I am reading "Using MPI (3rd edition)" from William Gropp, where in chapter 4 application section 4.13, it introduces an MPI application Nek5000/NekCEM which is based on spectral element method (SEM) coupled with high-order timestepping to solve the governing IVM/BVM problems.
There is one statement about SEM that I do not quite understand (Yes, it's not relevent to MPI):
"A central idea in the SEM is never to form local elemental matrices, which would in general have $(N+1)^6$ nonzeros per element, but to instead use preconditioned iterative solvers that requrie only the action of operators applied to functions."
I have some limited backgroud in FEM and SEM, and I know that SEM elemental matrices are usually much larger than FEM due to high polynomial orders, but I still can't get what the author is trying to conveying with this statement.
Could someone please explain it with more concrete descriptions? Any help would be greatly appreciated!