Higher-order discrete maximum principle for 1D diffusion–reaction problems

Sufficient conditions for the validity of the discrete maximum principle (DMP) for a 1D diffusion–reaction problem −u″+κ2u=f with homogeneous Dirichlet boundary conditions discretized by the higher-order finite element method are presented. It is proved that the DMP is satisfied if the lengths h of all elements are shorter then one-third of the length of the entire domain and if κ2h2 is small enough for all elements. In general, the bounds for κ2h2 depend on the polynomial degree of the elements, on h, and on the size of the domain. The obtained conditions are simple and easy to verify. A technical assumption (nonnegativity of certain rational functions) was verified by computer for polynomial degrees up to 10. The paper contains an analysis of the discrete Green's function which can be of independent interest.