A Hybrid High-Order method for the Cahn-Hilliard problem in mixed form

TL;DR

This paper introduces a fully implicit Hybrid High-Order method for the Cahn-Hilliard problem that supports complex meshes, arbitrary approximation orders, and offers efficient computation with proven stability and convergence.

Contribution

The paper presents a novel hybrid high-order discretization scheme for the Cahn-Hilliard equation that handles general meshes and provides optimal convergence with computational efficiency.

Findings

Supports polygonal and nonmatching meshes

Achieves optimal convergence rates

Enables local elimination of unknowns for efficiency

Abstract

In this work, we develop a fully implicit Hybrid High-Order algorithm for the Cahn-Hilliard problem in mixed form. The space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces. The proposed method has several assets: (i) It supports fairly general meshes possibly containing polygonal elements and nonmatching interfaces, (ii) it allows arbitrary approximation orders, (iii) it has a moderate computational cost thanks to the possibility of locally eliminating element-based unknowns by static condensation. We perform a detailed stability and convergence study, proving optimal convergence rates in energy-like norms. Numerical validation is also provided using some of the most common tests in the literature.