Volume 42 Issue 2
Feb.  2021
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ZHANG Maolin, RAN Jing, ZHANG Shougui. A Self-Adaptive Uzawa Block Relaxation Method for Stokes Problems With Slip Boundary Conditions[J]. Applied Mathematics and Mechanics, 2021, 42(2): 188-198. doi: 10.21656/1000-0887.410170
Citation: ZHANG Maolin, RAN Jing, ZHANG Shougui. A Self-Adaptive Uzawa Block Relaxation Method for Stokes Problems With Slip Boundary Conditions[J]. Applied Mathematics and Mechanics, 2021, 42(2): 188-198. doi: 10.21656/1000-0887.410170

A Self-Adaptive Uzawa Block Relaxation Method for Stokes Problems With Slip Boundary Conditions

doi: 10.21656/1000-0887.410170
Funds:  The National Natural Science Foundation of China(11971085)
  • Received Date: 2020-06-11
  • Rev Recd Date: 2020-07-25
  • Publish Date: 2021-02-01
  • A self-adaptive Uzawa block relaxation method was designed for Stokes problems under nonlinear slip boundary conditions. For the variational formulation of the problem, an auxiliary unknown was introduced to transform the problem into a saddle-point one based on an augmented Lagrangian function, which can be solved with the Uzawa block relaxation method. To improve the performance of the method, a self-adaptive rule was proposed with the proper penalty parameter chosen automatically. The main advantage of this method is that each iterative step consists of a linear problem while the auxiliary unknown can be computed explicitly. The convergence of the algorithm was analyzed. The numerical results show the feasibility and effectiveness of the proposed method.
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  • [1]
    黄鹏展, 何银年, 冯新龙. 解Stokes特征值问题的一种两水平稳定化有限元方法[J]. 应用数学和力学, 2012,33(5): 588-597.(HUANG Pengzhan, HE Yinnian, FENG Xinlong. A two-level stabilized finite element method for the Stokes eigenvalue problem[J]. Applied Mathematics and Mechanics,2012,33(5): 588-597.(in Chinese))
    [2]
    SHI D Y, PEI L F. Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L2 projection method[J]. Applied Mathematics and Mechanics (English Edition),2013,34(7): 861-874.
    [3]
    FUJITA H. A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions[J]. RIMS Kkyroku,1994,888(1): 199-216.
    [4]
    SAITO N. On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions[J]. Publications of the Research Institute for Mathematical Sciences,2004,40(2): 345-383.
    [5]
    LI Y, LI K T. Uzawa iteration method for Stokes type variational inequality of the second kind[J]. Acta Mathematicae Applicatae Sinica(English Series),2011,27(2): 303-316.
    [6]
    KASHIWABARA T. On a finite element approximation of the Stokes problem under leak or slip boundary conditions of friction type[J]. Japan Journal of Industrial and Applied Mathematics,2013,30(1): 227-261.
    [7]
    JING F F, LI J, YAN W J. Discontinuous Galerkin methods for a stationary Navier-Stokes problem with a nonlinear slip boundary condition of friction type[J]. Applied Mathematics Letters,2017,73(2): 113-119.
    [8]
    周康瑞, 尚月强. 带非线性滑移边界条件的Stokes方程的一种并行有限元算法[J]. 西南师范大学学报(自然科学版), 2020,45(5): 32-38.(ZHOU Kangrui, SHANG Yueqiang. A parallel finite element algorithm for Stokes equations with nonlinear slip boundary conditions[J]. Journal of Southwest China Normal University (Natural Science Edition),2020,45(5): 32-38.(in Chinese))
    [9]
    饶玲. 单调迭代结合虚拟区域法求解非线性障碍问题[J]. 应用数学和力学, 2018,39(4): 485-492.(RAO Ling. Monotone iterations combined with fictitious domain methods for numerical solution of nonlinear obstacle problems[J]. Applied Mathematics and Mechanics,2018,39(4): 485-492.(in Chinese))
    [10]
    GLOWINSKI R. Numerical Methods for Nonlinear Variational Problems [M]. Berlin: Springer-Verlag, 2008.
    [11]
    KOKO J. Uzawa block relaxation method for the unilateral contact problem[J]. Journal of Computational and Applied Mathematics,2011,235(8): 2343-2356.
    [12]
    DJOKO J K, KOKO J. Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip boundary conditions[J]. Computer Methods in Applied Mechanics and Engineering,2016,305(15): 936-958.
    [13]
    王光辉, 王烈衡. 基于对偶混合变分形式的Uzawa型算法[J]. 应用数学和力学, 2002,23(7): 682-688.(WANG Guanghui, WANG Lieheng. Uzawa type algorithm based on dual mixed variational formulation[J]. Applied Mathematics and Mechanics,2002,23(7): 682-688.(in Chinese))
    [14]
    HE B S. Self-adaptive operator splitting methods for monotone variational inequalities[J]. Numerische Mathematik,2013,94(4): 715-737.
    [15]
    郭楠馨, 张守贵. 自由边界问题的自适应Uzawa块松弛算法[J]. 应用数学和力学, 2019,40(6): 682-693.(GUO Nanxin, ZHANG Shougui. A self-adaptive Uzawa block relaxation algorithm for free boundary problems[J]. Applied Mathematics and Mechanics,2019,40(6): 682-693. (in Chinese))
    [16]
    ZHANG S G. Projection and self-adaptive projection methods for the Signorini problem the BEM[J]. Applied Mathematics and Computation,2017,74(6): 1262-1273.
    [17]
    ZHANG S G, LI X L. A self-adaptive projection method for contact problems with the BEM[J]. Applied Mathematical Modelling,2018,55: 145-159.
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