Mesh independent superlinear convergence estimates of the conjugate gradient method for some equivalent self-adjoint operators

A mesh independent bound is given for the superlinear convergence of the CGM for preconditioned self-adjoint linear elliptic problems using suitable equivalent operators. The results rely on K-condition numbers and related estimates for compact Hilbert-Schmidt operators in Hilbert space.

[1]  Arno B. J. Kuijlaars,et al.  Superlinear Convergence of Conjugate Gradients , 2001, SIAM J. Numer. Anal..

[2]  Jan Kadlec,et al.  О регулярности решения задачи Пуассона на области с границей подобной границе куба , 1963 .

[3]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[4]  J. Daniel.  The Conjugate Gradient Method for Linear and Nonlinear Operator Equations , 1967 .

[5]  Owe Axelsson,et al.  On the sublinear and superlinear rate of convergence of conjugate gradient methods , 2000, Numerical Algorithms.

[6]  R. Bank.  Marching Algorithms for Elliptic Boundary Value Problems. II: The Variable Coefficient Case , 1977 .

[7]  J. Kadlec,et al.  On the regularity of the solution of the Poisson problem on a domain with boundary locally similar t , 1964 .

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

[9]  John William Neuberger,et al.  Sobolev gradients and differential equations , 1997 .

[10]  Frédéric Riesz,et al.  Vorlesungen über Funktionalanalysis , 1982 .

[11]  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 .

[12]  O. Axelsson,et al.  ON THE RATE OF CONVERGENCE OF THE CONJUGATE GRADIENT METHOD FOR LINEAR OPERATORS IN HILBERT SPACE , 2002 .

[13]  István Faragó,et al.  Numerical solution of nonlinear elliptic problems via preconditioning operators : theory and applications , 2002 .

[14]  Tuomo Rossi,et al.  A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension , 1999, SIAM J. Sci. Comput..

[15]  R. Winther.  Some Superlinear Convergence Results for the Conjugate Gradient Method , 1980 .

[16]  D. Rose,et al.  Marching Algorithms for Elliptic Boundary Value Problems. I: The Constant Coefficient Case , 1977 .

[17]  I︠u︡. V. Vorobʹev.  Method of moments in applied mathematics , 1965 .

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

[19]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[20]  Tuomo Rossi,et al.  Parallel fictitious domain method for a non-linear elliptic neumann boundary value problem , 1999 .

[21]  Thomas A. Manteuffel,et al.  On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations , 1990 .

[22]  M. A. Kaashoek,et al.  Classes of Linear Operators Vol. I , 1990 .

[23]  Owe Axelsson,et al.  Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators , 2004, Numerische Mathematik.

[24]  Thomas A. Manteuffel,et al.  Optimal equivalent preconditioners , 1993 .

[25]  Zenon Fortuna.  Some Convergence Properties of the Conjugate Gradient Method in Hilbert Space , 1979 .

[26]  István Faragó,et al.  Variable Preconditioning via Quasi-Newton Methods for Nonlinear Problems in Hilbert Space , 2003, SIAM J. Numer. Anal..