Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems
Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems
复制标题
混合线性弹性和斯托克斯问题的标准粗空间混合和乘法重叠施瓦茨算法
DOI：
10.4208/cicp.020815.080316a
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
L. Pavarino
中科院分区：
文献类型：
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作者：
M. Cai;L. Pavarino
本文的目标是构造和研究混合和乘性的两层重叠施瓦茨算法，该算法具有标准的粗糙空间，适用于几乎不可压缩的线性弹性和Stokes系统，离散采用混合有限元和谱元方法，压力不连续。两种不同的方法被认为是解决所产生的鞍点系统：a）预处理共轭梯度（PCG）方法应用于对称正定的几乎不可压缩的线性弹性系统的重新制定，通过消除压力的未知数; B）GMRES方法与无限重叠施瓦茨预处理直接应用于鞍点制定的弹性和Stokes系统。对于几乎不可压缩弹性力学的正定重构问题，证明了所提出的混合和乘性重叠施瓦茨算法的条件数估计和收敛性.这些结果是基于我们以前的研究[8]，其中只考虑了几乎不可压缩弹性的加性施瓦茨预处理。大量的数值实验表明，建议的重叠施瓦茨预处理器是可扩展的，准最佳的未知数的数量在各个子域和强大的不连续性的材料参数在子域接口。结果表明，所提出的预条件保持了良好的性能时，准单调性假设，所要求的现有理论，不成立。
The goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.