An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition

Yesom Park; Jeongho Kim; Jinwook Jung; Euntaek Lee; Chohong Min

doi:10.1016/j.jcp.2017.11.028

An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition

Yesom Park, Jeongho Kim, Jinwook Jung, Euntaek Lee, Chohong Min

Department of Mathematics

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

2 Scopus citations

Abstract

MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O(h⁻²), which can be obtained from other ILU-type preconditioners, to O(h⁻¹). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O(h⁻¹) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O(h⁻¹) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU.

Original language	English
Pages (from-to)	115-126
Number of pages	12
Journal	Journal of Computational Physics
Volume	356
DOIs	https://doi.org/10.1016/j.jcp.2017.11.028
State	Published - 1 Mar 2018

Bibliographical note

Funding Information:
The research of C. Min and Y. Park was supported by Basic Science Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education ( 2017-006688 ) and the research of J. Kim, J. Jung and E. Lee was supported by the Samsung Science and Technology Foundation under Project (Number SSTF-BA1301-03 ).

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

Finite volume method
MILU preconditioning
Neumann boundary condition
Poisson equation
Purvis–Burkhalter method

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10.1016/j.jcp.2017.11.028

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title = "An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition",

abstract = "MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O(h−2), which can be obtained from other ILU-type preconditioners, to O(h−1). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O(h−1) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O(h−1) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU.",

keywords = "Finite volume method, MILU preconditioning, Neumann boundary condition, Poisson equation, Purvis–Burkhalter method",

author = "Yesom Park and Jeongho Kim and Jinwook Jung and Euntaek Lee and Chohong Min",

note = "Funding Information: The research of C. Min and Y. Park was supported by Basic Science Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education ( 2017-006688 ) and the research of J. Kim, J. Jung and E. Lee was supported by the Samsung Science and Technology Foundation under Project (Number SSTF-BA1301-03 ). Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

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AU - Jung, Jinwook

AU - Lee, Euntaek

AU - Min, Chohong

N1 - Funding Information: The research of C. Min and Y. Park was supported by Basic Science Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education ( 2017-006688 ) and the research of J. Kim, J. Jung and E. Lee was supported by the Samsung Science and Technology Foundation under Project (Number SSTF-BA1301-03 ). Publisher Copyright: © 2017 Elsevier Inc.

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N2 - MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O(h−2), which can be obtained from other ILU-type preconditioners, to O(h−1). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O(h−1) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O(h−1) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU.

AB - MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O(h−2), which can be obtained from other ILU-type preconditioners, to O(h−1). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O(h−1) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O(h−1) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU.

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An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition

Abstract

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Keywords

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