Abstract
Themultilevel adaptive iteration is an attempt to improve both the robustness and efficiency of iterative sparse system solvers. Unlike in most other iterative methods, the order of processing and sequence of operations is not determined a priori. The method consists of a relaxation scheme with an active set strategy and can be viewed as an efficient implementation of the Gauß-Southwell relaxation. With this strategy, computational work is focused on where it can efficiently improve the solution quality. To obtain full efficiency, the algorithm must be used on a multilevel structure. This algorithm is then closely related to multigrid or multilevel preconditioning algorithms, and can be shown to have asymptotically optimal convergence. In this paper the focus is on a variant that uses data structures with a locally uniform grid refinement. The resulting grid system consists of a collection of patches where each patch is a uniform rectangular grid and where adaptive refinement is accomplished by arranging the patches flexibly in space. This construction permits improved implementations that better exploit high performance computer designs. This will be demonstrated by numerical examples.
Similar content being viewed by others
References
M. J. Berger,Data structures for adaptive mesh refinement, in Adaptive Computational Methods for Partial Differential Equations, I. Babuska, J. Chandra, and J. E. Flaherty, eds., SIAM, Philadelphia, 1984.
A. Brandt,Multi-level adaptive solutions to boundary value problems, Math. Comp., 31 (1977), pp. 333–390.
C. C. Douglas,Caching in with multigrid algorithms: problems in two dimensions, Parallel Algorithms Appl., 9 (1996), pp. 195–204.
M. Griebel,Grid- and point-oriented multilevel algorithms, in Incomplete Decompositions (ILU)—Algorithms, Theory, and Applications, W. Hackbusch and G. Wittum, eds., Notes on Numerical Fluid Mechanics Vol. 41, Vieweg Verlag, Braunschweig, 1993, pp. 32–46. also as SFB Bericht, 342/14/92 A, Institut für Informatik, TU München, 1992.
W. Hackbusch,Multigrid Methods and Applications. Springer Verlag, Berlin, 1985.
H. Lötzbeyer.Parallele adaptive Mehrgitterverfahen: Ein objektorientierter Ansatz auf semistrukturierten Gittern, Dimplomarbeit, Institut für Informatik, TU München, 1996.
S. F. McCormick,Multilevel Adaptive Methods for Partial Differential Equations, Frontiers in Applied Mathematics, Vol. 6, SIAM, Philadelphia, PA, 1989.
P. Oswald,Multilevel Finite Element Approximation: Theory and Applications, Teubner Skripten zur Numerik. Teubner, Stuttgart, 1994.
U. Rüde,Mathematical and Computational Techniques for Multilevel Adaptive Methods, Frontiers in Applied Mathematics, Vol. 13, SIAM, Philadelphia, PA, 1993.
U. Rüde,Error estimators based on stable splittings, in Proceedings of the Seventh International Conference on Domain Decomposition in Science and Engineering Computing, D. Keyes and J. Xu, eds., Pennsylvania State University, Contemporary Mathematics, Vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 111–118.
R. V. Southwell,Stress-calculation in frameworks by the method of systematic relaxation of constraints, Parts I, II, Proc. Roy. Soc. (A), 151 (1935), pp. 56–95.
J. Xu,Iterative methods by space decomposition and subspace correction, SIAM Review, 34:4 (1992), pp. 581–613.
H. Yserentant,Old and new convergence proofs for multigrid methods, Acta Numerica, pp. 285–326, 1993.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lötzbeyer, H., Rüde, U. Patch-adaptive multilevel iteration. Bit Numer Math 37, 739–758 (1997). https://doi.org/10.1007/BF02510250
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02510250