An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations (Communications in Computational Physics, 2020)
The paper aims to develop an effective preconditioner and conduct the convergenceanalysis of the corresponding preconditioned GMRES for the solution of discreteproblems originating from multi-group radiation diffusion equations. We firstlyinvestigate the performances of the most widely used preconditioners (ILU(k) andAMG) and their combinations ($B_{co}$ and $\widetilde{B}_{co}$), and provide drawbacks on their feasibilities.Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzingthe features suitable for AMG using more detailed measurements on multiscalenature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we presentan adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along withits convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds thebest robustness and efficiency. At last, we analyze the convergence of GMRES withcombined preconditioning which not only provides a persuasive support for our proposedalgorithms, but also updates the existing estimation theory on condition numbersof combined preconditioned systems.