On Symmetric Part PCG for Mixed Elliptic Problems

The CGM is studied for nonsymmetric elliptic problems with mixed boundary conditions. The mesh independence of the convergence is in focus when symmetric part preconditioning is applied to the FEM discretizations of the BVP. Computations in 2 dimensions are presented to illustrate and complete the theoretical results.

[1]  O. Axelsson.  Iterative solution methods , 1995 .

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

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  Gene H. Golub,et al.  A generalized conjugate gradient method for non-symmetric systems of linear equations , 2007, Milestones in Matrix Computation.

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

[6]  János Karátson,et al.  Mesh independent superlinear convergence estimates of the conjugate gradient method for some equivalent self-adjoint operators , 2005 .

[7]  W. Hackbusch.  Elliptic Differential Equations , 1992 .

[8]  O. Widlund.  A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations , 1978 .

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

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

[11]  H. V. D. Vorst,et al.  Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems☆ , 1981 .

[12]  O. Widlund,et al.  On finite element domain imbedding methods , 1990 .

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

[14]  Y. Saad,et al.  Conjugate gradient-like algorithms for solving nonsymmetric linear systems , 1985 .

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

[16]  O. Axelsson.  A generalized conjugate gradient, least square method , 1987 .

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

[18]  J. Karátson,et al.  Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space , 2003 .

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