  The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness
matrix, implies discrete maximum principle, which is in general not true for
high order accurate schemes on unstructured meshes. On the other hand, it is
possible to construct high order accurate monotone schemes on structured
meshes. All previously known high order accurate inverse positive schemes are
fourth order accurate schemes, which is either an M-matrix or a product of two
M-matrices. For the $Q^3$ spectral element method for the two-dimensional
Laplacian, we prove its stiffness matrix is a product of four M-matrices thus
it is monotone. Such a scheme can be regarded as a fifth order accurate finite
difference scheme.
