Abstract:A block FETI–DP/BDDC preconditioner for mixed formulations of almost incompressible elasticity is constructed and analyzed; FETI–DP (Finite Element Tearing and Interconnecting Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. The saddle point problems of the mixed problems are discretized with mixed isogeometric analysis with continuous pressure fields. As in previous work by Tu and Li (2015) for finite element discretizations of the incompressible Stokes system, the proposed preconditioner is applied to a reduced positive definite system involving only the pressure interface variable and the Lagrange multiplier of the FETI–DP algorithm. In this work, we extend the theory to a wider class of saddle point problems and we propose a novel block-preconditioning strategy, which consists in using BDDC with deluxe scaling for the interface pressure block as well as deluxe scaling for the FETI–DP preconditioner for the Lagrange multiplier block. A convergence rate analysis is presented with a condition number bound for the preconditioned operator which depends on the inf-sup parameter of the fully assembled problem and the condition number of a closely related BDDC algorithm for compressible elasticity. This bound is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and element sizes, and robust with respect to material incompressibility. Parallel numerical experiments validate the theory, demonstrate robustness in the presence of discontinuities of the Lamé parameters, and indicate how the rate of convergence varies with respect to the spline polynomial degree and regularity and the deformation of the domain.

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中文翻译：

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块FETI–DP / BDDC预处理器，用于三维几乎不可压缩的弹性混合等距离散化

摘要：构建并分析了一种用于几乎不可压缩的弹性混合配方的块状FETI-DP / BDDC预处理剂；FETI-DP（有限元撕裂和互连双基元）和BDDC（受约束的平衡域分解）是解决各种椭圆问题的两种非常成功的域分解算法。混合问题的鞍点问题通过具有连续压力场的混合等几何分析离散化。正如Tu和Li（2015）先前关于不可压缩Stokes系统的有限元离散化的工作一样，拟议的预处理器被应用到只涉及压力接口变量和FETI-DP算法的Lagrange乘数的简化的正定系统。在这项工作中，我们将理论扩展到更广泛的鞍点问题类别，并提出了一种新颖的块预处理策略，该策略包括对接口压力块使用具有豪华比例缩放功能的BDDC以及针对Lagrange乘数的FETI-DP预处理器使用豪华比例缩放功能堵塞。提出了一个收敛速率分析，该条件的约束条件是预条件算子的条件数，该条件数取决于完全组装问题的inf-sup参数以及紧密相关的BDDC算法的可压缩弹性的条件数。该范围在子域的数量上是可扩展的，在子域和元素大小的比率上是多对数的，并且在材料不可压缩性方面是可靠的。并行数值实验验证了该理论，证明了在Lamé参数不连续的情况下的鲁棒性，

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