## DE Seminar: Andrei Draganescu

#### UMBC

**Title:**Relaxed sufficient conditions for strong discrete maximum principles in finite element solutions of linear and semilinear elliptic equations

**Speaker:**Andrei Draganescu

**Abstract:**The preservation of qualitative properties constitutes a central theme in the design of numerical methods for partial differential equations. Among those properties, maximum principles have captured the attention of many generations of numerical analysts, as they play an essential role in ensuring that solutions maintain their physical relevance. For example, it is not only desired, but sometimes critical that quantities representing concentrations lie in the interval [0,1], or that computed densities are positive, etc.

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.