Energy stable interior penalty discontinuous Galerkin finite element method for Cahn-Hilliard equation

TL;DR

This paper introduces an energy stable, conservative interior penalty discontinuous Galerkin method combined with an AVF time integrator for the Cahn-Hilliard equation, ensuring energy stability and accurate numerical solutions.

Contribution

It develops a novel energy stable discretization scheme for the Cahn-Hilliard equation using SIPG and AVF methods, ensuring energy decay and conservation.

Findings

Numerical results confirm theoretical convergence rates.

The method preserves the energy decreasing property.

Performance is validated through numerical experiments.

Abstract

An energy stable conservative method is developed for the Cahn--Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the CH equation. Numerical results confirm the theoretical convergence rates and the performance of the proposed approach.