Numerical analysis of energy stable weak Galerkin schemes for the Cahn–Hilliard equation

Abstract

In this paper, we construct and analyze the energy-stable weak Galerkin schemes for the Cahn–Hilliard equation in the mixed form. The energy stability depends on the newly defined discrete energy functional. We propose two robust weak Galerkin schemes with extra stabilizing terms in time: the first-order conditionally energy stable scheme of which the stability does not depend on spatial mesh size, and the second-order unconditionally energy stable scheme. Energy stabilities and error estimates for the fully discrete schemes are obtained. Numerical experiments are provided to validate the theoretical results. A classical phase transition model is presented to show the energy stable property of the schemes.