Title: Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Cahn-Hilliard type equation modeling microemulsions

Abstract: Microemulsions are thermodynamically stable, transparent, isotropic single-phase mixtures of two immiscible liquids stabilized primarily by surfactants. Recently, microemulsion systems have emerged as an effective tool in capturing the static properties of ternary oil-water-surfactant systems with widespread applications such as enhanced oil recovery processes, the development of environmentally-friendly solvents, consumer and commercial cleaning product formulations, and drug delivery systems. Despite its applications, a major challenge impeding the use of these equations has been, and continues to be, a lack of understanding of these complex systems. Microemulsions can be modeled by means of an initial-boundary value problem for a sixth-order Cahn-Hilliard equation. In this talk, we present a numerical scheme for approximating the solutions to these sixth-order equations. To numerically approximate this sixth-order evolutionary equation, we introduce the chemical potential as a dual variable and consider a Ciarlet-Raviart-type mixed formulation consisting of a linear second-order parabolic equation and a nonlinear fourth-order elliptic equation. Here, the spatial discretization relies on a continuous interior penalty Galerkin finite element method while for the temporal discretization, we propose a first-order accurate time-stepping scheme. Theoretical properties of convergence, unique solvability, and unconditional stability of the proposed scheme will be discussed and through extensive numerical experiments, we will demonstrate the performance of the proposed scheme.