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Abstract
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.
*This work has been supported by the Austrian Science Fund under the grand Fonds zur Förderung der wissenschaftlichen Forschung (FWF) - under the project SFB FOI3 Numerical and Symbolic Scientific Computing
*This work has been supported by the Austrian Science Fund under the grand Fonds zur Förderung der wissenschaftlichen Forschung (FWF) - under the project SFB FOI3 Numerical and Symbolic Scientific Computing
Notes
*This work has been supported by the Austrian Science Fund under the grand Fonds zur Förderung der wissenschaftlichen Forschung (FWF) - under the project SFB FOI3 Numerical and Symbolic Scientific Computing