# Accuracy and Monotonicity of Spectral Element Method on Structured Meshes

On rectangular meshes, the simplest spectral element method for elliptic equations is the classical Lagrangian

*Q*^{k}finite element method with only (*k*+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 (*k*+ 2)-th order accurate for*k*≥ 2, whereas*Q*^{k}spectral element method is usually considered as a (*k*+ 1)-th order accurate scheme in*L*-norm. This result can be extended to linear wave, parabolic and linear Schrödinger equations.^{2}Additionally, the

*Q*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^{k}*Q*Lagrange interpolants at the Gauss Lobatto points in each rectangular cell, which is also proven to be (^{k}*k*+ 2)-th order accurate.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.

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.