Convergence Analysis of the Standard Central Finite Difference Method for Poisson Equation

Gangjoon Yoon; Chohong Min

doi:10.1007/s10915-015-0096-2

Convergence Analysis of the Standard Central Finite Difference Method for Poisson Equation

Gangjoon Yoon, Chohong Min

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.

Original language	English
Pages (from-to)	602-617
Number of pages	16
Journal	Journal of Scientific Computing
Volume	67
Issue number	2
DOIs	https://doi.org/10.1007/s10915-015-0096-2
State	Published - 1 May 2016

Bibliographical note

Funding Information:
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827).

Publisher Copyright:
© 2015, Springer Science+Business Media New York.

Keywords

Central finite difference
Convergence analysis
Finite difference method
Poisson equation

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10.1007/s10915-015-0096-2

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abstract = "We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.",

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N2 - We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.

AB - We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.

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Convergence Analysis of the Standard Central Finite Difference Method for Poisson Equation

Abstract

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Keywords

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