On iterative methods based on Sherman-Morrison-Woodbury splitting

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 and especially for systems involving matrices that are perturbations of circulant or block circulant matrices. Such matrices typically 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, which are limiting factors for most classical iterative methods. 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. Remarkably, the new method was tested against very large matrices demonstrating extremely fast convergence. For these reasons it can be a valuable tool for the solution of various finite elements and finite differences discretizations of differential equations.