Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

Abstract

We consider a C0 interior penalty finite element approximation of a sixth-order Cahn-Hilliard type equation that models the dynamics of phase transitions in ternary oil-water-surfactant systems. The nonlinear sixth-order parabolic equation is expressed in a mixed form whereby a second-order (in space) parabolic equation and an algebraic fourth-order (in space) nonlinear equation are considered. The temporal discretization is chosen so that a discrete energy law can be established leading to unconditional energy stability. Additionally, we show that the numerical method is unconditionally uniquely solvable. We conclude with several numerical experiments demonstrating the unconditional stability and first-order accuracy of the proposed method.