Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems

On solving complex-symmetric eigenvalue problems arising in the design of axisymmetric VCSEL devices

“…This preconditioner was extremely successful in the solution of large scale real symmetric EVPs presented in [2], and in particular, was optimal in the sense that iteration counts did not depend on the mesh width. The unfavourable behaviour seems to be due to the so-called PML modes: the desired eigenvalues are surrounded by numerous non-physical ones corresponding to eigenfunctions that are not attenuated by the PML.…”

“…Clearly, this block structure is preserved when considering the matrix K = A − τ B. The main idea behind the two-level preconditioner [5], which has successfully been applied to real symmetric problems [2], consists in applying (several steps of) blocked fixed-point iterations to systems of the type Kz = f, which arise when preconditioning is applied during the execution of Krylov based iterative solvers. The matrix K is thereby split according to (19) and dealt with in block Jacobi or block symmetric-Gauss-Seidel fashion [27].…”

“…wheref p according to (2) denotes the coordinates of the meridian projection and the two-dimensional operators ∇ p , div p and curl p are derivatives in r and z only. Inserting the expressions (3) into (1) yields the time-harmonic Maxwell equation in cylindrical coordinates.…”

On large‐scale diagonalization techniques for the Anderson model of localization

“…Note that here the accuracy of the solution of (4.4) is uncritical until the approximate eigenpair converges [28]. This fact has been exploited in Jdbsym [4,32]. For an overview on Jacobi-Davidson methods for symmetric matrices see [33].…”

Block preconditioning for saddle point systems with indefinite (1, 1) block

“…Linear systems of the type (1) will be referred to as (generalized) "saddle point systems with indefinite (1, 1) block." Such linear systems arise in various areas of scientific computing, including the solution of eigenvalue problems in fluid mechanics [8,13] and electromagnetics [2] by shift-and-invert algorithms, and in certain time-harmonic wave propagation problems [12,15]. We emphasize that while numerous effective solution algorithms exist for the case of a positive definite or semidefinite (1, 1) block (corresponding to either β ≤ 0 or β > 0 but smaller than the real part of the eigenvalue of A of smallest magnitude), see [3,7,9], relatively little has been done for the case where the (1, 1) block is indefinite.…”