The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz's condition for proving monotonicity.

翻译：离散拉普拉斯算子的单调性蕴含离散最大值原理，该性质通常不适用于高阶格式。$Q^2$谱元方法已被证明在均匀矩形网格上具有单调性。本文证明在特定网格约束下，拟均匀矩形网格上$Q^2$谱元方法的单调性。特别地，我们提出一种用于证明单调性的松弛Lorenz条件。