The spectral element method is one of the major numerical methods in fluid mechanics. To apply this method, it is crucial to have an efficient iterative solver. The authors of this paper examine some existing algorithms, and introduce a new type of preconditioner.

A preconditioner is used in an iterative solver to reduce the condition number so that a higher convergence rate can be achieved. The condition number of a spectral element equation system is usually so high (due to its high-order polynomial bases) that, without a good preconditioner, an iterative solver might converge very slowly, or even diverge.

A Schwartz-based preconditioner is widely used in the spectral element method. This method takes advantage of the hierarchical structure in the spectral element method, and combines a local procedure (a direct method) and a global procedure so that local ill conditioning can be avoided. It is desirable to solve the local problem as fast as possible; the new preconditioner introduced in this paper employs the fast-diagonalization method to achieve this goal.

The new method is tested using a two-dimensional (2D) Poisson equation and a 2D Helmholtz equation. The convergence rate obtained using the new method is compared with those obtained using the finite element method and the spectral element method preconditioned by the finite element method. The authors’ tests show that the new method is quite promising, in general. The paper also contains a concise overview of the spectral element method, and the basics of iterative solvers and preconditioning methods, which is helpful for fast reference.

The paper lacks a theoretical analysis of the comparison (and thereby of the numerical results). This may raise questions regarding the performance of the new method in practice, since usually three-dimensional (3D) problems have higher condition numbers.