AI 中文总结
针对二阶椭圆方程离散的线性系统求解难题，提出AlgMortar这一完全代数多尺度Mortar预条件器，利用细网格矩阵信息，经图划分等构建局部系统并耦合，实验证明其用于共轭梯度法时可扩展性好且竞争力强。
AI 中文摘要
二阶椭圆方程离散产生的大规模对称正定线性系统的求解具有挑战性，特别是在系数高度非均匀的应用中。在此背景下，多尺度方法因其良好的并行可扩展性被用于加速Krylov子空间方法。本文提出AlgMortar，它是多尺度Mortar混合有限元方法的完全代数实现。该方法仅使用从细网格系统矩阵提取的信息，通过图划分定义子域分解，构建模拟狄利克雷问题的局部线性系统，并通过代数界面条件耦合局部解。证明了在细网格矩阵对称正定且非对角元素非正的情况下该方法适定。数值实验表明，作为共轭梯度法的预条件器，该方法具有良好的可扩展性，在具有挑战性的非均匀、高对比度测试案例中与现有代数多重网格方法具有竞争力。
英文摘要
The solution of large-scale symmetric positive definite linear systems arising from discretizations of second-order elliptic equations is challenging, especially in applications with highly heterogeneous coefficients, such as flow in porous media, which can lead to severely ill-conditioned systems. In this context, multiscale methods have recently been used to accelerate Krylov subspace methods, owing to their favorable parallel scalability. In this work, we present AlgMortar, a fully algebraic realization of the Multiscale Mortar Mixed Finite Element Method (MMMFEM). The method uses only information extracted from the fine-grid system matrix, which facilitates its implementation in existing solvers. AlgMortar uses graph partitioning to define a domain decomposition directly from the matrix graph. On each subdomain, it builds local linear systems that mimic Dirichlet problems, and couples the resulting local solutions through an algebraic interface condition that recovers the weak flux-continuity mechanism of MMMFEM. We prove that the method is well posed when the fine-grid matrix is symmetric positive definite and has nonpositive off-diagonal entries, a structure commonly arising from discretizations of elliptic problems. Numerical experiments on fine-grid linear systems arising from finite-volume discretizations of Darcy flow problems show that, when used as a preconditioner for the conjugate gradient method, the proposed approach exhibits good scalability and is competitive with state-of-the-art algebraic multigrid methods for challenging heterogeneous, high-contrast test cases, including highly irregular corner-point grids.
