On the Monotonicity of $Q^2$ Spectral Element Method for Laplacian on Quasi-Uniform Rectangular Meshes (Communications in Computational Physics, 2024)
The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral elementmethod has been proven monotone on a uniform rectangular mesh. In this paper weprove 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’scondition for proving monotonicity.
Keywords: Inverse positivity; discrete maximum principle; high order accuracy; monotonicity; discrete Laplacian; quasi uniform meshes; spectral element method.