Abstract
Finite element spaces are constructed that allow for different levels of refinement in different subdomains. In each subdomain the mesh is obtained by several steps of uniform refinement from an initial global coarse mesh. The approximation properties of the resulting discrete space are studied. Computationally feasible, bounded extension operators, from the interface into the subdomains, are constructed and used in the numerical experiments. These operators provide stable splitting of the composite (global) finite element space into local subdomain spaces (vanishing at the interior interfaces) and the “extended” interface finite element space. They also provide natural domain decomposition type preconditioners involving appropriate subdomain and interface preconditioners. Numerical experiments for 3-d elasticity illustrating the properties of the proposed discretization spaces and the algorithm for the solution of the respective linear system are also presented.
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References
Bramble, J.: Multigrid methods. Pitman Research Notes in Mathematics v. 294, Longman Scientific & Technical (1993).
J. H. Bramble, J. E. Pasciak and P. S. Vassilevski, “Computational scales of Sobolev norms with application to preconditioning”, Math. Comp. 69 (2000), 463–480.
V. Dobrev and P. S. Vassilevski, “Non-mortar finite elements for elliptic problems”, Proceedings of the Fourth Intern. Conference on Numerical Methods and Applications (NMA’98), “Recent Advances in Numerical Methods and Applications” (O. Iliev, M. Kaschiev, S. Margenov, Bl. Sendov and P. S. Vassilevski, eds.), World Scientific, Singapore, 1999, pp. 756–765.
G. Haase, U. Langer, A. Meyer, and S. V. Nepomnyaschikh, Hierarchical extension operators and local multigrid methods in domain decomposition preconditioners, East-West J. Numer. Math. 2(1994), 173–193.
S. V. Nepomnyaschikh, Optimal multilevel extension operators, Report SPC 95-3, Jan, 1995, Technische Universität Chemnitz-Zwickau, Germany.
P. S. Vassilevski and O. Axelsson, “A two-level stabilizing framework for interface domain decomposition preconditioners”, in: Proceedings of the Third International Conference O(h3), Sofia, Bulgaria, August 21-August 26, Sofia, Bulgaria, “Advances in Numerical Methods and Applications”, (I. T. Dimov, Bl. Sendov and P. S. Vassilevski, eds.), World Scientific, Singapore, New Jersey, ndon, Hong Kong, 1994, pp. 196–202.
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© 2001 Springer-Verlag Berlin Heidelberg
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Dobrev, V., Vassilevski, P. (2001). Local Refinement in Non-overlapping Domain Decomposition. In: Vulkov, L., Yalamov, P., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2000. Lecture Notes in Computer Science, vol 1988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45262-1_31
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DOI: https://doi.org/10.1007/3-540-45262-1_31
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