# On the superconvergence of a hydridizable discontinuous Galerkin method for the Cahn-Hilliard equation

## TL;DR

This paper introduces a hybridizable discontinuous Galerkin (HDG) method for the Cahn-Hilliard equation, demonstrating optimal convergence and superconvergence properties, supported by theoretical analysis and numerical validation.

## Contribution

The work develops a novel HDG Sobolev inequality and proves superconvergence of scalar variables in the HDG method for nonlinear PDEs.

## Key findings

- Optimal convergence rates in $L^2$ norm for all variables.
- Superconvergence of scalar variables in globally coupled degrees of freedom.
- Numerical results confirm theoretical convergence rates.

## Abstract

We propose a hydridizable discontinuous Galerkin (HDG) method for solving the Cahn-Hilliard equation. The temporal discretization can be based on either the backward Euler method or the convex-splitting method. We show that the fully discrete scheme admits a unique solution, and we establish optimal convergence rates for all variables in the $L^2$ norm for arbitrary polynomial orders. In terms of the globally coupled degrees of freedom, the scalar variables are superconvergent. Another theoretical contribution of this work is a novel HDG Sobolev inequality that is useful for HDG error analysis of nonlinear problems. Numerical results are reported to confirm the theoretical convergence rates.