Discrete maximum principle for higher-order finite elements in 1D

We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the hp-FEM. The DMP holds if a relative length of every element K in the mesh is bounded by a value H∗ rel(p) ∈ [0.9, 1], where p ≥ 1 is the polynomial degree of the element K. The values H∗ rel(p) are calculated for 1 ≤ p ≤ 100.