Abstract
This paper is concerned with the construction and analysis of multilevel Schwarz preconditioners for partition of unity methods applied to elliptic problems. We show under which conditions on a given multilevel partition of unity hierarchy (MPUM) one even obtains uniformly bounded condition numbers and how to realize such requirements. The main anlytical tools are certain norm equivalences based on two-level splits providing frames that are stable under taking subsets.
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Babuska, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach, Acta Numerica, Cambridge University Press, Cambridge, pp. 1–125 (2003)
Cohen A., Dahmen W. and DeVore R. (2002). Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math. 2: 203–245
Dahmen, W., Dekel, S., Petrushev, P.: Two-level-split decomposition of anisotropic Besov spaces, IGPM Report, RWTH Aachen, Jan. (2007)
Davydov O. and Petrushev P. (2003). Nonlinear approximation from differentiable piecewise polynomials. SIAM J. Math. Anal. 35: 708–758
Dryja, M., Widlund, O.: Towards a unified theory of domain decomposition for elliptic problems. In: Chan, T., Glowinski, R., Periaux, J., Widlund, O. (eds) 3rd International Symposium on Domain Decomp. Methods for PDEs, Houston (Texas), March 1989, SIAM (1990)
Dryja, M., Widlund, O.: Multilevel additive methods for elliptic finite element problems, In: Hackbusch, W. (ed) Parallel Algorithms for PDE, Proceedings 6th GAMM Seminar, Kiel 1990, Vieweg, Braunschweig (1991)
Griebel, M., Schweitzer, M.A.: Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Sciences and Engineering, Springer, Heidelberg, Vol. 26 (2002)
Griebel M. and Schweitzer M.A. (2002). A particle-partition of unity method—Part III: a multilevel solver. SIAM J. Sci. Comput. 24: 377–409
Griebel M. and Oswald P. (1995). Remarks on the abstract theory of additive and multiplicative Schwarz methods. Numer. Math. 70: 163–180
Griebel, M., Schweitzer, M.A.: Meshfree methods for partial differential equations II, Lecture Notes in Computational Sciences and Engineering, Springer, Heidelberg, Vol. 26 (2004)
Huerta, A., Belytschko, T., Fernández-Méndez, S., Rabczuk, T.: Meshfree methods. In: Stein, E., De Borst, R., Hughes, T.J.R (eds) Encyclopedia of Computational Mechanics, Vol. 1, Chap. 10, pp. 279–309. Wiley, New York (2004)
Karaivanov B. and Petrushev P. (2003). Nonlinear piecewise polynomial approximation beyond Besov spaces. Appl. Comput. Harmon. Anal. 15: 177–223
Triebel H. (2006). The Theory of Function Spaces III. Birkhäuser, Basel
Xu J. (1992). Iterative methods by space decomposition and subspace correction. SIAM Rev. 34: 581–613
Kyriazis, G., Park, K., Petrushev, P.: B-spaces and their characterization via anisotropic Franklin bases, approximation theory: a volume dedicated to Borislav Bojanov. In: Dimitrov, D.K., Nikolov, G., Uluchev, R. (eds) Marin Drinov, Academic, Sofia, pp. 145–162 (2004)
Oswald P. (1994). Multilevel Finite Element Approximation. Teubner Skripten zur Numerik, Teubner
Schweitzer, M.A.: A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations, Lecture Notes in Computational Sciences and Engineering, Springer, Heidelberg, Vol. 29 (2003)
Xu, J., Zikatanov, L.T.: On multigrid methods for generalized finite element methods. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations, Vol. 26 of Lecture notes in Computational Science and Engineering, Springer, Heidelberg, pp. 401–418 (2002)
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This work has been supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-202-00286 (BREAKING COMPLEXITY), by the Leibniz-Programme of the German Research Foundation (DFG), and by the SFB 401 funded by DFG.
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Dahmen, W., Dekel, S. & Petrushev, P. Multilevel preconditioning for partition of unity methods: some analytic concepts. Numer. Math. 107, 503–532 (2007). https://doi.org/10.1007/s00211-007-0089-7
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DOI: https://doi.org/10.1007/s00211-007-0089-7