Analysis of Mixed Interior Penalty Discontinuous Galerkin Methods for the Cahn-Hilliard Equation and the Hele-Shaw Flow

Numerical Analysis and Scientific Computing Preprint Seria C0 Interior Penalty Discontinuous Galerkin approximation of a sixth order Cahn-Hilliard equation modeling microemulsification processes

Microemulsions can be modeled by an initial-boundary value problem for a sixth order Cahn-Hilliard equation. Introducing the chemical potential as a dual variable, a Ciarlet-Raviart type mixed formulation yields a system consisting of a linear second order evolutionary equation and a nonlinear fourth order equation. The spatial discretization is done by a C0 Interior Penalty Discontinuous Galer...

A Posteriori Error Estimates for Finite Element Approximations of the Cahn-hilliard Equation and the Hele-shaw Flow

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut + ∆ ` ε∆u− ε−1f(u) ́ = 0. It is shown that the a posteriori error bounds depends on ε−1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm f...

An efficient fully-discrete local discontinuous Galerkin method for the Cahn-Hilliard-Hele-Shaw system

A Hybridized Crouziex-Raviart Nonconforming Finite Element and Discontinuous Galerkin Method for a Two-Phase Flow in the Porous Media

In this study, we present a numerical solution for the two-phase incompressible flow in the porous media under isothermal condition using a hybrid of the linear lower-order nonconforming finite element and the interior penalty discontinuous Galerkin (DG) method. This hybridization is developed for the first time in the two-phase modeling and considered as the main novelty of this research.The p...

A discontinuous Galerkin method for the Cahn-Hilliard equation

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite elemen...