We consider a new splitting based on the Sherman-Morrison-Woodbury formula, which is particularly effective with iterative methods for the numerical solution of large linear systems. These systems involve matrices that are perturbations of circulant or block circulant matrices, which commonly arise in the discretization of differential equations using finite element or finite difference methods. We prove the convergence of the new iteration without making any assumptions regarding the symmetry or diagonal-dominance of the matrix. To illustrate the efficacy of the new iteration we present various applications. These include extensions of the new iteration to block matrices that arise in certain saddle point problems as well as two-dimensional finite difference discretizations. The new method exhibits fast convergence in all of the test cases we used. It has minimal storage requirements, straightforward implementation and compatibility with nearly circulant matrices via the Fast Fourier Transform. For this reasons it can be a valuable tool for the solution of various finite element and finite difference discretizations of differential equations.

翻译：本文提出一种基于Sherman-Morrison-Woodbury公式的新型分裂方法，该方法与迭代法结合时，对求解大型线性系统的数值解尤为有效。这类系统涉及循环矩阵或块循环矩阵的扰动矩阵，这些矩阵常见于使用有限元或有限差分法离散微分方程的过程中。我们在不对矩阵的对称性或对角占优性做任何假设的前提下，证明了新迭代法的收敛性。为验证新迭代法的有效性，我们给出了多个应用实例，包括将该方法推广至特定鞍点问题及二维有限差分离散中出现的块矩阵。在所有测试案例中，新方法均展现出快速收敛性。该方法具有存储需求小、实现简单以及可通过快速傅里叶变换与近似循环矩阵兼容的优势，因此可成为求解多种微分方程有限元及有限差分离散问题的重要工具。