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

Source abstract

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 h p hp -FEM. The DMP holds if a relative length of every element K K in the mesh is bounded by a value H rel ∗ ( p ) ∈ [ 0.9 , 1 ] H^*_\textrm {rel}(p)\in [0.9,1] , where p ≥ 1 p\ge 1 is the polynomial degree of the element K K . The values H rel ∗ ( p ) H^*_\textrm {rel}(p) are calculated for 1 ≤ p ≤ 100 1 \le p \le 100 .