Uzawa block relaxation method for free boundary problem with unilateral obstacle

ABSTRACT

A Uzawa block relaxation method, based on augmented Lagrangian functional and adaptive rule, is designed and analysed for free boundary problems with unilateral obstacle. We introduce an auxiliary unknown and augmented Lagrangian functional to transform the problem into a saddle-point problem, which can be solved by the Uzawa block relaxation method, and each iterative step consists of a linear problem while the auxiliary unknown is computed explicitly. The convergence speed of the method depends on the parameter heavily, and it is difficult to choose a proper parameter for individual problems. To improve the efficiency of the method, we propose an adaptive rule which adjusts the parameter automatically per iteration. Numerical examples show the performance of the proposed method for 1D and 2D free boundary problems.

KEYWORDS:

• Free boundary
• unilateral obstacle
• Uzawa method
• augmented Lagrangian
• adaptive rule

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J86
• 65K15

Acknowledgements

This work was funded by the Program of Chongqing Innovation Research Group Project in University (Grant No. CXQT19018), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-M201800501), the National Natural Science Foundation of China (Grant No. 11971081), the Fundamental and Frontier Research Project of Chongqing (Grant No. cstc2018jcyjAX0144) and the School Foundation Project of Chongqing University of Science $\mathrm{\&}$ Technology of China [grant number CK2016Z07].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was funded by the Program of Chongqing Innovation Research Group Project in University [grant number CXQT19018), the Science and Technology Research Program of Chongqing Municipal Education Commission [grant number KJZD-M201800501], the National Natural Science Foundation of China [grant number 11971081), the Fundamental and Frontier Research Project of Chongqing [grant number cstc2018jcyjAX0144] and the School Foundation Project of Chongqing University of Science & Technology of China [grant number CK2016Z07].