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.

翻译：离散拉普拉斯算子的单调性（即刚度矩阵的逆正性）蕴含离散极大值原理，而高阶精确格式在非结构网格上通常不满足该性质。然而，在结构网格上构造高阶精确单调格式是可行的。已有已知的高阶精确逆正格式均为四阶精确格式，这些格式要么是M-矩阵，要么是两个M-矩阵的乘积。对于二维拉普拉斯算子的$Q^3$谱元法，我们证明其刚度矩阵是四个M-矩阵的乘积，因此具有单调性。该格式可视为五阶精确有限差分格式。