On the monotonicity of $Q^2$ spectral element method for Laplacian on quasi-uniform rectangular meshes

TL;DR

This paper proves the monotonicity of the $Q^2$ spectral element method for the Laplacian on quasi-uniform rectangular meshes, extending previous results from uniform meshes and introducing a relaxed Lorenz's condition.

Contribution

It establishes the monotonicity of the $Q^2$ spectral element method on quasi-uniform meshes under new relaxed conditions, advancing understanding of high-order scheme properties.

Findings

Monotonicity holds under certain mesh constraints.

A relaxed Lorenz's condition is proposed.

Extends previous uniform mesh results.

Abstract

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.