Title: Three-Level BDDC in Three Dimensions (NYU-CS-TR862) Author: Xuemin Tu Abstract: BDDC methods are nonoverlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems. Its coarse problem is given by a small number of continuity constraints which are enforced across the interface. The coarse problem matrix is generated and factored by direct solvers at the beginning of the computation and it can ultimately become a bottleneck, if the number of subdomains is very large. In this paper, two three-level BDDC methods are introduced for solving the coarse problem approximately in three dimensions. This is an extension of previous work for the two dimensional case and since vertex constraints alone do not suffice to obtain polylogarithmic condition number bound, edge constraints are considered in this paper. Some new technical tools are then needed in the analysis and this makes the three dimensional case more complicated than the two dimensional case. Estimates of the condition numbers are provided for two three-level BDDC methods and numerical experiments are also discussed.