We present a general framework for the numerical resolution of processes driven by geometric components such as the curvature of the domain boundary. In this context, the Laplace-Beltrami operator plays a crucial role. Two applications are discussed: Electrowetting on dielectric (EWOD) and the simulation of Biomembranes. The former refers to a parallel-plate micro-device that moves fluid droplets through electrically actuated surface tension effects. These devices have potential applications in biomedical `lab-on-a-chip' devices (such as automated DNA testing and cell separation) and controlled micro-fluidic transport (e.g. mixing and concentration control). We model the fluid dynamics using Hele-Shaw type equations (in 2-D) with a focus on including the relevant boundary phenomena. Specifically, we model contact line pinning as a static (Coulombic) friction effect that effectively becomes a variational inequality for the motion of the liquid-gas interface. We analyze this approach, present simulations and compare them to experimental videos of EWOD driven droplets. The latter applies to 2 mono-molecular forming an encapsulating bag called vesicle. Equilibrium shapes are obtained via the minimization of the Willmore energy under area and volume constraints. Physical dynamics are obtained by taking into account the effect of the inside (bulk) fluid. Forth order, highly nonlinear arising problems are solved using an adaptive mixed finite element method. Two and three dimensional simulations are presented. In particular, typical biconcave shape specific to red blood cells are obtained. This presentation is based on joint works with R. Nochetto, M. Pauletti, and S. Walker.