A Hybrid High-Order method for the Cahn-Hilliard problem in mixed form
It provides a robust numerical method for solving the Cahn-Hilliard equation, which is important for materials science and fluid dynamics, but the contribution is incremental as it extends existing HHO methods to a specific problem.
The paper develops a fully implicit Hybrid High-Order method for the Cahn-Hilliard problem in mixed form, achieving optimal convergence rates in energy-like norms on general meshes with polygonal elements and nonmatching interfaces.
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.