自由边界问题的自适应Uzawa块松弛算法

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 for Free Boundary Problems

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.