求解变分不等式的多重网格算法

多重网格法是求解椭圆型偏微分方程边值问题的一种快速、有效的数值方法.本文将多重网格算法应用于变分不等式问题的数值求解.将不动点法与多重网格过程相结合提出了求解变分不等式问题的一种多重网格算法.以障碍问题及其特例—弹、塑性杆的自由扭转问题为例,给出了求解所得的数值结果,讨论了这种算法的收敛性情况.实例表明,文中提出的算法保持了一般多重网格过程的主要特点.它具有远小于1的收敛比率;松弛因子的改变对收敛速率的影响很不灵敏;求解变分不等式问题的计算量接近或略小于相应的变分问题.

A Multigrid Algorithm for Solving Variational Inequalities

t The multigrid algorithm is applied to the numerical solution of the variational inequality problems. A multigrid scheme with a fixed point method for solving variational inequalities is proposed. As exampies the obstacle problems and elastic plastic free torsion problems are computed using the scheme and the computed results are given. The computing convergence is also discussed. These examples show that the multigrid scheme proposed in this paper conserves the main property of the general multigrid method. The convergence factor is considerably smaller than 1. The computing efficiency is essentially insensitive to the relaxation factor. The required number of computer steps of variational inequality problems are not larger than that of corresponding linear valiational problems.