On the Monotonicity of $Q^3$ Spectral Element Method for Laplacian

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 or can be regarded as fourth order accurate finite difference 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 unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of $Q^k$ spectral element methods will be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral element methods can be unconditionally monotone in three dimensions.