Iterative methods for cyclically reduced nonselfadjoint linear systems. II

Abstract

Abstract.We perform an analytic and experimental study of line iterative methods for solving linear systems arising from finite difference discretizations of non-self-adjoint elliptic partial differential equations on two-dimensional domains. The methods consist of performing one step of cyclic reduction, followed by solution of the resulting reduced system by line relaxation. We augment previous analyses of one-line methods, and we derive a new convergence analysis for two-line methods, showing that both classes of methods are highly effective for solving the convection-diffusion equation. In addition, we compare the experimental performance of several variants of these methods, and we show that the methods can be implemented efficiently on parallel architectures.

Protein Contacts and Ligand Binding in the Inward‐Facing Model of Human P‐Glycoprotein

“…As a result, when comparing the iterative solution procedure for the original system to that for the cyclically reduced system, the overall performance of solvers depends almost exclusively on the spectral structure of the reduced versus the unreduced operators, and not on the cost of a single iteration. The analysis and numerical experiments in 8–10 show that iterative solvers for the reduced system converge faster, and hence it pays off to perform one step of cyclic reduction in the 2D case.…”

“…In the early 1990s Elman and Golub offered a thorough analysis of the spectral properties and the convergence behavior of linear systems arising from a procedure of one step of cyclic reduction 8–10. Using red‐black ordering and eliminating all the unknowns corresponding to one of the two colors yields a linear system associated with a diamond‐shaped 9‐point computational molecule.…”

On the Application of Homotopy Perturbation Method for Solving Systems of Linear Equations

“…Also, the equidistant step-size h = 1/ n is used in the discretization and the natural lexicographic ordering is employed to the unknowns and the right hand side satisfies b ij = h 2 f ij ( x , y ). For details, we refer to [ 1 , 2 ] and the references therein. Convection-diffusion equations appear in many fields of science and engineering and there are some reliable methods for solving this class of problems ([ 1 , 2 ] and the references therein).…”

“…For details, we refer to [ 1 , 2 ] and the references therein. Convection-diffusion equations appear in many fields of science and engineering and there are some reliable methods for solving this class of problems ([ 1 , 2 ] and the references therein). Here, we use alternative approach to solve ( 2 ) based on homotopy perturbation method (HPM).…”