Rational Approximations in Robust Preconditioning of Multiphysics Problems
Rational Approximations in Robust Preconditioning of Multiphysics Problems
复制标题
多物理问题鲁棒预处理中的有理逼近
作者：
S. Harizanov;I. Lirkov;S. Margenov
多物理或多尺度问题自然涉及低维流形界面处的耦合。相关鞍点系统的块对角预处理是数值解决此类大规模问题的最有效方法之一。在操作员级别，预处理器的接口块是分数拉普拉斯算子。在离散层面，我们建议用其最佳均匀有理近似（BURA）代替分数拉普拉斯算子的逆。本文的目标是开发一个统一的框架来分析新型预处理迭代方法。作为最终结果，我们证明所提出的预处理器具有最佳计算复杂度 O(N)，其中 N 是耦合离散问题的未知数（自由度）的数量。主要的理论贡献是基于 BURA 的预处理器的条件数估计。值得注意的是，对于正分数幂和负分数幂，所获得的估计值完全相似。最后，对相对条件数行为的分析旨在描述所考虑的达西-斯托克斯和 3D-1D 耦合问题示例的最小 BURA 阶数的实际要求。
Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity O(N), where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems.