Fast auxiliary space preconditioners for linear elasticity in mixed form
Fast auxiliary space preconditioners for linear elasticity in mixed form
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DOI：
10.1090/mcom/3285
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发表时间：
2016-04
期刊：
影响因子：
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通讯作者：
Long Chen;Jun-Jue Hu;Xuehai Huang
中科院分区：
文献类型：
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作者：
Long Chen;Jun-Jue Hu;Xuehai Huang
针对线弹性的Hu-Zhang混合有限元法，开发了基于最小残差法的分块对角预处理器和基于广义最小残差法的分块三角预处理器。它们基于网格相关规范中鞍点系统的新稳定性结果。应力的网格相关范数对应于易于求逆的质量矩阵，而位移则与 Schur 补码谱等效。然后设计了一种基于线弹性问题的$H^1$一致线性元的快速辅助空间预处理器来求解Schur补。对于对角线和三角形预处理器，证明了预处理系统的条件数受一个独立于关键 Lam\'e 常数和网格尺寸的常数限制。给出了数值例子来支持理论结果。作为副产品，提出并分析了一种新的稳定低阶混合有限元方法，得到了Hu-Zhang单元的超收敛结果。
A block diagonal preconditioner with the minimal residual method and a block triangular preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods of linear elasticity. They are based on a new stability result of the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while the displacement it is spectral equivalent to Schur complement. A fast auxiliary space preconditioner based on the $H^1$ conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lam\'e constant and the mesh-size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results of Hu-Zhang element are obtained.