Monotonicity of $Q^3$ spectral element method for discrete Laplacian

TL;DR

This paper proves that the $Q^3$ spectral element method for the 2D Laplacian has a monotone stiffness matrix, making it a high-order accurate scheme that satisfies the discrete maximum principle.

Contribution

It demonstrates that the $Q^3$ spectral element method's stiffness matrix is a product of four M-matrices, establishing its monotonicity and high-order accuracy.

Findings

Stiffness matrix of $Q^3$ spectral element method is a product of four M-matrices.

The scheme is monotone and satisfies the discrete maximum principle.

Achieves fifth order accuracy on structured meshes.

Abstract

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.