代数多重网格求解器

Algebraic Multigrid Solver

多重网格法是求解由偏微分方程边值问题所导出的代数方程组的快速方法.普通的(又称几何的)多重网格法存在某些缺陷,影响它的推广应用.代数多重网格法可以避免这些缺陷.本文提出一种代数多重网格求解器,它适用于以大型稀疏矩阵为系数矩阵的代数方程组的求解.文中介绍了这种求解器的设计与实现方法;给出了求解有限差方程及有限元方程的若干算例;讨论了参数ζ的选取.实例表明,不论系数矩阵是否对称,代数多重网格求解器都具有多重网格迭代的收敛特性.

The multigrid method is a fast algorithm for solving algebraic equations, which arise from the discretization of the partial differential equation boundary-value problem. The usual (geometric) multigrid has some defects which restrict its application. The algebraic multigrid method can avoid those defects. In this paper an algebraic multigrid solver is proposed, which can be used to compute the large algebraic equations with the sparse matrix. The method of the design and implementation of the solver is introduced. Some numerical examples for solving the finite element equations and the finite difference equations are given. The selection of the parameter ξ is discussed. For those examples with the symmetric coefficient matrix or the non-symmetric coefficient matrix the solver yields an iterative method, which exhibits a convergence behavior typical for multigrid process.