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\title{Application of Multilevel BDDC Method to Elasticity Analysis}
\tocauthor{J. Sistek} \author{} \institute{}
{\large Jakub \v{S}\'{\i}stek}\\
Institute of Mathematics AS CR, Prague, Czech Republic\\
\\ \vspace{4mm}{\large Jan Mandel}\\
Department of Mathematical and Statistical Sciences, University of Colorado Denver, USA\\
\\ \vspace{4mm}{\large Bed\v{r}ich Soused\'{\i}k}\\
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, USA\\
The Balancing Domain Decomposition by Constraints (BDDC) method by Dohrmann \cite{Dohrmann-2003-PSC}
is one of the most advanced methods of iterative substructuring for the solution of large systems of linear algebraic
equations arising from discretization of boundary value problems.
However, in the case of many substructures, solving the coarse problem exactly becomes a bottleneck.
As was mentioned already in \cite{Dohrmann-2003-PSC},
for BDDC method, it is straightforward to substitute the exact solution of the coarse problem by another step of BDDC method
with subdomains playing the role of elements.
In this way, the algorithm of three-level BDDC method is obtained (studied e.g. in \cite{Tu-2007-TBT3D}).
If this approach is applied recursively, \emph{Multilevel BDDC} \cite{Mandel-2008-MMB} method is obtained.
Unlike for other methods, such extension is natural for BDDC, since the coarse problem has the same structure as the original problem.
Details of the three-level BDDC algorithm (as a special case of the Multilevel BDDC algorithm) can be found in \cite{Mandel-2008-MMB}.
We present a recently developed parallel implementation of the multilevel BDDC method.
The implementation has been tested on a large 3D problems of linear elasticity.
Results by the multilevel approach are compared to the standard (two-level) BDDC method.
Even these preliminary results suggest which drawbacks of the two-level implementation might be overcome by the extension to more levels.
{\sc C.R. Dohrmann}. {A preconditioner for substructuring based on constrained energy minimization }. SIAM J. Sci. Comput. 25, 1 (2003), 246--258.
{\sc J. Mandel and B. Soused{\'\i}k and C.R. Dohrmann}. {Multispace and multilevel {BDDC}}. Computing 83, 2-3 (2008), 55--85.
{\sc X. Tu}. {Three-level {BDDC} in three dimensions}. SIAM J. Sci. Comput. 29, 4 (2007), 1759--1780.