  In this paper, a parallel overlapping domain decomposition preconditioner is
proposed to solve the linear system of equations arising from the extended
finite element discretization of elastic crack problems. The algorithm
partitions the computational mesh into two types of subdomains: the regular
subdomains and the crack tip subdomains based on the observation that the crack
tips have a significant impact on the convergence of the iterative method while
the impact of the crack lines is not that different from those of regular mesh
points. The tip subdomains consist of mesh points at crack tips and all
neighboring points where the branch enrichment functions are applied. The
regular subdomains consist of all other mesh points, including those on the
crack lines. To overcome the mismatch between the number of subdomains and the
number of processor cores, the proposed method is divided into two steps: solve
the crack tip problem and then the regular subdomain problem during each
iteration. The proposed method was used to develop a parallel XFEM package
which is able to test different types of iterative methods. To achieve good
parallel efficiency, additional methods were introduced to reduce communication
and to maintain the load balance between processors. Numerical experiments
indicate that the proposed method significantly reduces the number of
iterations and the total computation time compared to the classical methods. In
addition, the method scales up to 8192 processor cores with over 70\% parallel
efficiency to solve problems with more than $2\times10^8$ degrees of freedom.
