A fourth order accurate approximation of the first and pure second derivatives of the Laplace equation on a rectangle
- Adiguzel A Dosiyev^{1}Email author and
- Hamid MM Sadeghi^{1}
https://doi.org/10.1186/s13662-015-0408-8
© Dosiyev and Sadeghi; licensee Springer. 2015
Received: 4 November 2014
Accepted: 9 February 2015
Published: 28 February 2015
Abstract
In this paper, we discuss an approximation of the first and pure second order derivatives for the solution of the Dirichlet problem on a rectangular domain. The boundary values on the sides of the rectangle are supposed to have the sixth derivatives satisfying the Hölder condition. On the vertices, besides the continuity condition, the compatibility conditions, which result from the Laplace equation for the second and fourth derivatives of the boundary values, given on the adjacent sides, are also satisfied. Under these conditions a uniform approximation of order \(O(h^{4})\) (h is the grid size) is obtained for the solution of the Dirichlet problem on a square grid, its first and pure second derivatives, by a simple difference scheme. Numerical experiments are illustrated to support the analysis made.
Keywords
finite difference method approximation of derivatives uniform error Laplace equation1 Introduction
Since the operation of differentiation is ill conditioned, to find a highly accurate approximation for the derivatives of the solution of a differential equation becomes problematic, especially when the smoothness is restricted.
In [1], it was proved that the higher order difference derivatives uniformly converge to the corresponding derivatives of the solution of the Laplace equation in any strictly interior subdomain, with the same order of h as which the difference solution converges on the given domain. In [2], by using the difference solution of the Dirichlet problem for the Laplace equation on a rectangle, the uniform convergence of its first and pure second divided difference over the whole grid domain to the corresponding derivatives of the exact solution with the rate \(O(h^{2})\) is proved. In [3], the difference schemes on a rectangular parallelepiped were constructed, where solutions approximate the Dirichlet problem for the Laplace equation and its first and second derivatives. Under the assumptions that the boundary functions belong to \(C^{4,\lambda }\), \(0<\lambda<1\), on the faces, are continuous on the edges, and their second-order derivatives satisfy the compatibility condition, the solution to their difference schemes converges uniformly on the grid with the rate of \(O(h^{2})\).
In this paper, we consider the Dirichlet problem for the Laplace equation on a rectangle, when the boundary values belong to \(C^{6,\lambda }\), \(0<\lambda <1\), on the sides of the rectangle and as a whole are continuous on the vertices. Also the 2τ, \(\tau=1,2\), order derivatives satisfy the compatibility conditions on the vertices which result from the Laplace equation. Under these conditions, we construct the difference problems, the solutions of which converge to the first and pure second derivatives of the exact solution with the order \(O(h^{4})\). Finally, numerical experiments are given in the last part of the paper to support the theoretical results.
2 The Dirichlet problem on rectangular domains
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 kth derivatives on D satisfying a Hölder condition with exponent \(\lambda\in(0,1)\).
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
Proof
Lemma 2.3
Proof
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
Proof
Lemma 2.5
The proof of Lemma 2.5 follows from the comparison theorem (see Chapter 4 in [7]).
Lemma 2.6
Proof
Theorem 2.7
Proof
3 Approximation of the first derivative
Lemma 3.1
Proof
Lemma 3.2
Proof
Theorem 3.3
Proof
4 Approximation of the pure second derivatives
Theorem 4.1
5 Numerical example
Let U be the exact solution and \(U_{h}\) be its approximate values on \(\overline{\Pi}^{h}\) of the Dirichlet problem on the rectangular domain Π. We denote \(\Vert U-U_{h}\Vert _{\overline{\Pi}^{h}}= \max_{\overline{\Pi}^{h}}\vert U-U_{h}\vert \), \(\Re _{U}^{m}=\frac{\Vert U-U_{2^{-m}}\Vert _{\overline{\Pi}^{h}}}{\Vert U-U_{2^{- ( m+1 ) }}\Vert _{\overline{\Pi}^{h}}}\).
The approximate results for the first derivative
h | \(\Vert v-v_{h}\Vert \) | \(\Re_{v}^{m}\) |
---|---|---|
\(\frac{1}{8}\) | 2.299996064764325009657E − 2 | |
\(\frac{1}{16}\) | 1.894059104568160525104E − 3 | 12.14 |
\(\frac{1}{32}\) | 1.344880793701474553783E − 4 | 14.08 |
\(\frac{1}{64}\) | 8.960663249977644986927E − 6 | 15.01 |
\(\frac{1}{128}\) | 5.796393863873542692774E − 7 | 15.46 |
The approximate results for the pure second derivative
h | \(\Vert \omega-\omega_{h}\Vert \) | \(\Re_{\omega}^{m}\) |
---|---|---|
\(\frac{1}{8}\) | 3.149059928597543772878E − 6 | |
\(\frac{1}{16}\) | 1.931058119052719414451E − 7 | 16.31 |
\(\frac{1}{32}\) | 1.180485369727342048019E − 8 | 16.36 |
\(\frac{1}{64}\) | 7.211217140499053022025E − 10 | 16.37 |
\(\frac{1}{128}\) | 4.404326492162507264392E − 11 | 16.37 |
The results show that the approximate solutions converge as \(O(h^{4})\).
6 Conclusion
The obtained results can be used to highly approximate the derivatives for the solution of Laplace’s equation by the finite difference method, in some version of domain decomposition methods, in composite grid methods, and in the combined methods for solving Laplace’s boundary value problems on polygons (see [10–13]).
Declarations
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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