Factorizing the factorization - a spectral-element solver for elliptic equations with linear operation count

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.