An efficient direct parallel elliptic solver based on the spectral
element discretization is developed. The direct solver is based on a
matrix decomposition approach which reduces multi-dimensional
separable problems to a sequence of one-dimensional problems that can
be efficiently handled by a static condensation process. Thanks to the
spectral accuracy and the localized nature of a spectral element
discretization, this elliptic solver is spectrally accurate and can be
efficiently parallelized, and it can serve as an essential building
block for large scale high-performance solvers in computational fluid
dynamics and computational materials science.