"Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Cahn-Hilliard type equation modeling microemulsions" by Natasha Sharma from University of Texas at El Paso 

Link  to AMC seminar. Zoom link is provided to those on the email list. 

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.

This is joint work with Amanda Diegel (Mississippi State University, USA). It is funded by National Science Foundation DMS 2110774 and the HPC resources are provided by Texas Advanced Computing Center (TACC).

Bio:

Natasha S. Sharma obtained her Ph.D. in 2011 in the area of adaptive finite element methods for Helmholtz equations from the Department of Mathematics at the University of Houston. Between 2012-2014 she was a teaching postdoctoral fellow at the Interdisciplinary Center for Scientific Computing at Heidelberg University in Heidelberg, Germany with her teaching focused on finite element methods using the C++ finite element library deal.II. In 2014, she joined the Department of Mathematical Sciences at the University of Texas at El Paso (UTEP) as a tenure-track assistant professor and received her tenure in 2022. Her current research focuses on developing efficient and accurate algorithms to approximate solutions to diffuse interface models with applications to material science and microemulsions.