Algebraic multigrid methods based on element preconditioning

• Published in: International journal of computer mathematics Vol. 78; no. 4; pp. 575 - 598
• Main Authors: Haase, G, Langer, U., Reitzinger, S., Schöberl, J.
• Format: Journal Article
• Language: English
• Published: Gordon and Breach Science Publishers 01.01.2001
• Subjects: Algebraic Multigrid; Element Preconditioning Technique; Finite Element Equations; G1.3; G1.8; Preconditioned Conjugate Gradient Method
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160108805133

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the "region of robustness" between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy. For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem. The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.