The computational kernel of scientific codes is often
a linear system solver. Krylov subspace methods
may lead to scalable linear system solvers when
provided with robust and efficient preconditioners.

We describe two parallel domain decomposition
preconditioners for linear systems of equations
arising from the piecewise Hermite bicubic
collocation discretization of second-order
elliptic PDEs defined on a rectangular domain
and with general boundary conditions.
The preconditioners use an original
three-level discretization scheme derived
from partitioning the domain into subdomains,
edges, and vertices.

We describe the parallel implementation and
evaluate the robustness and efficiency of the
preconditioners, using numerical experiments.

We outline the advantages and disadvantages of
selecting an OpenMP or MPI implementation, and
compare the performance of our approach with that
of a PETSc implementation.