## Isogeometric Schwarz preconditioners for the biharmonic problem

D. Cho, L. F. Pavarino, and S. Scacchi

### Abstract

A scalable overlapping Schwarz preconditioner for the biharmonic Dirichlet problem discretized by isogeometric analysis is constructed, and its convergence rate is analyzed. The proposed preconditioner is based on solving local biharmonic problems on overlapping subdomains that form a partition of the CAD domain of the problem and on solving an additional coarse biharmonic problem associated with the subdomain coarse mesh. An $h$-analysis of the preconditioner shows an optimal convergence rate bound that is scalable in the number of subdomains and is cubic in the ratio between subdomain and overlap sizes. Numerical results in 2D and 3D confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the isogeometric polynomial degree $p$ and regularity $k$.

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### Key words

domain decomposition methods, overlapping Schwarz, biharmonic problem, scalable preconditioners, isogeometric analysis, finite elements, B-splines, NURBS

### AMS subject classifications

65N55, 65N30, 65F10

### Links to the cited ETNA articles

 [17] Vol. 39 (2012), pp. 313-332 Susanne C. Brenner and Kening Wang: An Iterative Substructuring Algorithm for a ${C}^0$ Interior Penalty Method

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