The spectral element method is essentially a high-order finite element technique. This method is suited to problems for which the solutions can be expressed as a series of polynomial basis functions. The method may yield an equation system with a very large condition number. Using preconditioners to improve the computational efficiency is a standard practice. Parallelization is a must for industry-level use.

This paper focuses on the implementation issues of parallelizing a preconditioned conjugate gradient (PCG) solver for the spectral element method. The preconditioners discussed here include the additive Schwarz method and the Jacobi method. The authors found that the additive Schwarz method is more attractive, especially when the mesh is fine. The efficiency of the parallel solver, including the memory usage and scalability, is also discussed. The most valuable part of this paper is the detailed performance evaluation of the solver based on the numerical experiments.

This paper can be quite useful for researchers interested in applications. It is not recommended for those who are looking for breakthroughs in the field. The paper could have been made clearer and more concise if some of the implementation details, which have been extensively discussed by other researchers, had been included in an appendix, rather than in the body of the text. The authors should also have included some hp method papers in the reference list. The procedure of implementing a parallel solver for the spectral method is very similar to that in the hp method.