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

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.