In this talk, we employ a level-set method to define complex computational domains and propose a ghost nodal finite element strategy tailored for two distinct applications. In the first part, we introduce a model for a Poisson-Nernst-Planck system that accounts for the correlated motion of positive and negative ions through Coulomb interactions. For very short Debye lengths, one can adopt the so called Quasi-Neutral limit which drastically simplifies the system, reducing it to a diffusion equation for a single carriers with effective diffusion coefficient. This approach, while simplifying the mathematical model, can limit the scope of numerical simulations, as it may not capture the full range of behaviors near the Quasi-Neutral limit. Our goal is therefore to design an Asymptotic Preserving (AP)  to handle both regimes: the full system when the Debye length is small but non-negligible, and the Quasi-Neutral regime as the Debye length approaches zero. In the second part, we study the formation of biological transportation networks governed by a nonlinear elliptic equation for the pressure coupled with a reaction-diffusion parabolic equation for the conductivity tensor. We compute numerical solutions using the proposed ghost nodal finite element method, which shows that the network becomes highly intricate and its branches extend over large portions of the domain.

You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too.