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Volume 15, Issue 1-2
Preconditioning Techniques in Chebyshev Collocation Method for Elliptic Equations

Zhi-Wei Fang, Jie Shen & Hai-Wei Sun

Int. J. Numer. Anal. Mod., 15 (2018), pp. 277-287.

Published online: 2018-01

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  • Abstract

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and ill-conditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver for this preconditioner in multi-dimension. A modified preconditioner based on the approximate inverse technique is constructed in this paper. The computational cost of each iteration in solving the preconditioned system is $\mathcal{O}(\ell N_x N_y log N_x)$, where $N_x$, $N_y$ are the grid sizes in each direction and $\ell$ is a small integer. Numerical examples are given to demonstrate the efficiency of the proposed preconditioner.

  • Keywords

Chebyshe collocation method, elliptic equation, finite-difference preconditioner, approximate inverse.

  • AMS Subject Headings

35J25, 65F10, 65N22, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

fzw913@yeah.net (Zhi-Wei Fang)

shen7@purdue.edu (Jie Shen)

hsun@umac.mo (Hai-Wei Sun)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-277, author = {Fang , Zhi-WeiShen , Jie and Sun , Hai-Wei}, title = {Preconditioning Techniques in Chebyshev Collocation Method for Elliptic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {1-2}, pages = {277--287}, abstract = {

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and ill-conditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver for this preconditioner in multi-dimension. A modified preconditioner based on the approximate inverse technique is constructed in this paper. The computational cost of each iteration in solving the preconditioned system is $\mathcal{O}(\ell N_x N_y log N_x)$, where $N_x$, $N_y$ are the grid sizes in each direction and $\ell$ is a small integer. Numerical examples are given to demonstrate the efficiency of the proposed preconditioner.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10568.html} }
TY - JOUR T1 - Preconditioning Techniques in Chebyshev Collocation Method for Elliptic Equations AU - Fang , Zhi-Wei AU - Shen , Jie AU - Sun , Hai-Wei JO - International Journal of Numerical Analysis and Modeling VL - 1-2 SP - 277 EP - 287 PY - 2018 DA - 2018/01 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10568.html KW - Chebyshe collocation method, elliptic equation, finite-difference preconditioner, approximate inverse. AB -

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and ill-conditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver for this preconditioner in multi-dimension. A modified preconditioner based on the approximate inverse technique is constructed in this paper. The computational cost of each iteration in solving the preconditioned system is $\mathcal{O}(\ell N_x N_y log N_x)$, where $N_x$, $N_y$ are the grid sizes in each direction and $\ell$ is a small integer. Numerical examples are given to demonstrate the efficiency of the proposed preconditioner.

Fang , Zhi-WeiShen , Jie and Sun , Hai-Wei. (2018). Preconditioning Techniques in Chebyshev Collocation Method for Elliptic Equations. International Journal of Numerical Analysis and Modeling. 15 (1-2). 277-287. doi:
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