基于动界面问题的大规模结构系统的预条件算法研究

The moving interface problems have very wide applications and attract a lot of attention from numerical analysts over the years. Several analytic and numerical approaches have been developed in order to provide the solutions of various moving interface problems, such as the front tracking method, the level set method, the immersed boundary method, the immersed interface method and its augmented method. In simulating the moving interface problems by the above methods, the associated discrete systems are always obtained, which are sparse large-scale, and may be very ill-conditioned due to the presence of the interface and the discontinuity or non-smoothness in the solution. The project focuses on the preconditioned iterative algorithms for large sparse discrete systems from the interface problems. Some adaptive preconditioner will be proposed to solve some special cases, where the sub-blocks of the coefficient matrices are not explicitly formed. we will also pay attention to solve rectangular systems from the interface problems. The analysis of the stability and sensitivity will be considered too. It also allows to obtain the new schemes of constructing the Krylov subspace and derive the associated new theoretical results according to the characteristic of the discrete systems.

动界面问题有着十分广泛的应用背景，因而吸引了许多的数值工作者的注意。几种数值方法被用于模拟动界面问题，比如，界面跟踪法，水平集方法，浸入边界法，浸入界面法及其增广方法。所有这些方法在模拟界面问题时都会导致大而稀疏的离散系统的求解。由于界面的不规则及解的非连续性和不光滑性，离散系统可能是十分病态的。通常的迭代方法可能收敛极慢或者无法收敛，这反过来将影响整个数值模拟的效果。本项目的重点就是寻求这类源于动界面的大稀疏结构系统的高效预处理求解方法。针对各子块非显性已知的离散系统，自适应的预条件子将被设置去加速相应Krylov子空间方法的收敛性能。长方型大稀疏离散系统的求解也是我们的重点。我们将分析相应算法的稳定性和敏感性，并依据离散系统的结构特点，去修正相应的Krylov子空间算法，给出新的理论结果。

本项目着眼于研究源于动界面问题的结构线性系统的快速算法， 所获得的研究结果包含三方面内容：（1）给出了求解对偶问题的最小二乘增广侵入界面方法；（2）构造了新的分裂迭代算法和预处理算法去求解大型稀疏含绝对值的结构系统；（3）给出了结构矩阵特征值和特征空间的扰动分析。..在第一方面，通过增加网格细化分析，设计了新的离散格式以更好地逼近界面，使得所提出的最小二乘增广浸入界面法具有全局二阶收敛精度；..在第二方面，针对系数矩阵为H+矩阵和正定矩阵的情形，着眼于新的思路设计了松弛方法和预条件子，给出了拟最优参数的选取策略，并考虑了具有高并行效率的多分裂该算法；..在第三方面，研究了结构矩阵的特征值和特征空间的加法和乘法扰动界的问题，细化了界的选取，给出了更精确的统一的组合扰动界，并首次给出了组合扰动界中参数选取的策略，改善了现有的结果。