What is separable self-adjoint boundary value problem and why direct matrix solver is primarily restricted to this kind of problem? I encountered this claim when I am reading a book on multigrid solvers [1]: "Direct methods, of which Gaussian elimination is the prototype, determine a
solution exactly (up to machine precision) in a finite number of arithmetic steps. For systems such as (1.5) that arise from a two-dimensional elliptic equation, very efficient direct methods have been developed. They are usually based on the fast Fourier transform or the method of cyclic reduction. When applied to problems on an n x n grid, these methods require O(n^2log(n)) arithmetic operations. Because they approach the minimum operation count of O(n^2) operations, these methods are nearly optimal. However, they are also rather specialized and restricted primarily to systems that arise from separable self-adjoint boundary value problems." As no reference is given to it, could you please shed the light on me what is the separable self-adjoint boundary value problem and why the direct matrix solver is primarily restricted to this kind of problem?
[1] Briggs, William L., Van Emden Henson, and Steve F. McCormick. A multigrid tutorial. Society for Industrial and Applied Mathematics, 2000.
 A: The "they" in your quote refers to methods "based on the fast
Fourier transform or the method of cyclic reduction," not to direct linear solvers (which are a very general low-level tool used to solve all manner of problems!).
In this context my understanding of "separable self-adjoint boundary value problem" is that for linear boundary value problems involving the Laplacian, it's possible to split the PDE into separate 1D equations each of which can be solved using FFT. The sentence you quoted suggests this same recipe works for general self-adjoint operators beyond $\Delta$; I'm not an expert on PDEs but I'm skeptical since most operators don't have eigenfunctions that are as nice to work with as the Fourier basis.
But again, a vast diversity of PDEs (including initial value problems, higher-dimensional PDEs that do not separate, etc.) can be solved using e.g. the finite element method, where at the end of the day the central computational step is solving a linear equation using a direct linear solver.
