标题： 一种固有正则化方法用于几乎不可压缩线性渗透弹性及弹性问题的无参数预条件处理 显示英文标题

标题： An inherent regularization approach to parameter-free preconditioning for nearly incompressible linear poroelasticity and elasticity

摘要： 一种固有的正则化策略和块Schur补预条件方法被研究用于使用最低阶弱Galerkin有限元方法在空间上离散和隐式欧拉格式在时间上离散的线性渗透弹性问题。 在每个时间步，所得的鞍点系统在锁定区域中变得几乎奇异，其中固体几乎是不可压缩的。 这种近奇异性来源于对应的线性弹性系统的主块。 为了实现高效的迭代求解，这个几乎奇异的系统首先被重新表述为一个鞍点问题，然后通过向(2,2)块添加一项来进行正则化。 这种正则化保留了解，同时确保了新系统的非奇异。 结果，块Schur补预条件变得有效。 结果显示，预条件MINRES和GMRES的收敛性基本上与网格尺寸和锁定参数无关。 对于线性渗透弹性问题的迭代求解，考虑了两种和三种场的公式。 两种场公式的高效求解建立在对线性弹性问题的有效迭代求解之上。 对于这种情况，当使用块Schur补预条件时，MINRES和GMRES实现了无参数的收敛，其中主块的逆利用了线性弹性问题的高效求解器。 也可以通过在线性弹性部分引入一个数值压力变量，将渗透弹性问题重新表述为一个三场系统。 固有的正则化策略自然地扩展到这种形式，并且预条件MINRES和GMRES对于正则化系统也显示出无参数的收敛性。 二维和三维的数值实验验证了正则化策略的有效性和块预条件器的鲁棒性。 显示英文摘要

摘要： An inherent regularization strategy and block Schur complement preconditioning are studied for linear poroelasticity problems discretized using the lowest-order weak Galerkin FEM in space and the implicit Euler scheme in time. At each time step, the resulting saddle point system becomes nearly singular in the locking regime, where the solid is nearly incompressible. This near-singularity stems from the leading block, which corresponds to a linear elasticity system. To enable efficient iterative solution, this nearly singular system is first reformulated as a saddle point problem and then regularized by adding a term to the (2,2) block. This regularization preserves the solution while ensuring the non-singularity of the new system. As a result, block Schur complement preconditioning becomes effective. It is shown that the preconditioned MINRES and GMRES converge essentially independent of the mesh size and the locking parameter. Both two- and three-field formulations are considered for the iterative solution of the linear poroelasticity. The efficient solution of the two-field formulation builds upon the effective iterative solution of linear elasticity. For this case, MINRES and GMRES achieve parameter-free convergence when used with block Schur complement preconditioning, where the inverse of the leading block leverages efficient solvers for linear elasticity. The poroelasticity problem can also be reformulated as a three-field system by introducing a numerical pressure variable into the linear elasticity part. The inherent regularization strategy extends naturally to this formulation, and preconditioned MINRES and GMRES also show parameter-free convergence for the regularized system. Numerical experiments in both two and three dimensions confirm the effectiveness of the regularization strategy and the robustness of the block preconditioners.