In this work we focus on discrete maximum principles (DMP) for finite element solutions of linear and semilinear elliptic equations. While geometry-based, sufficient conditions for satisfying DMPs have been known since the early work of Ciarlet and Raviart, it is well understood that these conditions are not necessary. In fact, in practice, the DMP appears to be quite resilient to the violation of those classical mesh conditions, assuming the finite element spaces satisfy the standard approximation properties. Moreover, finding examples that provably violate the DMP as the mesh-size converges to zero can be quite challenging.

The main purpose of this work is to develop a novel class of relaxed sufficient conditions under which a strong version of the DMP (SDMP) holds. The majority of the classical arguments used to prove DMPs are based on the analysis of the stiffness matrix, and the common hypothesis is that its off-diagonal entries are nonpositive, a requirement which reduces to the classical geometric conditions on the mesh. In this talk we present a connectivity-based technique that is adapted from the continuous case, and which provides a different pathway to proving SDMPs. In short, we show that if the SDMP holds on a set of patches that cover the domain, then the connectivity of the mesh will extend the SDMP to the entire domain. We use this technique to give an alternative proof for SDMPs for monotone semilinear elliptic equations, and to prove SDMP for linear elliptic equations discretized using certain meshes that violated classical sufficient conditions.