Volume 23, Issue 2
Substructuring Preconditioners with a Simple Coarse Space for 2-D 3-T Radiation Diffusion Equations

Commun. Comput. Phys., 23 (2018), pp. 540-560.

Published online: 2018-02

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• Abstract

Inspired by [Q. Y. Hu, S. Shu and J. X. Wang, Math. Comput., 79 (272) (2010): 2059-2078], we firstly present two nonoverlapping domain decomposition (DD) preconditioners $B^a_h$ and $B^{sm}_h$ about the preserving-symmetry finite volume element (SFVE) scheme for solving two-dimensional three-temperature radiation diffusion equations with strongly discontinuous coefficients. It's worth mentioning that both $B^a_h$ and $B^{sm}_h$ involve a SFVE sub-system with respect to a simple coarse space and SFVE sub-systems which are self-similar to the original SFVE system but embarrassingly parallel. Next, the nearly optimal estimation $\mathcal{O}$((1+log$\frac{d}{h}$)3) on condition numbers is proved for the resulting preconditioned systems, where d and h respectively denote the maximum diameters in coarse and fine grids. Moreover, we present algebraic and parallel implementations of  $B^a_h$ and $B^{sm}_h$, develop parallel PCG solvers, and provide the numerical results validating the aforementioned theoretical estimations and stating the good algorithmic and parallel scalabilities.

• Keywords

2-D 3-T radiation diffusion equations, nonoverlapping domain decomposition, simple coarse space, condition number, parallel scalability.

65F10, 65F15, 65N55, 65Z05

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@Article{CiCP-23-540, author = {}, title = {Substructuring Preconditioners with a Simple Coarse Space for 2-D 3-T Radiation Diffusion Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {23}, number = {2}, pages = {540--560}, abstract = {

Inspired by [Q. Y. Hu, S. Shu and J. X. Wang, Math. Comput., 79 (272) (2010): 2059-2078], we firstly present two nonoverlapping domain decomposition (DD) preconditioners $B^a_h$ and $B^{sm}_h$ about the preserving-symmetry finite volume element (SFVE) scheme for solving two-dimensional three-temperature radiation diffusion equations with strongly discontinuous coefficients. It's worth mentioning that both $B^a_h$ and $B^{sm}_h$ involve a SFVE sub-system with respect to a simple coarse space and SFVE sub-systems which are self-similar to the original SFVE system but embarrassingly parallel. Next, the nearly optimal estimation $\mathcal{O}$((1+log$\frac{d}{h}$)3) on condition numbers is proved for the resulting preconditioned systems, where d and h respectively denote the maximum diameters in coarse and fine grids. Moreover, we present algebraic and parallel implementations of  $B^a_h$ and $B^{sm}_h$, develop parallel PCG solvers, and provide the numerical results validating the aforementioned theoretical estimations and stating the good algorithmic and parallel scalabilities.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0065}, url = {http://global-sci.org/intro/article_detail/cicp/10537.html} }
TY - JOUR T1 - Substructuring Preconditioners with a Simple Coarse Space for 2-D 3-T Radiation Diffusion Equations JO - Communications in Computational Physics VL - 2 SP - 540 EP - 560 PY - 2018 DA - 2018/02 SN - 23 DO - http://doi.org/10.4208/cicp.OA-2017-0065 UR - https://global-sci.org/intro/article_detail/cicp/10537.html KW - 2-D 3-T radiation diffusion equations, nonoverlapping domain decomposition, simple coarse space, condition number, parallel scalability. AB -

Inspired by [Q. Y. Hu, S. Shu and J. X. Wang, Math. Comput., 79 (272) (2010): 2059-2078], we firstly present two nonoverlapping domain decomposition (DD) preconditioners $B^a_h$ and $B^{sm}_h$ about the preserving-symmetry finite volume element (SFVE) scheme for solving two-dimensional three-temperature radiation diffusion equations with strongly discontinuous coefficients. It's worth mentioning that both $B^a_h$ and $B^{sm}_h$ involve a SFVE sub-system with respect to a simple coarse space and SFVE sub-systems which are self-similar to the original SFVE system but embarrassingly parallel. Next, the nearly optimal estimation $\mathcal{O}$((1+log$\frac{d}{h}$)3) on condition numbers is proved for the resulting preconditioned systems, where d and h respectively denote the maximum diameters in coarse and fine grids. Moreover, we present algebraic and parallel implementations of  $B^a_h$ and $B^{sm}_h$, develop parallel PCG solvers, and provide the numerical results validating the aforementioned theoretical estimations and stating the good algorithmic and parallel scalabilities.

Xiaoqiang Yue, Shi Shu, Junxian Wang & Zhiyang Zhou. (2020). Substructuring Preconditioners with a Simple Coarse Space for 2-D 3-T Radiation Diffusion Equations. Communications in Computational Physics. 23 (2). 540-560. doi:10.4208/cicp.OA-2017-0065
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