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

We present a proof of the discrete maximum principle (DMP) for the1D Poisson equation $−u''=f$ equipped with mixed Dirichlet-Neumann boundaryconditions. The problem is discretized using finite elements of arbitrary lengthsand polynomial degrees ($hp$-FEM). We show that the DMP holds on all mesheswith no limitations to the sizes and polynomial degrees of the elements.