Let \(\Pi= \{ (x,y):0< x<a,0<y<b \} \) be a rectangle, \(a/b\) be rational, \(\gamma_{j}\) (\(\gamma_{j}^{\prime}\)), \(j=1,2,3,4\), be the sides, including (excluding) the ends, enumerated counterclockwise starting from the left side (\(\gamma_{0}\equiv\gamma_{4}\), \(\gamma_{5}\equiv\gamma _{1}\)), and let \(\gamma=\bigcup_{j=1}^{4}\gamma_{j}\) be the boundary of Π. Denote by *s* the arclength, measured along *γ*, and by \(s_{j}\) the value of *s* at the beginning of \(\gamma_{j}\). We say that \(f\in C^{k,\lambda}(D)\), if *f* has *k*th derivatives on *D* satisfying a Hölder condition with exponent \(\lambda\in(0,1)\).

We consider the boundary value problem

$$ \Delta u=0 \quad\text{on } \Pi,\quad\quad u=\varphi_{j}(s) \quad\text{on } \gamma _{j}, j=1,2,3,4, $$

(1)

where \(\Delta\equiv\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}\), \(\varphi_{j}\) are given functions of *s*. Assume that

$$\begin{aligned}& \varphi_{j} \in C^{6,\lambda}(\gamma_{j}),\quad 0< \lambda <1, j=1,2,3,4, \end{aligned}$$

(2)

$$\begin{aligned}& \varphi_{j}^{(2q)}(s_{j}) = (-1)^{q} \varphi_{j-1}^{(2q)}(s_{j}),\quad q=0,1,2. \end{aligned}$$

(3)

### Lemma 2.1

*The solution*
*u*
*of problem* (1) *is from*
\(C^{5,\lambda }(\overline{\Pi})\).

The proof of Lemma 2.1 follows from Theorem 3.1 in [4].

### Lemma 2.2

*The inequality is true*

$$ \max_{0\leq p\leq3}\sup_{(x,y)\in\Pi}\biggl\vert \frac{\partial^{6}u}{\partial x^{2p}\,\partial y^{6-2p}}\biggr\vert < \infty, $$

(4)

*where*
*u*
*is the solution of problem* (1).

### Proof

From Lemma 2.1 it follows that the functions \(\frac{\partial ^{4}u}{\partial x^{4}}\) and \(\frac{\partial^{4}u}{\partial y^{4}}\) are continuous on \(\overline{\Pi}\). We put \(w=\frac{\partial^{4}u}{\partial x^{4}}\). The function *w* is harmonic in Π and is the solution of the problem

$$ \Delta w=0 \quad\text{on } \Pi,\quad\quad w=\Phi_{j}\quad\text{on } \gamma _{j}, j=1,2,3,4, $$

where

$$\begin{aligned}& \Phi_{\tau}=\frac{\partial^{4}\varphi_{\tau}}{\partial y^{4}},\quad \tau=1,3, \\& \Phi_{\nu}=\frac{\partial^{4}\varphi_{\nu}}{\partial x^{4}},\quad \nu=2,4. \end{aligned}$$

From the conditions (2) and (3) it follows that

$$ \Phi_{j}\in C^{2,\lambda}(\gamma_{j}),\quad 0< \lambda<1,\quad\quad \Phi_{j}(s_{j})=\Phi_{j-1}(s_{j}),\quad j=1,2,3,4. $$

Hence, on the basis of Theorem 6.1 in [5], we have

$$\begin{aligned}& \sup_{(x,y)\in\Pi}\biggl\vert \frac{\partial^{2}w}{\partial x^{2}}\biggr\vert = \sup_{(x,y)\in\Pi}\biggl\vert \frac{\partial ^{6}u}{\partial x^{6}}\biggr\vert < \infty, \end{aligned}$$

(5)

$$\begin{aligned}& \sup_{(x,y)\in\Pi}\biggl\vert \frac{\partial^{2}w}{\partial y^{2}}\biggr\vert = \sup_{(x,y)\in\Pi}\biggl\vert \frac{\partial ^{6}u}{\partial x^{4}\,\partial y^{2}}\biggr\vert < \infty. \end{aligned}$$

(6)

Similarly, it is proved that

$$ \sup_{(x,y)\in\Pi} \biggl\{ \biggl\vert \frac{\partial ^{6}u}{\partial y^{6}}\biggr\vert ,\biggl\vert \frac{\partial^{6}u}{\partial y^{4}\,\partial x^{2}}\biggr\vert \biggr\} < \infty. $$

(7)

From (5)-(7), estimation (4) follows. □

### Lemma 2.3

*Let*
\(\rho(x,y)\)
*be the distance from a current point of the open rectangle* Π *to its boundary and let*
\(\partial/\partial l\equiv \alpha \partial/\partial x+\beta\partial/\partial y\), \(\alpha^{2}+\beta^{2}=1\). *Then the next inequality holds*:

$$ \biggl\vert \frac{\partial^{8}u}{\partial l^{8}}\biggr\vert \leq c\rho^{-2}, $$

(8)

*where*
*c*
*is a constant independent of the direction of the derivative*
\(\partial/\partial l\), *u*
*is a solution of problem* (1).

### Proof

According to Lemma 2.2, we have

$$ \max_{0\leq p\leq3}\sup_{(x,y)\in\Pi}\biggl\vert \frac{\partial^{6}u}{\partial x^{2p}\,\partial y^{6-2p}}\biggr\vert \leq c< \infty. $$

Since any eighth order derivative can be obtained by two times differentiating some of the derivatives \(\partial^{6}/\partial x^{2p}\,\partial y^{6-2p}\), \(0\leq p\leq3\), on the basis of estimations (29) and (30) from [6], we obtain

$$ \max_{\nu+\mu=8}\biggl\vert \frac{\partial^{8}u}{\partial x^{\nu }\,\partial y^{\mu}}\biggr\vert \leq c_{1}\rho^{-2}(x,y)< \infty. $$

(9)

From (9), inequality (8) follows. □

Let \(h>0\), and \(a/h\geq6\), \(b/h\geq6\) be integers. We assign \(\Pi ^{h}\), a square net on Π, with step *h*, obtained by the lines \(x,y=0,h,2h,\ldots\) . Let \(\gamma_{j}^{h}\) be a set of nodes on the interior of \(\gamma_{j}\), and let

$$ \gamma^{h}=\bigcup_{j=1}^{4} \gamma_{j}^{h},\quad\quad \dot{\gamma_{j}}=\gamma _{j-1}\cap\gamma_{j},\quad\quad\overline{\gamma}^{h}= \bigcup_{j=1}^{4}\bigl(\gamma _{j}^{h}\cup\dot{\gamma_{j}}\bigr), \quad\quad\overline{ \Pi}^{h}=\Pi^{h}\cup\overline{\gamma}^{h}. $$

Let the operator *B* be defined as follows:

$$\begin{aligned} Bu(x,y) =&\bigl(u(x+h,y)+u(x-h,y)+u(x,y+h)+u(x,y-h)\bigr)/5 \\ &{}+\bigl(u(x+h,y+h)+u(x+h,y-h) \\ &{}+u(x-h,y+h)+u(x-h,y-h)\bigr)/20. \end{aligned}$$

(10)

We consider the classical 9-point finite difference approximation of problem (1):

$$ u_{h}=Bu_{h}\quad\text{on }\Pi^{h},\quad\quad u_{h}=\varphi_{j}\quad\text{on }\gamma _{j}^{h}\cup\dot{\gamma_{j}}, j=1,2,3,4. $$

(11)

By the maximum principle, problem (11) has a unique solution.

In what follows, and for simplicity, we will denote by \(c,c_{1},c_{2},\ldots\) constants which are independent of *h* and the nearest factor, and identical notation will be used for various constants.

Let \(\Pi^{1h}\) be the set of nodes of the grid \(\Pi^{h}\) that are at a distance *h* from *γ*, and let \(\Pi^{2h}=\Pi^{h}\backslash\Pi ^{1h}\).

### Lemma 2.4

*The following inequality holds*:

$$ \max_{(x,y)\in ( \Pi^{1h}\cup\Pi^{2h} ) }\vert Bu-u\vert \leq ch^{6}, $$

(12)

*where*
*u*
*is a solution of problem* (1).

### Proof

Let \((x_{0},y_{0})\) be a point of \(\Pi^{1h}\), and let

$$ R_{0}= \bigl\{ (x,y): \vert x-x_{0}\vert < h,\vert y-y_{0}\vert <h \bigr\} $$

(13)

be an elementary square, some sides of which lie on the boundary of the rectangle Π. On the vertices of \(R_{0}\) and on the mid-points of its sides lie the nodes of which the function values are used to evaluate \(Bu(x_{0},y_{0})\).

We represent a solution of problem (1) in some neighborhood of \((x_{0},y_{0})\in\Pi^{1h}\) by Taylor’s formula

$$ u(x,y)=p_{7}(x,y)+r_{8}(x,y), $$

(14)

where \(p_{7}(x,y)\) is the seventh order Taylor’s polynomial, \(r_{8}(x,y)\) is the remainder term. Taking into account that the function *u* is harmonic, by exhaustive calculations, we have

$$ Bp_{7}(x_{0},y_{0})=u(x_{0},y_{0}) . $$

(15)

Now, we estimate \(r_{8}\) at the nodes of the operator *B*. We take a node \((x_{0}+h,y_{0}+h)\) which is one of the eight nodes of *B*, and we consider the function

$$ \widetilde{u}(s)=u \biggl( x_{0}+\frac{s}{\sqrt{2}},y_{0}+ \frac{s}{\sqrt{2}}\biggr) , \quad-\sqrt{2}h\leq s\leq\sqrt{2}h $$

(16)

of one variable *s*. By virtue of Lemma 2.3, we have

$$ \biggl\vert \frac{d^{8}\widetilde{u}(s)}{ds^{8}}\biggr\vert \leq c(\sqrt{2}h-s)^{-2},\quad 0\leq s< \sqrt{2}h. $$

(17)

We represent function (16) around the point \(s=0\) by Taylor’s formula

$$ \widetilde{u}(s)=\widetilde{p}_{7}(s)+\widetilde{r}_{8}(s), $$

where \(\widetilde{p}_{7}(s)\equiv p_{7} ( x_{0}+\frac{s}{\sqrt{2}},y_{0}+\frac{s}{\sqrt{2}} ) \) is the seventh order Taylor’s polynomial of the variable *s*, and

$$ \widetilde{r}_{8}(s)\equiv r_{8} \biggl( x_{0}+ \frac{s}{\sqrt{2}},y_{0}+\frac{s}{\sqrt{2}} \biggr) ,\quad 0\leq \vert s \vert < \sqrt{2}h , $$

(18)

is the remainder term.

On the basis of (17) and the integral form of the remainder term of Taylor’s formula, we have

$$ \bigl\vert \widetilde{r}_{8}(\sqrt{2}h-\varepsilon)\bigr\vert \leq c\frac{1}{7!}\int_{0}^{\sqrt{2}h-\varepsilon} ( \sqrt{2}h- \varepsilon-t ) ^{7}(\sqrt{2}h-t)^{-2}\,dt\leq c_{1}h^{6}, \quad 0< \varepsilon\leq\frac{h}{\sqrt{2}}. $$

(19)

Taking into account the continuity of the function \(\widetilde {r}_{8}(s)\) on \([ -\sqrt{2}h,\sqrt{2}h ] \), from (18) and (19), we obtain

$$ \bigl\vert r_{8} ( x_{0}+h,y_{0}+h ) \bigr\vert \leq c_{1}h^{6}, $$

(20)

where \(c_{1}\) is a constant independent of the point taken, \((x_{0},y_{0})\) on \(\Pi^{1h}\).

Estimation (20) is obtained analogously for the remaining seven nodes of the operator *B*. Since the norm of the operator is equal to 1 in the uniform metric, by using (20), we have

$$ \bigl\vert Br_{8} ( x_{0},y_{0} ) \bigr\vert \leq c_{2}h^{6}. $$

(21)

Hence, on the basis of (14), (15), (17), and linearity of the operator *B*, we obtain

$$ \bigl\vert Bu(x_{0},y_{0})-u ( x_{0},y_{0} ) \bigr\vert \leq ch^{6}, $$

for any \((x_{0},y_{0})\in\Pi^{1h}\).

Now, let \((x_{0},y_{0})\) be a point of \(\Pi^{2h}\), and let in the Taylor formula (14) corresponding to this point, the remainder term \(r_{8}(x,y)\) be represented in the Lagrange form. Then \(Br_{8}(x_{0},y_{0})\) contains eighth order derivatives of the solution of problem (1) at some points of the open square \(R_{0}\) defined by (13), when \((x_{0},y_{0})\in\Pi^{2h}\). The square \(R_{0}\) lies at a distance from the boundary *γ* of the rectangle Π; it is not less than *h*. Therefore, on the basis of Lemma 2.3, we obtain

$$ \bigl\vert Br_{8} ( x_{0},y_{0} ) \bigr\vert \leq c_{3}h^{6}, $$

(22)

where \(c_{3}\) is a constant independent of the point \(( x_{0},y_{0} ) \in\Pi^{2h}\). Again, on the basis of (14), (15), and (22) follows estimation (12) at any point \((x_{0},y_{0})\in\Pi^{2h}\). Lemma 2.4 is proved. □

We present two more lemmas. Consider the following systems:

$$\begin{aligned}& q_{h} = Bq_{h}+g_{h}\quad\text{on }\Pi ^{h},\quad\quad q_{h}=0\quad\text{on }\gamma^{h}, \end{aligned}$$

(23)

$$\begin{aligned}& \overline{q}_{h} = B\overline{q}_{h}+ \overline{g}_{h}\quad\text{on }\Pi^{h}, \quad\quad \overline{q}_{h}\geq0\quad\text{on }\gamma^{h}, \end{aligned}$$

(24)

where \(g_{h}\) and \(\overline{g}_{h}\) are given functions, and \(\vert g_{h}\vert \leq\overline{g}_{h}\) on \(\Pi^{h}\).

### Lemma 2.5

*The solutions*
\(q_{h}\)
*and*
\(\overline{q}_{h}\)
*of systems* (23) *and* (24) *satisfy the inequality*

$$ \vert q_{h}\vert \leq\overline{q}_{h}\quad\textit{on } \overline{\Pi}^{h}. $$

The proof of Lemma 2.5 follows from the comparison theorem (see Chapter 4 in [7]).

### Lemma 2.6

*For the solution of the problem*

$$ q_{h}=Bq_{h}+h^{6}\quad\textit{on }\Pi ^{h}, \quad\quad q_{h}=0\quad\textit{on }\gamma^{h}, $$

(25)

*the following inequality holds*:

$$ q_{h}\leq\frac{5}{3}\rho dh^{4}\quad\textit{on } \overline{\Pi}^{h}, $$

*where*
\(d=\max\{a,b\}\), \(\rho=\rho(x,y)\)
*is the distance from the current point*
\((x,y)\in\overline{\Pi}^{h}\)
*to the boundary of the rectangle* Π.

### Proof

We consider the functions

$$ \overline{q}_{h}^{(1)}(x,y)=\frac{5}{3}h^{4} \bigl(ax-x^{2}\bigr)\geq0,\quad\quad \overline{q}_{h}^{(2)}(x,y)= \frac{5}{3}h^{4}\bigl(by-y^{2}\bigr)\geq0\quad\text{on }\overline{\Pi}, $$

which are solutions of the equation \(\overline{q}_{h}=B\overline{q}_{h}+h^{6} \) on \(\Pi^{h}\). By virtue of Lemma 2.5, we obtain

$$ q_{h}\leq\min_{i=1,2}\overline{q}_{h}^{(i)}(x,y) \leq\frac{5}{3}\rho dh^{4}\quad\text{on }\overline{\Pi }^{h}. $$

□

### Theorem 2.7

*Assume that the boundary functions*
\(\varphi_{j}\), \(j=1,2,3,4\)
*satisfy conditions* (2) *and* (3). *Then*

$$ \max_{\overline{\Pi}^{h}}\vert u_{h}-u\vert \leq c\rho h^{4}, $$

(26)

*where*
\(u_{h}\)
*is the solution of the finite difference problem* (11), *and*
*u*
*is the exact solution of problem* (1).

### Proof

Let

$$ \varepsilon_{h}=u_{h}-u\quad\text{on }\overline{\Pi }^{h}. $$

(27)

It is obvious that

$$ \varepsilon_{h}=B\varepsilon_{h}+(Bu-u)\quad\text{on }\Pi ^{h},\quad\quad \varepsilon_{h}=0\quad\text{on }\gamma^{h}. $$

(28)

By virtue of estimation (12) for \((Bu-u)\) and by applying Lemma 2.5 to the problems (25) and (28), on the basis of Lemma 2.6 we obtain

$$ \max_{\overline{\Pi}^{h}}\vert \varepsilon_{h}\vert \leq c \rho h^{4}. $$

(29)

From (27) and (29) follows the proof of Theorem 2.7. □