AI 中文总结
研究有限元质量矩阵求解及近似逆问题，提出破碎空间加法 Schwarz（BRAS）质量逆近似方法，证明其对预处理质量矩阵的谱界，通过数值实验表明该方法能降低谱条件数等，在多种有限元及系统上有加速效果。
AI 中文摘要
有限元质量矩阵求解和近似逆在显式时间积分以及基于舒尔补的块预处理中经常出现。质量矩阵的对角近似虽成本低且应用广泛，但可能是较差的近似，尤其对于高阶单元。本文引入一种破碎空间加法 Schwarz（BRAS）质量逆近似，通过在破碎有限元空间应用精确的单元局部逆质量矩阵并将结果平均回到协调空间形成。其构造使用与标准质量组装相同的单元矩阵和局部到全局映射，具有与协调质量矩阵相同的单元邻接稀疏图，对任何协调空间、网格几何和多项式基都是对称正定的。我们证明了预处理质量矩阵的谱界，在二维和三维单纯形网格上对\(H^1\)、\(H(\text{curl})\)和\(H(\text{div})\)有限元进行的广泛数值实验表明，与对角预处理相比，BRAS 降低了谱条件数、Krylov 迭代次数和求解时间。对于预处理共轭梯度（CG），在有限元阶数\(p\in[1,4]\)的所有测试案例中，BRAS 比对角/雅可比预处理产生 1.1 - 4.7 倍的加速。此外，理论和数值实验表明，在块预处理情况下，BRAS 可以改善舒尔补近似并减少外部求解时间。在混合泊松和双调和系统上，BRAS 比基于标准对角的预处理方法在求解时间上产生 1.5 - 3 倍的加速。
英文摘要
Finite-element mass matrix solves and approximate inverses arise often in explicit time integration as well as Schur-complement-based block preconditioning. A diagonal approximation of the mass matrix is cheap and widely used, but can be a poor approximation, particularly for high-order elements. This paper introduces a broken-space additive Schwarz (BRAS) mass inverse approximation, formed by applying exact element-local inverse mass matrices on the broken finite-element space and averaging the result back to the conforming space. The construction uses the same element matrices and local-to-global maps as standard mass assembly, has the same element-adjacency sparsity graph as the conforming mass matrix, and is symmetric positive definite for any conforming space, mesh geometry, and polynomial basis. We prove spectral bounds for the preconditioned mass matrix, and wide-ranging numerical experiments for \(H^1\), \(H(\operatorname{curl})\), and \(H(\operatorname{div})\) finite elements on two- and three-dimensional simplicial meshes show that BRAS reduces spectral condition numbers, Krylov iterations, and solve times relative to diagonal preconditioning. For preconditioned conjugate gradient (CG), BRAS yields a 1.1--4.7$\times$ speedup over diagonal/Jacobi preconditioning across all cases tested over finite-element orders $p\in[1,4]$. Further, theory and numerical experiments show that in the block-preconditioning case, BRAS can improve Schur-complement approximations and reduce outer solve times. On mixed Poisson and biharmonic systems, BRAS yields a 1.5--3$\times$ speedup in time-to-solution over a standard diagonal-based preconditioning approach.
