自由边界问题的自适应Uzawa块松弛算法

郭楠馨; 张守贵

doi:10.21656/1000-0887.390347

自由边界问题的自适应Uzawa块松弛算法

doi: 10.21656/1000-0887.390347

重庆师范大学数学科学学院, 重庆 401331

基金项目: 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)

详细信息

作者简介:
郭楠馨(1993—)，女，硕士生（E-mail: guonx1419@163.com）;张守贵(1973—)，男，副教授，博士(通讯作者. E-mail: shgzhang@cqnu.edu.cn).

中图分类号: O221.6
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- 被引次数: 0
出版历程
- 收稿日期: 2018-12-10
- 修回日期: 2019-03-12
- 刊出日期: 2019-06-01

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

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R.China

Funds: The National Natural Science Foundation of China(General Program)(11471063)

摘要

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.
- 自由边界 /
- 互补问题 /
- Uzawa块松弛算法 /
- 增广Lagrange函数 /
- 自适应法则
Abstract: 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.
- free boundary /
- linear complementarity /
- Uzawa block relaxation algorithm /
- augmented Lagrangian function /
- selfadaptive rule

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参考文献(16)

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