_{i+\frac{1}{2}}\frac{(u_{i+1}-ui)}{h_{i+1}}-p_{i\frac{1}{2}}-\frac{(u_{i}-ui-1)}{h_{i}}]/(\frac{h_{i}+h_{i+1}}{2})$(3.6), then the resulting matrix is not necessarily symmetric
where
tridiagonal. The iterations (2.1) and (2.2), essentially due to BukhbergerEmel’yanenko [3], correspond to (3.4) and (3.5), respectively. In particular,, then the discretized system
if we consider the caseis$p_{i+\frac{1}{2}}=p(x_{i+\frac{1}{2}})$$\alpha_{1}=\beta_{1}=1,$$\alpha_{2}=\beta_{2}=0$$(DA+Q)U=f$ ,where$b_{2}^{-b_{2}}..+.b_{3}$$A=(^{-b_{2}}b_{1}+b_{2}$$-..\cdot.b_{3}..\sim-\cdot..b_{n}\sim.$.

$D=$$U=(U_{1,\ldots,n}U)^{t}$,,$b_{n}+b_{n+}-b_{n}1$),$b_{i}= \frac{p_{i-\frac{1}{2}}}{h_{i}}$$Q=$$f=(f_{1}, \ldots, f_{n})t$

.

,

,

21
Hence, if we denote by$\tau_{i}$the local truncation error of the approximation$-[ \frac{p_{i+\frac{1}{2}}\frac{(u_{i+1^{-}}u_{i})}{h_{i+1}}-p_{i}-\frac{1}{2}\frac{(u_{i}-u_{i-1})}{h_{i}}}{\frac{h_{i+}h_{i+1}}{2}}]+qiui=fi$

at$x_{i}$, then$(DA+Q)(u-U)=\tau$where $u=(u_{1}, \ldots, u_{n})^{t}$ and$\tau=(\tau_{1}, \ldots, \tau_{n})^{t}$. It follows that$|u-U|\leq(DA+Q)^{-1}|\tau|\leq(DA)^{-1}|\tau|=A^{-1}D^{-1}|\tau|$since $DA$ is an $M$-matrix and$(|u_{1}-U1|, \ldots , |u_{n}-U_{n}|)^{t}$ andorem 2.5 then yields$A^{-1}=(g_{ij})$$Q$

is nonnegative diagonal, where $|u-U|=$. An application of The-$|\tau|=(|\tau_{1}|, \ldots, |\tau_{n}|)^{t}$,$( \sum_{\lambda=}^{n+1}1\frac{h_{\lambda}}{p_{\lambda-2}1})^{-}1(\sum_{\mu=1}^{i}\frac{h_{\mu}}{p_{\mu-2}1})(\sum_{\nu j+}^{n+}=11^{\frac{h_{\nu}}{\mathrm{P}_{\nu-}12}})$$(i\leq.

j)$

(3.7)

$g_{ij}=\{$$(i\geq j)$$( \sum_{\lambda=}^{n+1}1\frac{h_{\lambda}}{p_{\lambda-2}1})^{-}1(\sum_{\mu=1}^{j}\frac{h_{\mu}}{p_{\mu-2}1})(\sum_{\nu=i}^{n+}1+1\frac{h_{\nu}}{p_{\nu-2}1})$$( \int_{a}^{b}\frac{ds}{p(s)})-1(\int_{a}^{x_{i}}\frac{ds}{p(s)})(\int_{x_{j}}^{b}\frac{ds}{p(s)})$$(i\leq j)$$( \int_{a}^{b_{\frac{ds}{p(s)})}}-1(\int_{a}^{x_{j}}\frac{ds}{p(s)})(\int_{x_{i}}^{b}\frac{ds}{p(s)})$$(i\geq j)$$.=.\{$$=G(x_{i,j}x)$,where $G(x, \xi)$ is the Green function for$-(pu’)’=f$ ,$a<x<b$ ,$u(a)=u(b)=0$ .

Therefore, we obtain$|u_{i}-U_{i}| \leq\sum_{j=1}^{n}gij\frac{h_{j}+h_{j+1}}{2}|\mathcal{T}j|$

On the other hand, if
the use of fictitious nodeandin $(3.2)-(3.3)$ , then, with(cf. [1]), the difference equation at $x_{n+1}=b$ is$\alpha_{1}=\beta_{2}=1$$x_{n+2}$.

$\alpha_{2}=\beta_{1}=0$given by,$\frac{-2p_{n+\frac{1}{2}}}{h_{n+1}^{2}}U_{n}+\frac{2p_{n+\frac{1}{2}}}{h_{n+1}^{2}}Un+1+qn+1Un+1=fn+1$

22

so that we again have$|u-U|\leq(\hat{D}\hat{A}+Q)^{-1}|\hat{\mathcal{T}}|\leq\hat{A}^{-1}\hat{D}^{-1}|_{\hat{\mathcal{T}}1}$

since$\hat{D}\hat{A}$,is an $M$ -matrix, where$b_{2}^{-b_{2}}..+.b_{3}$$\hat{A}=(^{b_{1}+}-b2b_{2}$.

.

$-..b_{3}b_{n}$$b_{n}+\cdot.b_{n+1}-b_{n+1}$$-b_{n+1}b_{n+1}$$\hat{D}=$,$\hat{\tau}=(\hat{\mathcal{T}}_{1}, \ldots,\hat{\tau}n)^{t}$andstands for the local truncation error at) to
Theorem 2.6 (replacing $a_{n+1}+b_{n+1}$ by$\hat{\tau}_{i}$$b_{n+1}$$\hat{A}^{-1}=$ ,In this case, the Green function$- \frac{d^{2}}{dx^{2}}$$\hat{G}(x, \xi)$),$b_{i}= \frac{p_{i-\frac{1}{2}}}{h_{i}}$$x_{i}$. Then, an application ofyield$\hat{A}^{-1}$$z_{i}= \sum_{\lambda=1}\frac{h_{\lambda}}{p_{\lambda-\frac{1}{2}}}=i..\int_{a}^{x_{i}}\frac{ds}{p(s)}$for the operator: $\hat{\mathscr{D}}=\{u\in C^{2}[a, b]|u(a)=u’(b)=0\}arrow C[a, b]$

is given by$\int_{a}^{x}\frac{ds}{p(s)}$$(x\leq\xi)$$\int_{a}^{\xi}\frac{ds}{p(s)}$$(x\geq\xi)$$\hat{G}(X, \xi)=\{$

Hence$\hat{A}^{-1}$.

is also a Green’s matrix which approximates$\hat{G}(x, \xi)$

.

.

23

4

Superconvergence and Nonsuperconvergenceof the S-W Approximation

Consider the boundary value problem$- \frac{d}{dx}(p(x)\frac{du}{dx})+q(x)u=f(x),$$a<x<b$ ,$u(a)=\alpha,$$u(b)=\beta$

.

(4.1)

Then, as was already seen in \S 3, the FDM which employs (3.6) leads to theestimate$|u_{i}-U_{i}| \leq(A^{-1}D^{-1}|\mathcal{T}|)_{i}=\sum gij\frac{h_{j}+h_{j+1}}{2}j=1n|_{\mathcal{T}_{j}}|$

.

(4.2)

If $p(x)=1$ , then it follows from (3.7) that$g_{ij}=$$=G(x_{i}, x_{j})$$(i\leq j)(i\geq j)$,

where $G(x, \xi)$ is the Green function for$- \frac{d^{2}}{dx^{2}}$: $\mathscr{D}=\{u\in C^{2}[a, b]|u(a)=u(b)=0\}arrow C[a, b]$ .

Therefore, if the solution $u(x)$ of (4.1) belongs to the class $C^{4}[a, b]$ andand,are equidistant, i.e., $h_{i}=h\forall i$ , then
the nodes
every.
yieldsatfrom (4.2), the error of FDM solution
Even if $h_{1}\neq h$ or $h_{n+1}\neq h$ and $h_{i}=h$ for $i\neq 1,$ $n+1$ , this estimate istrue. More precisely we have$\sup_{i}|\tau_{i}|\leq \mathrm{O}(h^{2})$$\{x_{i}\}$$\mathrm{O}(h^{2})$$\{U_{i}\}$$\mathrm{O}(h^{3})$$u_{i}-U_{i}=\{$$\mathrm{O}(h^{2})$near $x=a$ or $x=b$
atotherwise.
$x_{i}$$x_{i}$(4.3)

This is a special case of recent results due to Yamamoto [9] and MatsunagaYamamoto [7], which holds for the Dirichlet problem$\triangle u=f(x, y, u)$in$u=g(_{X}, y)$on$\Omega$,$f_{u}\geq 0$$\Gamma=\partial\Omega$

(4.4)
(4.5)

. The property (4.3) is called “superconverin a bounded domain
gence”. It is reported in [6] that the Collatz approximation does not have$\Omega\subset \mathbb{R}^{2}$

24, for example, if $u\in C[a, b]\cap C4(a, b)$ , then doessuch a property. If“superconvergence” occur? It has been shown in Yamamoto-Fang-Chen [12]that different situations may happen. For example, if we apply the centered$n+1$ to the problem
finite difference method with $h_{i}=h= \frac{1}{n+1},$ $i=1,2,$$u\not\in C^{4}[a, b]$$\ldots,$, $0<x<1$ ,$u(0)=u(1)=0$ ,$- \frac{d^{2}}{dx^{2}}u=f(x)$$\{$and the exact solution$u$satisfies $u\in C[0,1]\cap C^{4}(0,1)$ and$\sup_{x\in(0,1)}\frac{x^{4}(1-X)^{4}|u(4)(x)|}{x^{\rho}(1-x)^{\sigma}}<\infty$with some constants
we have$\rho,$(4.6)$\sigma\in(0,2),$$\rho\neq 1,$$\sigma\neq 1$$|u_{i}-U_{i}|=\mathrm{O}(h\theta)$,

then, putting$\forall i$$\theta=\min\{\rho, \sigma\}$

.

,

(4.7)

converges to the exact solution as
That is, the FDM solution,
although and. But, superconvergence does not occur
asat any . If the solution satisfies $u\in C[0,1]\cap C^{4}(0,1)$ and$\{U_{i}\}$$\tau_{1}$$x_{i}$$\tau_{n}arrow+\infty$$u_{i}$$harrow \mathrm{O}$$harrow \mathrm{O}$$u$$\sup_{x\in(0,1)}\frac{x^{4}|u^{(4)}(x)|}{x^{\rho}}<\infty$with some constant $\rho\in(0,2)$ and$\rho\neq 1$$\mathrm{O}(h^{\rho 1}+)$$|u_{i}-U_{i}|\leq\{$$\mathrm{O}(h^{\rho})$, thenatnear $x=1$
otherwise.
$x_{i}$

That is, superconvergence occurs near $x=1$ . These results can be extended
to two dimensional problems like $(4.4)-(4.5)$ .
This suggests us that FDM is a quite simple and natural approximation$1\leq m\leq 3$ andmethod for solving boundary value problems inmight approximate the exact solution in many cases, even if the truncation
error does not converges to zero at a point. The modern numerical analysis
has not well grasped such a harmonic relation between the problem (4.1) andtheir discretization by FDM.
$\Omega\subset \mathbb{R}^{m},$5

A Mesh Refinement Technique

Again consider the FDM applied to the problem $(3.1)-(3.3)$ , with a meshspacing$a=x_{0}<x_{1}<\cdots<x_{n+1}=b$,$h_{i}=x_{i}-X_{i1}-$,$i=1,2,$$\ldots,$$n+1$ .

25

Without loss of generality, we may assume $a=0$ and $b=1$ . Then$\mathcal{T}(X_{i})\overline{=}-[p_{i+\frac{1}{2}}\frac{(u_{i+1}-u_{i})}{h_{i+1}}-p_{i-}\frac{1}{2}\frac{(u_{i}-u_{i-}1)}{h_{i}}]/(\frac{h_{i}+h_{i+1}}{2})+q_{i}u_{i}-fi$$=(h_{i+1}-h_{i})( \frac{1}{2}p_{i}^{\prime J\prime}ui+\frac{1}{3}p_{i}u_{i}^{J}/)-\frac{1}{6}(h_{i}^{2}+1^{-h_{i}h_{i}}+1+h_{i}^{2})p_{i}^{\prime J;}u_{i}’+\cdots$.

If, we put $x=\varphi(t)$ with $\varphi\in C^{2}[0,1]$ and set$t_{i}=ih$,$i=0,1,2,$$\ldots,$$n+1$ ,

$h= \frac{1}{n+1}$,then$h_{i+1}-h_{i\varphi_{i}}=1-+2\varphi_{i}+\varphi_{i-1}=h^{2}\varphi^{\prime J}(\zeta)$,$t_{\mathrm{i}-1}<\zeta<t_{i+1}$.

Therefore, if $u\in C^{4}[0,1]$ anddenotes the FDM solution, then $u_{i}-U_{i}=$. The superconvergence property also holds near the end points$x=0$ and $x=1$ . Even iffor some constant $\kappa>0$ , theconvergence of
may be expected. In fact, let
as$\{U_{i}\}$$\mathrm{O}(h^{2})\forall i$$\tau(x_{i})=\mathrm{O}(h^{-\kappa})$$harrow \mathrm{O}$$\{U_{i}\}$$f(x)= \frac{1}{4}\{x(1-X)\}-\frac{3}{2}$$- \frac{d^{2}u}{dx^{2}}=f(x)$,$0<x<1$ ,,$0<x<1$ .

$u(\mathrm{O})=u(1)=0$

for near $x=0$ or $x=1$ . In this case, choosing
and, we canso as to satisfy $\varphi’(t)=\{t(1-t)\}3$ , i.e.,
prove that$\tau(x_{i})=\mathrm{O}(h^{\frac{1}{2}-2})$$\varphi(t)$$x_{i}$$\varphi(t)=t^{4}(\frac{1}{4}-\frac{3}{5}t+\frac{3}{2}t^{2}-\frac{1}{7}t^{3})$$|u_{i}-U_{i}|= \mathrm{O}(h2\log\frac{1}{h})(=\mathrm{O}(h^{2-\epsilon})\forall\epsilon>0)\forall i$.

.
This is a marked improvement over the estimate (4.7) with
It is now easy to apply this technique to refine finite difference net near
Furthermore, it is also
finite numbers of grid points$..,$ $a_{m}\in(0,1)$ .
possible to extend the technique to 2 or 3-dimensional boundary value problem $(4.4)-(4.5)$ . The detail will be reported elsewhere, together with numerical examples.
$\theta=\frac{1}{2}$$a_{1},$$,$

26

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[2] W. W. Barrett, A theorem on inverses of tridiagonal matrices, Linear
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10-20[4] F. R. Gantmakher and M. G. Krein, Oszillationsmatrizen, Oszillationskerne und Kleine Schwingungen Mechanischer Systeme, AkademieVerlag, Berlin, 1960

[5] Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and Appl.
24 (1979), 93-97[6] N. Matsunaga, Comparison of three finite difference approximations for
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135-142[10] T. Yamamoto, A new insight of the Shortley-Weller approximation for
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