标题： 三场混合有限元离散化Biot固结模型的块状BDDC/FETI-DP预条件器 显示英文标题

标题： Block BDDC/FETI-DP Preconditioners for Three-Field mixed finite element Discretizations of Biot's consolidation model

摘要： 本文中，我们为基于位移、压力和总压形式的三场混合有限元方法近似的Biot固结模型构建并分析了一个块状对偶-原始预条件子。 将区域分解为非重叠子域，并通过引入拉格朗日乘子来强制执行位移分量在子域界面处的连续性。 消去所有的位移变量以及压力和总压的独立子域内部分量后，问题被简化为一个关于子域界面压力、总压和拉格朗日乘子的对称正定线性系统。 该降阶系统通过预条件共轭梯度法求解，使用了块状对偶-原始预条件子，其中包含平衡约束区域分解（BDDC）预条件子，用于界面总压块和界面压力块，以及有限元撕裂与互联-对偶-原始（FETI-DP）预条件子，用于拉格朗日乘子块。 通过对与界面压力和总压分量相关的预条件子子系统的条件数进行分析，我们得到了预条件系统的条件数界，该界在子域数量上是可扩展的，在子域尺寸与网格尺寸之比上是多项式对数增长的，并且对模型参数具有鲁棒性。 广泛的数值实验验证了所提出算法的理论结果。 显示英文摘要

摘要： In this paper, we construct and analyze a block dual-primal preconditioner for Biot's consolidation model approximated by three-field mixed finite elements based on a displacement, pressure, and total pressure formulation. The domain is decomposed into nonoverlapping subdomains, and the continuity of the displacement component across the subdomain interface is enforced by introducing a Lagrange multiplier. After eliminating all displacement variables and the independent subdomain interior components of pressure and total pressure, the problem is reduced to a symmetric positive definite linear system for the subdomain interface pressure, total pressure, and the Lagrange multiplier. This reduced system is solved by a preconditioned conjugate gradient method, with a block dual-primal preconditioner using a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for both the interface total pressure block and the interface pressure blocks, as well as a Finite Element Tearing and Interconnecting-Dual Primal (FETI-DP) preconditioner for the Lagrange multiplier block. By analyzing the conditioning of the preconditioned subsystem associated with the interface pressure and total pressure components, we obtain a condition number bound of the preconditioned system, which is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and mesh sizes, and robust with respect to the parameters of the model. Extensive numerical experiments confirm the theoretical result of the proposed algorithm.