Abstract
The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem that are presented demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relaxation on the finest level.
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References
Brandt, A. (1984). Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics. Monograph, 191 pages. Available as GMD Studien 85 from GMD-AIW, Postfach 1240, D-5205, St. Augustin 1, West Germany.
Brandt, A., and Ta'asan, S. (1985). Multigrid method for nearly singular and slightly indefinite problems. InMultigrid Methods II, W. Hackbusch and U. Trottenber (eds.), Lecture Notes in Mathematics 1228, Springer-Verlag, New York, pp. 99–121.
Brandt, A., McCormick, S., and Ruge, J. (1983). Multigrid algorithms for differential eigenproblems.SIAM J. Sci. Stat. Components 4, 244–260.
Drazin, P. G., and Reid, W. H. (1981).The Theory of Hydrodynamic Stability, Cambridge University Press, New York.
Goldhirsch, I., Orszag, S. A., Maulik, B. K. (1987). An Efficient Method for Computing Leading Eigenvalues and Eigenvectors of Large asymmetric Matrices,J. of Sci. Comp. 2(1), 33–58.
Stanley, H. E. (1971).Introduction to Phase-Transition and Critical Phenomena, Oxford University Press, New York.
Wilkinson, J. H. (1965).The Algebraic Eigenvalue Problem, Oxford University Press, Clarendon.
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Ta'asan, S. Multigrid method for stability problems. J Sci Comput 3, 261–274 (1988). https://doi.org/10.1007/BF01061286
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DOI: https://doi.org/10.1007/BF01061286