Fourier Analysis of Iterative Methods for Elliptic pr

his paper presents a Fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems. As in the von Neumann stability analysis of hyperbolic and parabolic problems, the approach is easier to apply, reveals more details about convergence properties than about standard techniques, and can be applied in a systematic way to a wide class of numerical methods. Although the analysis is applicable only to periodic problems, the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions, such as the Dirichlet problem. In addition, they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the Dirichlet case

[1]  P. Swarztrauber.  THE METHODS OF CYCLIC REDUCTION, FOURIER ANALYSIS AND THE FACR ALGORITHM FOR THE DISCRETE SOLUTION OF POISSON'S EQUATION ON A RECTANGLE* , 1977 .

[2]  George E. Forsythe,et al.  Finite-Difference Methods for Partial Differential Equations , 1961 .

[3]  D. Young.  Iterative methods for solving partial difference equations of elliptic type , 1954 .

[4]  M. Schultz,et al.  Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations , 1986 .

[5]  H. Elman.  A stability analysis of incomplete LU factorizations , 1986 .

[6]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[7]  R. Kettler.  Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods , 1982 .

[8]  Y. Saad,et al.  Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems , 1984 .

[9]  C. J. Kuot,et al.  A LOCAL RELAXATION METHOD FOR SOLVING ELLIPTIC PDEs ON MESH-CONNECTED ARRAYS * , 2022 .

[10]  Todd F. Dupont,et al.  A Factorization Procedure for the Solution of Elliptic Difference Equations , 1968 .

[11]  R. Chandra.  Conjugate gradient methods for partial differential equations. , 1978 .

[12]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[13]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[14]  Antony Jameson,et al.  Multigrid algorithms for compressible flow calculations , 1986 .

[15]  Gene H. Golub,et al.  Some History of the Conjugate Gradient and Lanczos Algorithms: 1948-1976 , 1989, SIAM Rev..

[16]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[17]  L. Trefethen,et al.  Fourier Analysis of the SOR Iteration , 1988 .

[18]  O. Axelsson.  Solution of linear systems of equations: Iterative methods , 1977 .

[19]  R. P. Kendall,et al.  An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations , 1968 .

[20]  Jack J. Dongarra,et al.  Matrix Eigensystem Routines — EISPACK Guide Extension , 1977, Lecture Notes in Computer Science.

[21]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[22]  Wolfgang Hackbusch,et al.  Multigrid convergence for a singular perturbation problem , 1984 .

[23]  C.-C. Jay Kuo.  Parallel algorithms and architectures for solving elliptic partial differential equations , 1985 .

[24]  D. Young,et al.  Analysis of the SOR iteration for the 9-point Laplacian , 1988 .

[25]  Tony F. Chan,et al.  Alternating-Direction Incomplete Factorizations , 1983 .

[26]  G. Golub,et al.  Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. , 1972 .

[27]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[28]  John C. Strikwerda,et al.  Interior Regularity Estimates for Elliptic Systems of Difference Equations , 1983 .

[29]  Werner Liniger,et al.  On factored discretizations of the laplacian for the fast solution of Poisson's equation on general regions , 1984 .