- Research
- Open Access
Error estimate of a high accuracy difference scheme for Poisson equation with two integral boundary conditions
- Liping Zhou^{1, 2} and
- Haiyuan Yu^{1}Email author
https://doi.org/10.1186/s13662-018-1682-z
© The Author(s) 2018
- Received: 7 February 2018
- Accepted: 7 June 2018
- Published: 28 June 2018
Abstract
Partial differential equations with nonlocal boundary conditions have been widely applied in various fields of science and engineering. In this work, we first build a high accuracy difference scheme for Poisson equation with two integral boundary conditions. Then, we prove that the scheme can reach the asymptotic optimal error estimate in the maximum norm through applying the discrete Fourier transformation. In the end, numerical experiments validate the correctness of theoretical results and show the stability of the scheme.
Keywords
- Poisson equation
- Integral boundary condition
- Finite difference scheme
- Discrete Fourier transformation
- Asymptotic optimal error estimate
1 Introduction
Partial differential equations with nonlocal boundary conditions have been widely used to build mathematical models in various fields of science and engineering such as thermoelasticity, physics, medical science, chemical engineering, and so on (see [1–6]).
FDM is preferred by many researchers because of its simple format and easy programming. Recently, Sapagovas [7] presented a difference scheme of fourth-order approximation for Poisson equation with two integral boundary conditions. The author also studied its solvability and justified an iteration method for solving the corresponding difference system. Berikelashvili [8] constructed some difference schemes for Poisson problem with one integral condition and obtained its estimate of the convergence rate. For Poisson equation with Bitsadze–Samarskii nonlocal boundary, a new method was developed [9] which used the five-point difference scheme to discretize Laplace operator. There are also some literature works on nonlinear and high order elliptic problems with nonlocal boundary conditions. In [10, 11], the authors presented some iterative methods for the system of difference equations to solve nonlinear elliptic equation with integral condition. Pao and Wang [12, 13] used finite difference method to construct a coupled system of two second-order equations for fourth-order elliptic equations with nonlocal boundary conditions.
In recent years, the radial basis function (RBF) collocation method is very popular for PDEs to seek numerical solution, especially for elliptic equations with nonlocal boundary [14–16]. However, the numerical results of RBF collocation method often suffer from shape parameter and condition number of the collocation matrix. As for some other numerical methods for elliptic equations with nonlocal boundary conditions, e.g., FEM, we refer the reader to [17–19].
To our knowledge, few studies not only focus on building high accuracy difference schemes which are of optimal or asymptotic optimal order for error estimation and showing theoretical proofs, but also on displaying corresponding numerical tests for Poisson problem with nonlocal boundary conditions. Therefore, how to design a high accuracy scheme and prove that it is of optimal or asymptotic optimal order for error estimation is a great challenge for us. In this work, we consider a two-dimensional Poisson problem with two integral conditions. The first novel idea is that we build a high accuracy difference scheme by introducing the equivalent relations which are convenient to discretize two nonlocal conditions. The second one is that we ingeniously apply the discrete Fourier transformation (DFT) to transform the two-dimensional problem to a one-dimensional problem for error analysis. Besides, we prove that the difference scheme can reach the asymptotic optimal error estimate in the maximum norm. Numerical examples confirm the correctness of theoretical results.
This work is organized as follows. In Sect. 2, we present a finite difference scheme for Problem (1.1a)–(1.1e). In Sect. 3, the error equations of the scheme are analyzed with the DFT and the corresponding error estimates are presented. In Sect. 4, we show numerical results to support our conclusions. Finally, a summary of this article and future work in this field are discussed.
2 The finite difference discretization
For convenience of discretizing the integral boundary conditions, we can easily prove the equivalent relations as follows.
Lemma 2.1
Proof
Similarly, from (1.1a)–(1.1c) and (2.2), we can also derive that (1.1e) is equivalent to (2.4). Thus, the proof of this lemma is completed. □
3 Error estimate
Let \(\Vert \widehat{\alpha}_{k}\Vert =\max_{j=1,\ldots,N-1} |\widehat{\alpha}_{k,j}| \). Now, we can obtain the following estimates.
Lemma 3.1
Proof
Now we present the convergence theorem for Problem (1.1a)–(1.1e).
Theorem 3.1
Proof
\(C_{1} = \frac{h(\widetilde{\beta}_{2})_{k}}{(1-\lambda ^{N_{1}-N_{2}-N}_{k})(\lambda^{N}_{k}\overline{\eta}+\lambda^{N_{2}}_{k} \eta)}\) and \(C_{2} = \frac{h(\widetilde{\beta}_{1})_{k}}{(1-\lambda ^{N+N_{2}-N_{1}}_{k})(\lambda^{N_{1}}_{k}\overline{\eta}+\eta)}\).
4 Numerical experiments
In this section, we present two typical examples to demonstrate the theoretical results and compare the numerical results with the RBF collocation method [15].
Example 4.1
The errors for finite difference method in two norms
h | \(\|u-U\|_{2}\) | ratio | \(\|u-U\|_{\infty}\) | ratio |
---|---|---|---|---|
1/16 | 6.504E–03 | 1.724E–02 | ||
1/32 | 1.728E–03 | 3.76 | 4.545E–03 | 3.79 |
1/64 | 4.451E–04 | 3.88 | 1.166E–03 | 3.90 |
1/128 | 1.130E–04 | 3.94 | 2.953E–04 | 3.95 |
The errors for finite difference method in the sense of pointwise
(x,y) | (0.25, 0.25) | (0.25, 0.5) | (0.5, 0.25) | (0.5, 0.5) | ||||
---|---|---|---|---|---|---|---|---|
h | ratio | ratio | ratio | ratio | ||||
1/16 | 4.241E–03 | 5.471E–03 | 5.998E–03 | 7.737E–03 | ||||
1/32 | 1.165E–03 | 3.64 | 1.437E–03 | 3.80 | 1.647E–03 | 3.64 | 2.035E–03 | 3.80 |
1/64 | 3.043E–04 | 3.83 | 3.688E–04 | 3.90 | 4.303E–04 | 3.83 | 5.216E–04 | 3.90 |
1/128 | 7.773E–05 | 3.91 | 9.337E–05 | 3.95 | 1.099E–04 | 3.92 | 1.320E–04 | 3.95 |
The errors for the RBF collocation method in two norms
h | κ(A) | \(\|u-U_{\mathrm{RBF}}\|_{2}\) | ratio | \(\|u-U_{\mathrm{RBF}}\|_{\infty}\) | ratio |
---|---|---|---|---|---|
1/16 | 9.8519e+16 | 2.6519e–05 | – | 1.4564e–04 | – |
1/32 | 2.0368e+20 | 2.0121e–06 | 13.18 | 1.3284e–05 | 10.96 |
1/64 | 3.9795e+21 | 1.0321e–05 | 0.19 | 6.7160e–05 | 0.20 |
1/128 | 9.1311e+22 | 5.1418e–05 | 0.20 | 3.4878e–04 | 0.19 |
The errors for the RBF collocation method in the sense of pointwise
(x,y) | (0.25, 0.25) | (0.25, 0.5) | (0.5, 0.25) | (0.5, 0.5) | ||||
---|---|---|---|---|---|---|---|---|
h | ratio | ratio | ratio | ratio | ||||
1/16 | 1.1372e–05 | – | 6.8717e–06 | – | 1.8715e–05 | – | 1.0472e–05 | – |
1/32 | 2.2706e–06 | 5.01 | 1.5711e–06 | 4.37 | 5.3550e–07 | 34.95 | 1.5465e–06 | 6.77 |
1/64 | 1.3262e–06 | 1.71 | 9.3458e–06 | 0.17 | 5.7546e–06 | 0.09 | 6.5863e–06 | 0.23 |
1/128 | 3.5495e–05 | 0.04 | 2.1038e–05 | 0.44 | 2.2841e–05 | 0.25 | 9.6709e–06 | 0.68 |
Example 4.2
The errors for finite difference solutions in two norms
h | \(\|u-U\|_{2}\) | ratio | \(\|u-U\|_{\infty}\) | ratio |
---|---|---|---|---|
1/16 | 2.394E–04 | 3.716E–04 | ||
1/32 | 6.077E–05 | 3.94 | 9.298E–05 | 4.00 |
1/64 | 1.531E–05 | 3.97 | 2.327E–05 | 4.00 |
1/128 | 3.842E–06 | 3.98 | 5.817E–06 | 4.00 |
The errors for finite difference solutions in the sense of pointwise
(x,y) | (0.25, 0.25) | (0.25, 0.5) | (0.5, 0.25) | (0.5, 0.5) | ||||
---|---|---|---|---|---|---|---|---|
h | ratio | ratio | ratio | ratio | ||||
1/16 | 2.168E–04 | 2.269E–04 | 3.187E–04 | 3.333E–04 | ||||
1/32 | 5.422E–05 | 4.00 | 5.674E–05 | 4.00 | 7.972E–05 | 4.00 | 8.337E–05 | 4.00 |
1/64 | 1.356E–05 | 4.00 | 1.419E–05 | 4.00 | 1.993E–05 | 4.00 | 2.085E–05 | 4.00 |
1/128 | 3.389E–06 | 4.00 | 3.547E–06 | 4.00 | 4.984E–06 | 4.00 | 5.212E–06 | 4.00 |
The errors for the RBF collocation method in two norms
h | κ(A) | \(\|u-U_{\mathrm{RBF}}\|_{2}\) | ratio | \(\|u-U_{\mathrm{RBF}}\|_{\infty}\) | ratio |
---|---|---|---|---|---|
1/16 | 1.4646e+09 | 1.3794e–03 | – | 1.7503e–02 | – |
1/32 | 3.9473e+14 | 6.9637e–05 | 19.81 | 1.5305e–03 | 11.44 |
1/64 | 3.5329e+21 | 1.0098e–06 | 68.96 | 2.0507e–05 | 74.63 |
1/128 | 1.8454e+22 | 8.4855e–04 | 0.00 | 8.2442e–03 | 0.00 |
The errors for the RBF collocation method in the sense of pointwise
(x,y) | (0.25, 0.25) | (0.25, 0.5) | (0.5, 0.25) | (0.5, 0.5) | ||||
---|---|---|---|---|---|---|---|---|
h | ratio | ratio | ratio | ratio | ||||
1/16 | 2.1454e–04 | – | 4.2667e–05 | – | 3.3242e-04 | – | 9.1856e–05 | – |
1/32 | 1.0465e–05 | 20.50 | 2.0434e–06 | 20.88 | 1.5989e–05 | 20.79 | 4.6344e–06 | 19.82 |
1/64 | 4.7127e–08 | 222.06 | 8.0976e–08 | 25.23 | 6.8216e–08 | 234.39 | 7.1355e–008 | 64.95 |
1/128 | 4.7203e–05 | 0.00 | 5.1627e–04 | 0.00 | 1.9326e–04 | 0.00 | 1.2537e–05 | 0.01 |
5 Summary and conclusions
In this paper, we construct a high accuracy difference scheme for Poisson equation with two integral boundary conditions and prove that the scheme can reach the asymptotic optimal error estimate. Numerical results verify the correctness of theoretical analysis. In the future, we will work on designing some other high order difference schemes (e.g., fourth-order nonstandard compact finite difference [20], or sixth-order implicit finite difference [21]) for Poisson problem with other nonlocal boundary conditions. Besides, we will also try to apply some other analytic methods for error estimation, e.g., homotopy analysis transform method [22, 23], or Lie symmetry analysis method [24, 25].
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their useful suggestions and comments that improved the presentation of this paper.
Funding
This work is partially supported by NSFC Project (Grant Nos. 11571293, 11171281 and 61603322), the Key Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (Grant No. 2011FJ2011), Hunan Provincial Natural Science Foundation of China (Grant No. 2016JJ2129), Hunan Provincial Civil Military Integration Industrial Development Project “Adaptive Multilevel Solver and Its Application in ICF Numerical Simulation” and Open Foundation of Guangdong Provincial Engineering Technology Research Center for Data Science (Grant No. 2016KF03).
Authors’ contributions
All authors have participated in the sequence alignment and drafted the manuscript. They have approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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