  Local Fourier analysis is a commonly used tool for the analysis of multigrid
and other multilevel algorithms, providing both insight into observed
convergence rates and predictive analysis of the performance of many
algorithms. In this paper, for the first time, we adapt local Fourier analysis
to examine variants of two- and three-level balancing domain decomposition by
constraints (BDDC) algorithms, to better understand the eigenvalue
distributions and condition number bounds on these preconditioned operators.
This adaptation is based on a modified choice of basis for the space of Fourier
harmonics that greatly simplifies the application of local Fourier analysis in
this setting. The local Fourier analysis is validated by considering the two
dimensional Laplacian and predicting the condition numbers of the
preconditioned operators with different sizes of subdomains. Several variants
are analyzed, showing the two- and three-level performance of the "lumped"
variant can be greatly improved when used in multiplicative combination with a
weighted diagonal scaling preconditioner, with weight optimized through the use
of LFA.

Numerical Analysis;Numerical Analysis