On a highly accurate approximation of the first and pure second derivatives of the Laplace equation in a rectangular parallelpiped

Abstract

We propose and justify difference schemes for the approximation of the first and pure second derivatives of a solution of the Dirichlet problem in a rectangular parallelepiped. The boundary values on the faces of the parallelepiped are supposed to have six derivatives satisfying the Hölder condition, to be continuous on the edges, and to have second- and fourth-order derivatives satisfying the compatibility conditions resulting from the Laplace equation. We prove that the solutions of the proposed difference schemes converge uniformly on the cubic grid of order $O(h^{4})$, where h is a grid step. Numerical experiments are presented to illustrate and support the analysis made.

1 Introduction

A highly accurate method is one of the powerful tools reducing the number of unknowns, which is the main problem in the numerical solution of differential equations, to get reasonable results. This becomes more valuable in 3D problems when we are looking for the derivatives of the unknown solution by the finite difference or finite element methods for a small discretization parameter h.

The derivative problem was investigated in [1], in which it was proved that the high-order difference derivatives uniformly converge to the corresponding derivatives of the solution for the 2D Laplace equation in any strictly interior subdomain with the same order h as in the given domain. The uniform convergence of the difference derivatives over the whole grid domain to the corresponding derivatives of the solution for the 2D Laplace equation with order $O(h^{2})$ was proved in [2]. In [3], for the first and pure second derivatives of the solution for the 2D Laplace equation, special finite difference problems were investigated. It is proved that the solution of these problems converge to the exact derivatives with order $O(h^{4})$.

In [4], for the 3D Laplace equation, the convergence of order $O(h^{2})$ of the difference derivatives to the corresponding first-order derivatives of the exact solution is proved. It was assumed that the boundary values have the third derivatives on the faces and satisfy the Hölder condition. Furthermore, they are continuous on the edges, and their second derivatives satisfy the compatibility condition that is implied by the Laplace equation. Whereas in [5], when the boundary values on the faces of a parallelepiped are supposed to have the fourth derivatives satisfying the Hölder condition, the constructed difference schemes converge with order $O(h^{2})$ to the first and pure second derivatives of the exact solution.

In this paper, we consider the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. We assume that the boundary values on the faces have the sixth-order derivatives satisfying the Hölder condition, and the second- and fourth-order derivatives satisfy some compatibility conditions on the edges. We construct three different schemes on a cubic grid with mesh size h, whose solutions separately approximate the solution of the Dirichlet problem with order $O(h^{6}\vert \ln h\vert )$ and its first and pure second derivatives with order $O(h^{4})$. We show that, for the same boundary functions, if we use fifth-order numerical differentiation formulae to construct the finite-difference problem for the first derivatives, then the accuracy can be increased up to $O(h^{5}\vert \ln h\vert )$. Finally, numerical experiments are given to support the theoretical results.

2 The Dirichlet problem on a rectangular parallelepiped

Let $R= \{ (x_{1},x_{2},x_{3}):0< x_{i}< a_{i},i=1,2,3 \}$ be an open rectangular parallelepiped, $\Gamma_{j}$ ($j=1,2,\ldots,6$) be its faces including the edges such that $\Gamma_{j}$ for $j=1,2,3$ (for $j=4,5,6$) belong to the plane $x_{j}=0$ (to the plane $x_{j-3}=a_{j-3}$), let $\Gamma=\bigcup_{j=1}^{6}\Gamma_{j}$ be the boundary of R, and let $\gamma_{\mu \nu}=\Gamma_{\mu}\cap\Gamma_{\nu}$ be the edges of the parallelepiped R. We say that $f\in C^{k,\lambda}(D)$ if f has kth derivatives on D satisfying the Hölder condition with exponent $\lambda\in ( 0,1 )$.

We consider the boundary value problem

$$ \Delta u=0\quad \text{on } R,\qquad u=\varphi_{j}\quad \text{on } \Gamma _{j}, j=1,2,\ldots,6, $$

(2.1)

where $\Delta\equiv\partial^{2}/\partial x_{1}^{2}+\partial ^{2}/\partial x_{2}^{2}+\partial^{2}/\partial x_{3}^{2}$, and $\varphi_{j}$ are given functions. Assume that

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

(2.2)

$$\begin{aligned}& \varphi_{\mu}=\varphi_{\nu}\quad \text{on } \gamma_{\mu\nu}, \end{aligned}$$

(2.3)

$$\begin{aligned}& \frac{\partial^{2}\varphi_{\mu}}{\partial t_{\mu}^{2}}+\frac {\partial ^{2}\varphi_{\nu}}{\partial t_{\nu}^{2}}+\frac{\partial^{2}\varphi _{\mu }}{\partial t_{\mu\nu}^{2}}=0\quad \text{on } \gamma_{\mu\nu}, \end{aligned}$$

(2.4)

$$\begin{aligned}& \frac{\partial^{4}\varphi_{\mu}}{\partial t_{\mu}^{4}}+\frac {\partial ^{4}\varphi_{\mu}}{\partial t_{\mu}^{2}\, \partial t_{\mu\nu }^{2}}=\frac{\partial^{4}\varphi_{\nu}}{\partial t_{\nu}^{4}}+\frac{\partial ^{4}\varphi_{\nu}}{\partial t_{\nu}^{2}\, \partial t_{\nu\mu }^{2}}\quad \text{on }\gamma_{\mu\nu}, \end{aligned}$$

(2.5)

where $1\leq\mu<\nu\leq6$, $\nu-\mu\neq3$, $t_{\mu\nu}$ is an element in $\gamma_{\mu\nu}$, and $t_{\mu}$ and $t_{\nu}$ are elements of the normal to $\gamma_{\mu\nu}$ on the face $\Gamma _{\mu}$ and $\Gamma_{\nu}$, respectively.

Lemma 2.1

The solution u of problem (2.1) is from $C^{5,\lambda }(\overline{R})$.

The proof of Lemma 2.1 follows from Theorem 2.1 in [6].

Lemma 2.2

We have the inequality

$$ \max_{0\leq p\leq3}\max_{0\leq q\leq3-p}\sup _{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{6}u}{\partial x_{1}^{2p}\, \partial x_{2}^{2q}\, \partial x_{3}^{6-2p-2q}}\biggr\vert \leq c< \infty, $$

(2.6)

where u is the solution of problem (2.1).

Proof

From Lemma 2.1 it follows that the functions $\frac{\partial ^{4}u}{\partial x_{1}^{4}}$, $\frac{\partial^{4}u}{\partial x_{2}^{4}}$, and $\frac{\partial^{4}u}{\partial x_{3}^{4}}$ are continuous on R̅. We put $w=\frac{\partial^{4}u}{\partial x_{1}^{4}}$. The function w is harmonic in R and is the solution of the problem

$$ \Delta w=0\quad \text{on } R,\qquad w=\Psi_{j}\quad \text{on } \Gamma _{j}, j=1,2,\ldots,6, $$

where

$$\begin{aligned}& \Psi_{\tau}=\frac{\partial^{4}\varphi_{\tau}}{\partial x_{2}^{4}}+\frac{\partial^{4}\varphi_{\tau}}{\partial x_{3}^{4}}+2\frac{\partial ^{4}\varphi_{\tau}}{\partial x_{2}^{2}\, \partial x_{3}^{2}}, \quad \tau =1,4, \\& \Psi_{\nu}=\frac{\partial^{4}\varphi_{\nu}}{\partial x_{1}^{4}},\quad \nu=2,3,5,6. \end{aligned}$$

From conditions (2.2)-(2.5) it follows that

$$\begin{aligned}& \Psi_{j} \in C^{2,\lambda}(\Gamma_{j}),\quad 0< \lambda< 1, j=1,2,\ldots,6, \\& \Psi_{\mu} = \Psi_{\nu}\quad \text{on }\gamma_{\mu\nu}, 1\leq\mu < \nu\leq6, \nu-\mu\neq3. \end{aligned}$$

Hence, by Theorem 4.1 in [6] we have

$$\begin{aligned}& \sup_{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{6}u}{\partial x_{1}^{6}}\biggr\vert =\sup _{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{2}w}{\partial x_{1}^{2}}\biggr\vert < \infty, \end{aligned}$$

(2.7)

$$\begin{aligned}& \sup_{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{6}u}{\partial x_{1}^{4}\, \partial x_{2}^{2}}\biggr\vert =\sup _{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{2}w}{\partial x_{2}^{2}}\biggr\vert < \infty, \end{aligned}$$

(2.8)

$$\begin{aligned}& \sup_{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{6}u}{\partial x_{1}^{4}\, \partial x_{3}^{2}}\biggr\vert =\sup _{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{2}w}{\partial x_{3}^{2}}\biggr\vert < \infty. \end{aligned}$$

(2.9)

Similarly, it is proved that

$$ \sup_{(x_{1},x_{2},x_{3})\in R} \biggl\{ \biggl\vert \frac{\partial^{6}u}{ \partial x_{2}^{6}}\biggr\vert ,\biggl\vert \frac{\partial ^{6}u}{\partial x_{3}^{6}}\biggr\vert ,\biggl\vert \frac{\partial^{6}u}{\partial x_{1}^{2}\, \partial x_{2}^{4}}\biggr\vert ,\biggl\vert \frac{\partial ^{6}u}{\partial x_{1}^{2}\, \partial x_{3}^{4}}\biggr\vert , \biggl\vert \frac {\partial ^{6}u}{\partial x_{2}^{2}\, \partial x_{3}^{4}}\biggr\vert ,\biggl\vert \frac{\partial^{6}u}{\partial x_{2}^{4}\, \partial x_{3}^{2}} \biggr\vert \biggr\} < \infty. $$

(2.10)

From (2.7)-(2.10) estimate (2.6) follows. □

Lemma 2.3

Let $\rho(x_{1},x_{2},x_{3})$ be the distance from the current point of the open parallelepiped R to its boundary, and let $\partial /\partial l\equiv\alpha_{1}\partial/\partial x_{1}+\alpha _{2}\partial /\partial x_{2}+\alpha_{3}\partial/\partial x_{3}$, $\alpha _{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}=1$. Then we have the inequality

$$ \biggl\vert \frac{\partial^{8}u(x_{1},x_{2},x_{3})}{\partial l^{8}}\biggr\vert \leq c \rho^{-2}(x_{1},x_{2},x_{3}), \quad (x_{1},x_{2},x_{3})\in R, $$

(2.11)

where c is a constant independent of the direction of differentiation $\partial/\partial l$, and u is a solution of problem (2.1).

Proof

Since the sixth-order derivatives of the solution u of the form $\partial^{6}/\partial x_{1}^{2p}\, \partial x_{2}^{2q}\, \partial x_{3}^{6-2p-2q}$, $p+q+s=3$, are harmonic and by Lemma 2.2 are bounded in R, any eighth-order derivative can be obtained by twice differentiating some of these derivatives, on the basis of Lemma 3 from [7] (see Chapter 4, Section 3), we have

$$ \max_{0\leq\mu\leq8}\max_{0\leq\nu\leq8-\mu}\biggl\vert \frac {\partial ^{8}u(x_{1},x_{2},x_{3})}{\partial x_{1}^{\mu}\, \partial x_{2}^{\upsilon }\, \partial x_{3}^{8-\mu-\upsilon}}\biggr\vert \leq c_{1}\rho ^{-2}(x_{1},x_{2},x_{3}), \quad (x_{1},x_{2},x_{3})\in R. $$

(2.12)

Inequality (2.11) follows from inequality (2.12). □

Let $h>0$ and $a_{i}/h\geq6$, $i=1,2,3$. We assign $R^{h}$, a cubic grid on R, with step h, obtained by the planes $x_{i}=0, h, 2h,\ldots$ , $i=1,2,3$. Let $D_{h}$ be the set of nodes of this grid, $R_{h}=R\cap D_{h}$, $\Gamma_{jh}=\Gamma_{j}\cap D_{h}$, and $\Gamma _{h}=\Gamma_{1h}\cup\Gamma_{2h}\cup\cdots\cup\Gamma_{6h}$.

Let the operator ℜ be defined as follows (see [8]):

$$ \Re u(x_{1},x_{2},x_{3})=\frac{1}{128} \Biggl( 14 \sum_{p=1}^{6}{\vphantom{\frac{1}{1}}}_{(1)}u_{p}+3 \sum_{q=7}^{18}{\vphantom{\frac{1}{1}}}_{(2)}u_{p}+ \sum_{r=19}^{26}{\vphantom{\frac{1}{1}}}_{(3)}u_{r} \Biggr) ,\quad (x_{1},x_{2},x_{3})\in R, $$

(2.13)

where the sum $\sum_{(k)}$ is taken over the grid nodes that are at a distance of $\sqrt{k}h$ from the point $(x_{1},x_{2},x_{3})$, and $u_{p}$, $u_{q}$, and $u_{r}$ are the values of u at the corresponding grid points.

We consider the following finite difference approximations of problem (2.1):

$$ u_{h}=\Re u_{h}\quad \text{on }R^{h},\qquad u_{h}=\varphi_{j}\quad \text{on }\Gamma_{jh}, j=1,2,\ldots,6. $$

(2.14)

By the maximum principle (see [9], Chapter 4), problem (2.14) has a unique solution.

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

Let $R^{kh}$ be the set of nodes of the grid $R^{h}$ whose distance from Γ is kh. It is obvious that $1\leq k\leq N(h)$, where

$$ N(h)= \bigl[ \min \{ a_{1},a_{2},a_{3} \} /(2h) \bigr] . $$

(2.15)

We define, for $1\leq k\leq N(h)$,

$$ f_{h}^{k}= \textstyle\begin{cases} 1, &(x_{1},x_{2},x_{3})\in R^{kh}, \\ 0,& (x_{1},x_{2},x_{3})\in R^{h}\backslash R^{kh}.\end{cases} $$

Lemma 2.4

The solution of the system

$$ v_{h}^{k}=\Re v_{h}^{k}+f_{h}^{k} \quad \textit{on }R^{h},\qquad v_{h}^{k}=0\quad \textit{on }\Gamma_{h}, $$

satisfies the inequality

$$ \max_{(x_{1},x_{2},x_{3})\in R^{h}}v_{h}^{k}\leq6k,\quad 1\leq k \leq N(h). $$

Proof

For the proof, see Lemma 2 in [10]. □

Lemma 2.5

Let u be a solution of problem (2.1). Then

$$ \max_{(x_{1},x_{2},x_{3})\in R^{kh}}\vert \Re u-u\vert \leq c \frac{h^{6}}{k^{2}}, \quad k=1,2,\ldots,N(h). $$

(2.16)

Proof

Let $(x_{10},x_{20},x_{30})$ be a point of $R^{1h}$, and let

$$ R_{0}= \bigl\{ (x_{1},x_{2},x_{3}): \vert x_{i}-x_{i0} \vert < h,i=1,2,3 \bigr\} $$

(2.17)

be an elementary cube, some faces of which lie on the boundary of the rectangular parallelepiped R. On the vertices of $R_{0}$ and on the center of its faces and edges. there lie the nodes of which the function values are used to evaluate $\Re u(x_{10},x_{20},x_{30})$. We represent a solution of problem (2.1) in some neighborhood of $x_{0}=(x_{10},x_{20},x_{30})\in R^{1h}$ by Taylor’s formula

$$ u(x_{1},x_{2},x_{3})=p_{7}(x_{1},x_{2},x_{3};x_{0})+r_{8}(x_{1},x_{2},x_{3};x_{0}), $$

(2.18)

where $p_{7}(x_{1},x_{2},x_{3})$ is the seventh-order Taylor polynomial, and $r_{8}(x_{1},x_{2},x_{3})$ is the remainder term. Taking into account that the function u is harmonic, we have

$$ \Re p_{7}(x_{10},x_{20},x_{30};x_{0})=u(x_{10},x_{20},x_{30}). $$

(2.19)

Now, we estimate $r_{8}$ at the nodes of the operator ℜ. We take a node $(x_{10}+h,x_{20},x_{30}+h)$, which is one of the twenty six nodes of ℜ, and consider the function

$$ \tilde{u}(s)=u \biggl( x_{10}+\frac{s}{\sqrt{2}},x_{20},x_{30}+ \frac {s}{\sqrt{2}} \biggr) , \quad -\sqrt{2}h\leq s\leq\sqrt{2}h, $$

(2.20)

of single variable s, which is the arc length along the straight line through the points $(x_{10}-h,x_{20},x_{30}-h)$ and $(x_{10}+h,x_{20},x_{30}+h)$. By Lemma 2.3 we have

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

(2.21)

We represent the function (2.20) around the point $s=0$ by Taylor’s formula

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

where $\tilde{p}_{7}(s)\equiv p_{7} ( x_{10}+\frac{s}{\sqrt {2}},x_{20},x_{30}+\frac{s}{\sqrt{2}} )$ is the seventh-order Taylor polynomial of the variable s, and

$$ \tilde{r}_{8}(s)\equiv r_{8} \biggl( x_{10}+ \frac{s}{\sqrt{2}},x_{20},x_{30}+ \frac{s}{\sqrt{2}};x_{0} \biggr) , \quad \vert s\vert < \sqrt{2}h, $$

(2.22)

is the remainder term. By the continuity of $\tilde{r}_{8}(s)$ on the interval $[ -\sqrt{2}h,\sqrt{2}h ]$ and estimate (2.21) we obtain

$$\begin{aligned}& r_{8} ( x_{10}+h,x_{20},x_{30}h;x_{0} ) \\& \quad = \lim_{\varepsilon \rightarrow+0}\tilde{r}_{8}(\sqrt{2}h-\varepsilon) \\& \quad \leq \lim_{\varepsilon\rightarrow+0} \biggl[ c\frac{1}{7!} \int_{0}^{\sqrt{2}h-\varepsilon} ( \sqrt{2}h-\varepsilon-t ) ^{7}(\sqrt {3}h-t)^{-2}\, dt \biggr] \\& \quad \leq c_{1}h^{6}, \quad 0< \varepsilon\leq \frac{\sqrt{2}h}{2}, \end{aligned}$$

(2.23)

where $c_{1}$ is a constant independent of the choice of $(x_{10},x_{20},x_{30})\in R^{kh}$.

Estimate (2.23) is obtained analogously for the remaining twenty five nodes on the closed cube (2.17). Since the norm of the operator ℜ in the uniform metric is equal to one, by (2.23) we have

$$ \bigl\vert \Re r_{8} ( x_{10},x_{20},x_{30} ) \bigr\vert \leq c_{2}h^{6}. $$

(2.24)

From (2.18), (2.19), and (2.24) we obtain

$$ \bigl\vert \Re u(x_{10},x_{20},x_{30})-u ( x_{10},x_{20},x_{30} ) \bigr\vert \leq ch^{6} $$

for any $(x_{10},x_{20},x_{30})\in R^{1h}$.

Now, let $(x_{10},x_{20},x_{30})$ be a point of $R^{kh}$ for $2\leq k\leq N(h)$. By Lemma 2.3 for any k, $2\leq k\leq N(h)$, we obtain

$$ \bigl\vert \Re r_{8} ( x_{10},x_{20},x_{30} ) \bigr\vert \leq c_{3}\frac{h^{6}}{k^{2}}, $$

(2.25)

where $c_{3}$ is a constant independent of k, $2\leq k\leq N(h)$, and the choice of $(x_{10},x_{20},x_{30})\in R^{kh}$. From (2.18), (2.19), and (2.25) estimate (2.16) follows. □

Lemma 2.6

Assume that the boundary functions $\varphi_{j}$, $j=1,2,\ldots,6$, satisfy conditions (2.2)-(2.5). Then

$$ \max_{\overline{R}^{h}}\vert u_{h}-u\vert \leq ch^{6}\bigl(1+\vert \ln h\vert \bigr), $$

(2.26)

where $u_{h}$ is the solution of the finite difference problem (2.14), and u is the exact solution of problem (2.1).

Proof

Let

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

(2.27)

By (2.14) and (2.27) the error function satisfies the system of equations

$$ \varepsilon_{h}=\Re\varepsilon_{h}+(\Re u-u)\quad \text{on }R^{h},\qquad \varepsilon_{h}=0\quad \text{on } \Gamma^{h}. $$

(2.28)

We represent a solution of system (2.28) as follows:

$$ \varepsilon_{h}=\sum_{k=1}^{N(h)} \varepsilon_{h}^{k}, $$

(2.29)

where $\varepsilon_{h}^{k}$, $1\leq k\leq N(h)$, with $N(h)$ defined by (2.15), is a solution of the system

$$ \varepsilon_{h}^{k}=\Re\varepsilon_{h}^{k}+ \nu^{k}\quad \text{on }R^{h},\qquad \varepsilon_{h}^{k}=0\quad \text{on }\Gamma^{h}, $$

(2.30)

where

$$ \nu^{k}= \textstyle\begin{cases} \Re u-u&\text{on }R^{kh}, \\ 0&\text{on }R^{h}\backslash R^{kh}.\end{cases} $$

Then for the solution of (2.30) by applying Lemmas 2.4 and 2.5 we have

$$ \max_{(x_{1},x_{2},x_{3})\in R^{h}}\bigl\vert \varepsilon _{h}^{k} \bigr\vert \leq c\frac{h^{6}}{k},\quad 1\leq k\leq N(h). $$

(2.31)

By (2.27), (2.29), and (2.31) we obtain

$$ \max_{(x_{1},x_{2},x_{3})\in R^{h}}\vert u_{h}-u\vert \leq ch^{6} \bigl( 1+\vert \ln h\vert \bigr) . $$

 □

Let ω be a solution of the problem

$$ \Delta\omega=0 \quad \text{on } R, \qquad \omega=\psi_{j}\quad \text{on } \Gamma _{j}, j=1,2,\ldots,6, $$

(2.32)

where $\psi_{j}$, $j=1,2,\ldots,6$, are given functions, and

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

(2.33)

$$\begin{aligned}& \psi_{\mu} = \psi_{\nu}\quad \text{on }\gamma_{\mu\nu}, \end{aligned}$$

(2.34)

$$\begin{aligned}& \frac{\partial^{2}\psi_{\mu}}{\partial t_{\mu}^{2}}+\frac{\partial ^{2}\psi_{\nu}}{\partial t_{\nu}^{2}}+\frac{\partial^{2}\psi_{\mu }}{\partial t_{\mu\nu}^{2}} = 0\quad \text{on } \gamma_{\mu\nu}. \end{aligned}$$

(2.35)

Lemma 2.7

We have the estimate

$$ \max_{\overline{R}^{h}}\vert \omega_{h}-\omega \vert \leq ch^{4}, $$

(2.36)

where ω is the exact solution of problem (2.32), and $\omega_{h}$ is the exact solution of the finite difference problem

$$ \omega_{h}=\Re\omega_{h}\quad \textit{on }R^{h}, \qquad \omega_{h}=\psi _{j}\quad \textit{on }\Gamma_{jh}, j=1,2,\ldots,6. $$

(2.37)

Proof

It follows from Lemma 1.2 in [5] that

$$ \max_{0\leq p\leq q}\max_{0\leq q\leq2-p}\sup _{(x_{1},x_{2},x_{3})\in R}\biggl\vert \frac{\partial^{4}\omega(x_{1},x_{2},x_{3})}{\partial x_{1}^{2p}\, \partial x_{2}^{2q}\, \partial x_{3}^{4-2p-2q}}\biggr\vert < \infty, $$

where u is the solution of problem (2.32). Then, instead of inequality (2.12), we have

$$ \max_{0\leq\mu\leq8}\max_{0\leq\nu\leq8-\mu}\biggl\vert \frac {\partial ^{8}\omega(x_{1},x_{2},x_{3})}{\partial x_{1}^{\mu}\, \partial x_{2}^{\nu }\, \partial x_{3}^{8-\mu-\nu}}\biggr\vert \leq c\rho ^{-4}(x_{1},x_{2},x_{3}), \quad (x_{1},x_{2},x_{3})\in R, $$

(2.38)

where $\rho(x_{1},x_{2},x_{3})$ is the distance from $(x_{1},x_{2},x_{3})\in R$ to the boundary Γ.

By estimate (2.38) and Taylor’s formula, by analogy with the proof of Lemma 2.5 we have

$$ \max_{(x_{1},x_{2},x_{3})\in R^{kh}}\vert \Re\omega-\omega \vert \leq c \frac{h^{4}}{k^{4}},\quad k=1,2,\ldots,N(h). $$

We put

$$ \epsilon_{h}=\omega_{h}-\omega\quad \text{on }R^{h}\cup\Gamma_{h}. $$

Then, as in the proof of Lemma 2.6, we obtain

$$ \max_{\overline{R}^{h}}\vert \omega_{h}-\omega \vert \leq c_{4}h^{4}\sum_{k=1}^{N(h)} \frac{1}{k^{3}}\leq ch^{4}. $$

 □

3 Approximation of the first derivative

Let $v=\frac{\partial u}{\partial x_{1}}$, and let $\Phi_{j}=\frac {\partial u}{\partial x_{1}}$ on $\Gamma_{j}$, $j=1,2,\ldots,6$, and consider the boundary value problem

$$ \Delta v=0\quad \text{on }R,\qquad v=\Phi_{j}\quad \text{on } \Gamma_{j}, j=1,2,\ldots,6, $$

(3.1)

where u is a solution of the boundary value problem (2.1).

We define the following operators $\Phi_{\nu h}$, $\nu=1,2,\ldots,6$:

$$\begin{aligned}& \Phi_{1h}(u_{h}) = \frac{1}{12h}\bigl( -25\varphi _{1}(x_{2},x_{3})+48u_{h}(h,x_{2},x_{3})-36u_{h}(2h,x_{2},x_{3}) \\& \hphantom{\Phi_{1h}(u_{h}) ={}}{}+ 16u_{h}(3h,x_{2},x_{3})-3u_{h}(4h,x_{2},x_{3}) \bigr) \quad \text{on }\Gamma_{1}^{h}, \end{aligned}$$

(3.2)

$$\begin{aligned}& \Phi_{4h}(u_{h}) = \frac{1}{12h}\bigl( 25\varphi _{4}(x_{2},x_{3})-48u_{h}(a_{1}-h,x_{2},x_{3})+36u_{h}(a_{1}-2h,x_{2},x_{3}) \\& \hphantom{\Phi_{4h}(u_{h}) ={}}{}- 16u_{h}(a_{1}-3h,x_{2},x_{3})+3u_{h}(a_{1}-4h,x_{2},x_{3}) \bigr)\quad \text{on }\Gamma_{4}^{h}, \end{aligned}$$

(3.3)

$$\begin{aligned}& \Phi_{ph}(u_{h}) = \frac{\partial\varphi_{p}}{\partial x_{1}}\quad \text{on } \Gamma_{p}^{h}, p=2,3,5,6, \end{aligned}$$

(3.4)

where $u_{h}$ is the solution of finite difference problem (2.14).

Lemma 3.1

We have the inequality

$$ \bigl\vert \Phi_{kh}(u_{h})-\Phi_{kh}(u)\bigr\vert \leq c_{3}h^{5}\bigl(1+\vert \ln h\vert \bigr), \quad k=1,4, $$

(3.5)

where $u_{h}$ is the solution of problem (2.14), and u is the solution of problem (2.1).

Proof

It is obvious that $\Phi_{ph}(u_{h})-\Phi_{ph}(u)=0$ for $p=2,3,5,6$. For $k=1$, by (3.2) and Lemma 2.6 we have

$$\begin{aligned} \bigl\vert \Phi_{1h}(u_{h})-\Phi_{1h}(u)\bigr\vert \leq&\frac {1}{12h}\bigl( 48\bigl\vert u_{h}(h,x_{2},x_{3})-u(h,x_{2},x_{3}) \bigr\vert \\ &{}+ 36\bigl\vert u_{h}(2h,x_{2},x_{3})-u(2h,x_{2},x_{3}) \bigr\vert \\ &{}+16\bigl\vert u_{h}(3h,x_{2},x_{3})-u(3h,x_{2},x_{3}) \bigr\vert \\ &{}+3\bigl\vert u_{h}(4h,x_{2},x_{3})-u(4h,x_{2},x_{3}) \bigr\vert \bigr) \\ \leq& c_{5}h^{5}\bigl(1+\vert \ln h\vert \bigr). \end{aligned}$$

The same inequality is also true when $k=4$. □

Lemma 3.2

We have the inequality

$$ \max_{(x_{1},x_{2},x_{3})\in\Gamma_{k}^{h}}\bigl\vert \Phi _{kh}(u_{h})- \Phi_{k}\bigr\vert \leq c_{4}h^{4},\quad k=1,4, $$

(3.6)

where $\Phi_{kh}$, $k=1,4$, are defined by (3.2), (3.3), and $\Phi_{k}=\frac{\partial u}{\partial x_{1}}$ on $\Gamma_{k}$, $k=1,4$.

Proof

From Lemma 2.1 it follows that $u\in C^{5,0}(\overline{R})$. Then, at the end points $(0,\nu h,\omega h)\in\Gamma_{1}^{h}$ and $(a_{1},\nu h,\omega h)\in\Gamma_{4}^{h}$ of each line segment

$$\bigl\{ (x_{1},x_{2},x_{3}):0\leq x_{1}\leq a_{1},0< x_{2}=\nu h< a_{2},0< x_{3}= \omega h< a_{3} \bigr\} , $$

expressions (3.2) and (3.3) give the fourth-order approximation of $\frac{\partial u}{\partial x_{1}}$, respectively. From the truncation error formulas (see [11]) it follows that

$$ \max_{(x_{1},x_{2},x_{3})\in\Gamma_{k}^{h}}\bigl\vert \Phi(u)-\Phi _{k}\bigr\vert \leq c_{5}h^{4},\quad k=1,4. $$

(3.7)

By Lemma 3.1 and estimate (3.7), (3.6) follows. □

We consider the finite difference boundary value problem

$$ v_{h}=\Re v_{h}\quad \text{on }R^{h},\qquad v_{h}=\Phi_{jh}\quad \text{on }\Gamma_{j}^{h}, j=1,2,\ldots,6, $$

(3.8)

where $\Psi_{jh}$, $j=1,2,\ldots,6$, are defined by (3.2)-(3.4).

Theorem 3.3

We have the estimate

$$ \max_{(x_{1},x_{2},x_{3})\in\overline{R}^{h}}\biggl\vert v_{h}-\frac{\partial u}{\partial x_{1}} \biggr\vert \leq ch^{4}, $$

(3.9)

where u is the solution of problem (2.1), and $v_{h}$ is the solution of the finite difference problem (3.8).

Proof

Let

$$ \varepsilon_{h}=v_{h}-v\quad \text{on } \overline{R}^{h}, $$

(3.10)

where $v=\frac{\partial u}{\partial x_{1}}$. From (3.8) and (3.10) we have

$$\begin{aligned}& \varepsilon_{h} = \Re\varepsilon_{h}+(\Re v-v)\quad \text{on }R^{h}, \\& \varepsilon_{h} = \Phi_{kh}(u_{h})-v\quad \text{on }\Gamma_{k}^{h}, k=1,4,\qquad \varepsilon_{h}=0\quad \text{on }\Gamma_{p}^{h}, p=2,3,5,6. \end{aligned}$$

We represent

$$ \varepsilon_{h}=\varepsilon_{h}^{1}+ \varepsilon_{h}^{2}, $$

(3.11)

where

$$\begin{aligned}& \varepsilon_{h}^{1} = \Re\varepsilon_{h}^{1} \quad \text{on }R^{h}, \end{aligned}$$

(3.12)

$$\begin{aligned}& \varepsilon_{h}^{1} = \Phi_{kh}(u_{h})-v \quad \text{on }\Gamma_{k}^{h},k=1,4, \qquad \varepsilon_{h}^{1}=0\quad \text{on }\Gamma_{p}^{h},p=2,3,5,6; \end{aligned}$$

(3.13)

$$\begin{aligned}& \varepsilon_{h}^{2} = \Re\varepsilon_{h}^{2}+( \Re v-v)\quad \text{on }R^{h},\qquad \varepsilon_{h}^{2}=0 \quad \text{on }\Gamma_{j}^{h}, j=1,2,\ldots,6. \end{aligned}$$

(3.14)

By Lemma 3.2 and by the maximum principle, for the solution of system (3.12)-(3.13), we have

$$ \max_{(x_{1},x_{2},x_{3})\in\overline{R}^{h}}\bigl\vert \varepsilon _{h}^{1} \bigr\vert \leq\max_{q=1,4}\max_{(x_{1},x_{2},x_{3})\in\Gamma _{q}^{h}}\bigl\vert \Phi_{qh}(u_{h})-v\bigr\vert \leq c_{4}h^{4}. $$

(3.15)

The solution $\varepsilon_{h}^{2}$ of system (3.14) is the error of the approximate solution obtained by the finite difference method for problem (3.1) when on the boundary nodes $\Gamma_{jh}$, the approximate values are defined as the exact values of the functions $\Phi _{j}$ in (3.1). It is obvious that $\Phi_{j}$, $j=1,2,\ldots,6$, satisfy the conditions

$$\begin{aligned}& \Phi_{j}\in C^{5,\lambda}(\Gamma_{j}),\quad 0< \lambda< 1, j=1,2,\ldots,6, \end{aligned}$$

(3.16)

$$\begin{aligned}& \Phi_{\mu}=\Phi_{\nu}\quad \text{on }\gamma_{\mu\nu}, \end{aligned}$$

(3.17)

$$\begin{aligned}& \frac{\partial_{\mu}^{2}\Phi}{\partial t_{\mu}^{2}}+\frac{\partial _{\nu }^{2}\Phi}{\partial t_{\nu}^{2}}+\frac{\partial_{\mu}^{2}\Phi }{\partial t_{\mu\nu}^{2}}=0\quad \text{on } \gamma_{\mu\nu}. \end{aligned}$$

(3.18)

Since the function $v=\frac{\partial u}{\partial x_{1}}$ is harmonic on R with the boundary functions $\Psi_{j}$, $j=1,2,\ldots,6$, by (3.16)- (3.18) and Lemma 2.7 we obtain

$$ \max_{(x_{1},x_{2},x_{3})\in\overline{R}^{h}}\bigl\vert \epsilon _{h}^{2} \bigr\vert \leq c_{6}h^{4}. $$

(3.19)

By (3.11), (3.15), and (3.19), inequality (3.9) follows. □

Remark 1

By Lemma 2.2 the sixth-order pure derivatives are bounded in R. Therefore, if we replace formulae (3.2) and (3.3) by the fifth-order forward and backward numerical differentiation formulae (see Chapter 2 in [12]), then by analogy with the proof of estimate (3.9) we obtain

$$ \max_{(x_{1},x_{2},x_{3})\in\overline{R}^{h}}\biggl\vert v_{h}-\frac{\partial u}{\partial x_{1}} \biggr\vert \leq ch^{5}\bigl(1+\vert \ln h\vert \bigr). $$

4 Approximation of the pure second derivatives

We denote by $\omega=\frac{\partial^{2}u}{\partial x_{1}^{2}}$. The function ω is harmonic on R, by Lemma 2.1 is continuous on R̅, and is a solution of the following Dirichlet problem:

$$ \Delta\omega=0\quad \text{on }R,\qquad \omega=\chi_{j}\quad \text{on }\Gamma_{j}, j=1,2,\ldots,6, $$

(4.1)

where

$$\begin{aligned}& \chi_{\tau} = \frac{\partial^{2}\varphi_{\tau}}{\partial x_{1}^{2}},\quad \tau=2,3,5,6, \end{aligned}$$

(4.2)

$$\begin{aligned}& \chi_{\nu} = - \biggl( \frac{\partial^{2}\varphi_{\nu}}{\partial x_{2}^{2}}+\frac{\partial^{2}\varphi_{\nu}}{\partial x_{3}^{2}} \biggr) , \quad \nu=1,4. \end{aligned}$$

(4.3)

Let $\omega_{h}$ be the solution of the finite difference problem

$$ \omega_{h}=\Re\omega_{h}\quad \text{on }R^{h}, \qquad \omega_{h}=\chi _{j}\quad \text{on } \Gamma_{j}^{h}, j=1,2,\ldots,6, $$

(4.4)

where $\chi_{j}$, $j=1,2,\ldots,6$, are the functions determined by (4.2) and (4.3).

Theorem 4.1

We have the estimate

$$ \max_{\overline{R}^{h}}\vert \omega_{h}-\omega \vert \leq ch^{4}, $$

(4.5)

where $\omega=\frac{\partial^{2}u}{\partial x_{1}^{2}}$, u is the solution of problem (2.1), and $\omega_{h}$ is the solution of the finite difference problem (4.4).

Proof

By the continuity of the function ω on R̅, from (2.2)-(2.5) and (4.2), (4.3) it follows that

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

(4.6)

$$\begin{aligned}& \chi_{\mu}=\chi_{\nu}\quad \text{on }\gamma_{\mu\nu}, \end{aligned}$$

(4.7)

$$\begin{aligned}& \frac{\partial^{2}\chi_{\mu}}{\partial t_{\mu}^{2}}+\frac{\partial ^{2}\chi_{\nu}}{\partial t_{\nu}^{2}}+\frac{\partial^{2}\chi_{\mu }}{\partial t_{\mu\nu}^{2}}=0\quad \text{on } \gamma_{\mu\nu}. \end{aligned}$$

(4.8)

The boundary functions $\chi_{j}$, $j=1,2,\ldots,6$, in (4.1) by (4.6)-(4.8) satisfy all conditions of Lemma 2.7, from which the proof of the error estimate (4.5) follows. □

5 Numerical results

Let $R= \{ ( x_{1},x_{2},x_{3} ) :0< x_{i}<1,i=1,2,3 \}$, and let Γ be the boundary of R. We consider the following problem:

$$ \Delta u=0 \quad \text{on } R, \qquad u=\varphi ( x_{1},x_{2},x_{3} ) \quad \text{on } \Gamma_{j}, j=1,2,\ldots,6, $$

(5.1)

where

$$\begin{aligned}& \theta = \arctan \biggl( \frac{x_{2}}{x_{1}} \biggr) , \\& \varphi(x_{1},x_{2},x_{3}) = \biggl( x_{3}-\frac{1}{2} \biggr) ^{2}- \biggl( \frac{x_{1}^{2}+x_{2}^{2}}{2} \biggr) + \bigl( x_{1}^{2}+x_{2}^{2} \bigr) ^{\frac{ ( 6+\frac{1}{30} ) }{2}}\cdot\cos \biggl( 6+\frac {1}{30} \biggr) \theta \end{aligned}$$

is the exact solution of this problem.

Let U be the exact solution of the continuous problem, and $U_{h}$ be its approximate values on $\overline{R}^{h}$. We denote $\Vert U-U_{h}\Vert _{\overline{R}^{h}}=\max_{\overline{R}^{h}}\vert U-U_{h}\vert$ and $E_{U}^{m}=\frac{\Vert U-U_{2^{-m}}\Vert _{\overline{R}^{h}}}{\Vert U-U_{2^{- ( m+1 ) }}\Vert _{\overline{R}^{h}}}$.

In Tables 1 and 2, the maximum errors and the order of convergence of the approximate solution for different step sizes h are given, which corresponds to order of accuracy $O(h^{6}\vert \ln h\vert )$. In Tables 2 and 3, the results for the first and pure second derivatives of problem (5.1) are presented, which correspond to $O(h^{4})$. The results presented in Table 4 show that the accuracy is improved by using the fifth-order accurate formulae for the same conditions imposed on the given boundary functions.

Table 1 Results for the solution

Table 2 First derivative approximation results with the fourth-order accurate formulae

Table 3 Second pure derivative approximation results

Table 4 First derivative approximation results with the fifth-order accurate formulae

6 Conclusion

A highly accurate difference schemes are proposed and investigated under the conditions imposed on the given boundary values to approximate the solution of the 3D Laplace equation and its first and pure second derivatives on a cubic grid. The uniform convergence for the approximate solution at the rate of $O(h^{6}\vert \ln h\vert )$ and for the first and pure second derivatives at the rate of $O(h^{4})$ is proved. It is shown that the accuracy for the approximate value of the first derivatives can be improved up to $O(h^{5}\vert \ln h\vert )$ for the same boundary functions by using the fifth-order formulae on some faces of the parallelepiped.

The obtained results can be used to justify finding the above-mentioned derivatives of the solution of 3D Laplace boundary value problems on domains described as unions or as intersections of a finite number of rectangular parallelepipeds by the difference method, using the Schwarz or Schwarz-Neumann iterations (see [13–19]).