On fast domain decomposition solving procedures for hp-discretizations of 3-d elliptic problems

A DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp-discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp-discretization, obtain a fast preconditioner-solver for faces by K-interpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently. The relative condition number, provided by the DD preconditioner under consideration, is $O((1+\log p)^{3.5})$ and its total arithmetic cost is $O((1+\log p)^{1.75}[(1+\log p)(1+\log(1+\log p))p^3 R+ p R^2])$, where $R$ is the number of finite elements. The term $p R^2$ is due to the solver for the wire basket subsystem. We outline, how the cost of this component may be reduced to $ O(p R)$. The presented DD algorithms are highly parallelizable