Multilevel preconditioning for partition of unity methods: some analytic concepts

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. Mathematics Subject Classification (2000) 46E35 · 65F35 · 65F10 · 65N30 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. W. Dahmen (B) Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany e-mail: [email protected] URL: http://www.igpm.rwth-aachen.de/∼dahmen/ S. Dekel GE Healthcare, 6 Hamasger St., Or-Yehuda 60408, Israel e-mail: [email protected] URL: http://shaidekel.tripod.com/ P. Petrushev Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA e-mail: [email protected] URL: http://www.math.sc.edu/∼pencho/