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On a highly accurate approximation of the first and pure second derivatives of the Laplace equation in a rectangular parallelpiped
- Adiguzel A Dosiyev^{1}Email author and
- Hamid Mir-Mohammad Sadeghi^{2}
https://doi.org/10.1186/s13662-016-0868-5
© Dosiyev and Sadeghi 2016
Received: 10 February 2016
Accepted: 19 May 2016
Published: 3 June 2016
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.
Keywords
- finite difference method
- approximation of derivatives
- uniform error
- Laplace equation
MSC
- 65M06
- 65M12
- 65M22
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 \(\lambda\in ( 0,1 ) \).
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
Proof
Lemma 2.3
Proof
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 \(\Gamma _{h}=\Gamma_{1h}\cup\Gamma_{2h}\cup\cdots\cup\Gamma_{6h}\).
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.
Lemma 2.4
Proof
For the proof, see Lemma 2 in [10]. □
Lemma 2.5
Proof
Lemma 2.6
Proof
Lemma 2.7
Proof
3 Approximation of the first derivative
Lemma 3.1
Proof
Lemma 3.2
Proof
Theorem 3.3
Proof
Remark 1
4 Approximation of the pure second derivatives
Theorem 4.1
Proof
5 Numerical results
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 \(E_{U}^{m}=\frac{\Vert U-U_{2^{-m}}\Vert _{\overline{R}^{h}}}{\Vert U-U_{2^{- ( m+1 ) }}\Vert _{\overline{R}^{h}}}\).
Results for the solution
\(\boldsymbol{\frac{1}{h}}\) | \(\boldsymbol{\Vert u-u_{h}\Vert _{\overline{R}^{h}}}\) | \(\boldsymbol{E_{u}^{m}}\) |
---|---|---|
\(\frac{1}{8}\) | 1.3642E − 9 | 54.95 |
\(\frac{1}{16}\) | 2.4828E − 11 | 62.64 |
\(\frac{1}{32}\) | 3.9637E − 13 | 63.14 |
\(\frac{1}{64}\) | 6.2773E − 15 | 63.77 |
\(\frac{1}{128}\) | 9.8437E − 17 |
First derivative approximation results with the fourth-order accurate formulae
\(\boldsymbol{\frac{1}{h}}\) | \(\boldsymbol{\Vert v-v_{h}\Vert _{\overline{R}^{h}}}\) | \(\boldsymbol{E_{v}^{m}}\) |
---|---|---|
\(\frac{1}{8}\) | 1.4993E − 2 | 9.78 |
\(\frac{1}{16}\) | 1.5327E − 3 | 12.93 |
\(\frac{1}{32}\) | 1.1854E − 4 | 14.50 |
\(\frac{1}{64}\) | 8.1771E − 6 | 15.25 |
\(\frac{1}{128}\) | 5.3605E − 7 |
Second pure derivative approximation results
\(\boldsymbol{\frac{1}{h}}\) | \(\boldsymbol{\Vert w-w_{h}\Vert _{\overline{R}^{h}}}\) | \(\boldsymbol{E_{w}^{m}}\) |
---|---|---|
\(\frac{1}{8}\) | 9.8243E − 7 | 15.21 |
\(\frac{1}{16}\) | 6.4587E − 8 | 16.21 |
\(\frac{1}{32}\) | 3.9850E − 9 | 16.36 |
\(\frac{1}{64}\) | 2.4361E − 10 | 16.37 |
\(\frac{1}{128}\) | 1.4879E − 11 |
First derivative approximation results with the fifth-order accurate formulae
\(\boldsymbol{\frac{1}{h}}\) | \(\boldsymbol{\Vert v-v_{h}\Vert _{\overline{R}^{h}}}\) | \(\boldsymbol{E_{v}^{m}}\) |
---|---|---|
\(\frac{1}{8}\) | 2.0469E − 3 | 22.08 |
\(\frac{1}{16}\) | 9.2725E − 5 | 27.35 |
\(\frac{1}{32}\) | 3.3903E − 6 | 29.78 |
\(\frac{1}{64}\) | 1.1382E − 7 | 30.91 |
\(\frac{1}{128}\) | 3.6823E − 9 |
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]).
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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