  The spectral element method constructed by the $Q^k$ ($k\geq 2$) continuous
finite element method with $(k+1)$-point Gauss-Lobatto quadrature on
rectangular meshes is a popular high order scheme for solving wave equations in
various applications. It can also be regarded as a finite difference scheme on
all Gauss-Lobatto points. We prove that this finite difference scheme is
$(k+2)$-order accurate in discrete 2-norm for smooth solutions. The same proof
can be extended to the spectral element method solving linear parabolic and
Schr\"odinger equations. The main result also applies to the spectral element
method on curvilinear meshes that can be smoothly mapped to rectangular meshes
on the unit square.
