# Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn-Hilliard problem

## TL;DR

This paper presents a comprehensive numerical analysis of a hybridized discontinuous Galerkin method applied to the Cahn-Hilliard problem, establishing stability, convergence, and new discrete inequalities for the method.

## Contribution

It introduces a stable, convergent hybridizable discontinuous Galerkin scheme for the Cahn-Hilliard equations with new discrete inequalities and error estimates.

## Key findings

- Proved unconditional stability of the scheme.
- Derived a priori error estimates for the method.
- Established discrete Agmon and Gagliardo-Nirenberg inequalities.

## Abstract

The mixed form of the Cahn-Hilliard equations is discretized by the hybridizable discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg-Lindau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo-Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.