Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the related constant coefficient Helmholtz equation with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions of the equation. The proposed Fourier method, combined with proper eigensolver, results in an efficient and clear approach for computing the Stekloff eigenvalues.Comment: 12 pages, 4 figure

MODIFICATION AND COMPENSATION STRATEGIES FOR THRESHOLD-BASED INCOMPLETE FACTORIZATIONS ∗

Abstract. Standard (single-level) incomplete factorization preconditioners are known to successfully accelerate Krylov subspace iterations for many linear systems. The classical Modified Incomplete LU (MILU) factorization approach improves the acceleration given by (standard) ILU approaches, by modifying the non-unit diagonal in the factorization to match the action of the system matrix on a given vector, typically the constant vector. Here, we examine the role of similar modifications within the dual-threshold ILUT algorithm. We introduce column and row variants of the modified ILUT algorithm and discuss optimal ways of modifying the columns or rows of the computed factors to improve their accuracy and stability. Modifications are considered for both the diagonal and offdiagonal entries of the factors, based on one or many vectors, chosen a priori or through an Arnoldi iteration. Numerical results are presented to support our findings. Key words. Incomplete factorization preconditioners, algebraic preconditioners, ILUT, modified IL