Discrete Maximum Principle for Poisson Equation with Mixed Boundary Conditions Solved by $hp$-FEM

Abstract

We present a proof of the discrete maximum principle (DMP) for the 1D Poisson equation $−u''=f$ equipped with mixed Dirichlet-Neumann boundary conditions. The problem is discretized using finite elements of arbitrary lengths and polynomial degrees ($hp$-FEM). We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.