Balancing domain decomposition by constraints (BDDC) algorithms are 
non-overlapping domain decomposition methods for solutions of large sparse linear algebraic systems arising from the 
discretization of boundary value problems. They are suitable for parallel computation. The coarse problem matrix of 
BDDC algorithms is generated and factored by a direct solver at the beginning of the computation and it will be a 
bottleneck when the computer systems with a large number of processors are used. In this talk, an inexact coarse solver 
for BDDC algorithms is introduced and analyzed. This solver helps remove the bottleneck.  At the same time, a good 
convergence rate is maintained. We will also talk about the extensions of these inexact BDDC algorithms to saddle 
point problems and problems with mortar finite element discretization.