<div>On rectangular meshes, the simplest spectral element method for elliptic equations is the classical Lagrangian <i>Q</i>^k finite element method with only (<i>k</i>+1)-point Gauss-Lobatto quadrature, which can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is (<i>k</i> + 2)-th order accurate for <i>k</i> ≥ 2, whereas <i>Q</i>^<i>k</i> spectral element method is usually considered as a (<i>k</i> + 1)-th order accurate scheme in <i>L²</i>-norm. This result can be extended to linear wave, parabolic and linear Schrödinger equations.</div><div><br></div><div><div>Additionally, the <i>Q^k</i> finite element method for elliptic problems can also be viewed as a finite difference scheme on all Gauss-Lobatto points if the variable coefficients are replaced by their piecewise <i>Q^k </i>Lagrange interpolants at the Gauss Lobatto points in each rectangular cell, which is also proven to be (<i>k</i> + 2)-th order accurate.</div></div><div><br></div><div><div>Moreover, the monotonicity and discrete maximum principle can be proven for the fourth order accurate Q2 scheme for solving a variable coefficient Poisson equation, which is the first monotone and high order accurate scheme for a variable coefficient elliptic operator.</div></div><div><br></div><div><div>Last but not the least, we proved that certain high order accurate compact finite difference methods for convection diffusion problems satisfy weak monotonicity. Then a simple limiter can be designed to enforce the bound-preserving property when solving convection diffusion equations without losing conservation and high order accuracy.</div><div><br></div></div>