Three-level BDDC in two dimensions

Balancing Domain Decomposition by Constraints (BDDC) methods are non-overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems. They are similar to the balancing Neumann–Neumann algorithm. However, in BDDC methods, a small number of continuity constraints are enforced across the interface, and these constraints form a new coarse, global component. An important advantage of using such constraints is that the Schur complements that arise in the computation will all be strictly positive definite. The matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation. However, this problem 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 two-dimensional space, while still maintaining a good convergence rate. Estimates of the condition numbers are provided for the two three-level BDDC methods and numerical experiments are also discussed. Copyright 2006 John Wiley & Sons, Ltd.