We first briefly review some recently proven new results about \(Q^k\) spectral element method for second order linear PDEs, including its order of accuracy as a finite difference method in \(\ell ^2\) -norm and monotonicity, both of which are special properties of \(Q^k\) spectral element method on structured meshes. We discuss some extensions or applications of these two special properties, including the accuracy for the Helmholtz equation and applications of monotone discrete Laplacian to a semi-linear problem. In particular, the \(Q^2\) spectral element method gives a fourth order accurate monotone discrete Laplacian, with which one can obtain explicit convergence rates of Picard and Newton iterations for solving a special second order semilinear PDE.