# Need clarification on a piece of book excerpt about spectral element method!

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!

• In most iterative solvers (eg, Krylov subspace methods), you never need the actual matrix $A$ when solving $Ax=b$. Rather, you only need to be able to compute the action of that matrix on a vector. It is possible to write finite element methods in such a way that you compute the action of the stiffness matrix on a vector rather than assembling the full stiffness matrix and then calling the solver. When you do this, you no longer have to be able to store the system matrix in memory. Jul 17 '17 at 20:14