自由边界问题的自适应Uzawa块松弛算法  

A Self-Adaptive Uzawa Block Relaxation Algorithm for Free Boundary Problems

作　　者：郭楠馨 张守贵 GUO Nanxin;ZHANG Shougui(School of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,P.R.China)

机构地区：[1]重庆师范大学数学科学学院

出　　处：《应用数学和力学》2019年第6期682-693,共12页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(面上项目)(11471063);重庆市基础科学与前沿技术研究项目(cstc2017jcyjAX0316)~~

摘　　要：利用增广Lagrange乘子法和自适应法则,得到求解单侧障碍自由边界问题的自适应Uzawa块松弛法.单侧障碍自由边界问题离散为有限维线性互补问题,等价于一个用辅助变量和增广Lagrange函数表示的鞍点问题.采用Uzawa块松弛算法求解该问题得到一个两步迭代法,主要的子问题为一个线性问题,同时能显式求解辅助变量.由于Uzawa块松弛算法的收敛速度显著依赖于罚参数,而且对具体问题很难选择合适的罚参数.为提高算法的性能,提出了自适应法则,该方法自动调整每次迭代所需的罚参数.数值结果验证了该算法的理论分析.A self-adaptive Uzawa block relaxation algorithm,based on the augmented Lagrangian multiplier method and the self-adaptive rule,was designed and analyzed for free boundary problems with unilateral obstacle.The problem was discretized as a finite-dimensional linear complementary problem which is equivalent to a saddle-point one with an augmented Lagrangian function and an auxiliary unknown.With the Uzawa block relaxation method for the problem,a 2-step iterative method was got with a linear problem as a main subproblem while the auxiliary unknown was computed explicitly.The convergence speed of the method highly depends on the penalty parameter,and it is difficult to choose a proper parameter for an individual problem.To improve the performance of the method,a self-adaptive rule was proposed to adjust the parameter automatically per iteration.Numerical results confirm the theoretical analysis of the proposed method.

关 键 词：自由边界 互补问题 Uzawa块松弛算法 增广LAGRANGE函数 自适应法则 

分 类 号：O221.6[理学—运筹学与控制论]