# Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?

For numerical methods of the Stokes equations, with appropriate boundary:

$$-\nabla^{2} \vec{u}+\nabla p=\overrightarrow{0}$$

$$\nabla \cdot \vec{u}=0$$

one may use FDM (finite difference method) or FEM (finite element method) to discretize it, which can both be tranformed into a saddle points system as follows:

$$\left[\begin{array}{ll}{A} & {B^{T}} \\ {B} & {O}\end{array}\right]\left[\begin{array}{l}{\mathbf{u}} \\ {\mathbf{p}}\end{array}\right]=\left[\begin{array}{l}{\mathbf{f}} \\ {\mathbf{g}}\end{array}\right]$$

Where $$A$$ is vector laplacian, and $$B$$ is full row rank.

1, If we use FEM, then for the large saddle points system, we can use Krylov methods, such as MINRES with block-diagonal preconditioner M = blkdiag(A,S), where $$A$$ can be approximated by AMG, and the Schur complement S can be approximated by pressure mass matrix (because the pressure mass matrix has been proved that it is spectrally equivalent to Schur complement S), which will result an optimal iteration results, i.e., iteration number independent of mesh size. So in this case, I think there is no other better preconditioner than this block-diagonal preconditioner. Anyone know is there some other better choice?

2, If we use FDM method, then there will be no such a concept of mass matrix, so we can not use some preconditioner with the Schur complement form S, and in this case, only the matrix A can be approximated by AMG, so we cannot get an optimal iteration results (independent of mesh size).

My question is: In my opinion, for the stokes equations, the best method at the present(though I should know that there is no best) is FEM with block-diagonal preconditioner, right? But I still find many preconditioners such as HSS(Hermitian and skew-Hermitian splitting), and some variants: AHSS, PHSS, RHSS, and shift-splitting, generalized shift-splitting and so on in the literature. If one preconditioner (denoted by M1) performs poorer than the FEM with block-diagonal preconditioner, what sense does that M1 have? (because I have seen that in many publications, the new methods presented by the author does not perform better than the FEM with block-diagonal preconditioner, e.g., they use Cholesky to approximate stiffness matrix $$A$$ which is so expensive) I am really confused by the so many methods in the literature. Because I think those are meanless unless they can perform better than the FEM method with block-diagonal preconditioner.