  High-order methods gain more and more attention in computational fluid
dynamics. However, the potential advantage of these methods depends critically
on the availability of efficient elliptic solvers. With spectral-element
methods, static condensation is a common approach to reduce the number of
degree of freedoms and to improve the condition of the algebraic equations. The
resulting system is block-structured and the face-based operator well suited
for matrix-matrix multiplications. However, a straight-forward implementation
scales super-linearly with the number of unknowns and, therefore, prohibits the
application to high polynomial degrees. This paper proposes a novel
factorization technique, which yields a linear operation count of just 13N
multiplications, where N is the total number of unknowns. In comparison to
previous work it saves a factor larger than 3 and clearly outpaces unfactored
variants for all polynomial degrees. Using the new technique as a building
block for a preconditioned conjugate gradient method resulted in a runtime
scaling linearly with N for polynomial degrees $2 \leq p \leq 32$ . Moreover
the solver proved remarkably robust for aspect ratios up to 128.
