一种变系数扩散问题有限体积格式的高效预条件子

AN EFFICIENT PRECONDITIONER FOR FINITE VOLUME APPROXIMATION OF DIFFUSION PROBLEMS WITH VARIABLE AND JUMP COEFFICIENTS

本文针对一类含变跳系数的扩散问题,在矩形网格下构造了一种节点型MACH类有限体积格式.将相应常跳系数辅助扩散问题离散格式的系数矩阵的逆作为其预条件子.利用该系数矩阵的特殊代数结构,通过降维处理技术和快速傅里叶变换等,为预条件子（该矩阵逆）的数学行为设计了一种低运算复杂度（O（Nln（N）））的直接法.数值实验验证了基于该预条件子的PCG算法的高效性和稳健性.

In this paper, we firstly present a vertex-centered MACH-like finite volume method for the diffusion problems with variable and jump coefficients on rectangular grids. Then, its corresponding auxiliary diffusion problem with constant and jump coefficients is introduced,and the inverse of the coefficient matrix of the auxiliary problem＇s discrete system is used as the preconditioner. By using the special algebraic structure of the inverse matrix and together with a dimensionality reduction technique and the fast fourier transform, a direct method with low computational complexity（O（Nln（N））） is designed to achieve the behavior of our preconditioner. Numerical results show that the PCG algorithm with this preconditioner is efficient and robust.