Year: 2018
Author: Zhi-Wei Fang, Jie Shen, Hai-Wei Sun
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 1-2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2018-IJNAM-10568
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 1-2 : pp. 277–287
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 11
Keywords: Chebyshe collocation method elliptic equation finite-difference preconditioner approximate inverse.