A Stabilized Trace FEM for Surface Cahn--Hilliard Equations: Analysis and Simulations

TL;DR

This paper introduces a stabilized trace finite element method combined with an implicit-explicit scheme for solving surface Cahn--Hilliard equations, providing stability analysis, error estimates, and numerical validation on various surfaces.

Contribution

It develops a novel stabilized trace FEM approach with proven stability, optimal error estimates, and demonstrates its effectiveness through comprehensive numerical experiments.

Findings

The method is unconditionally stable with an energy dissipation law.

Optimal-order error estimates are derived for the scheme.

Numerical experiments confirm robustness and convergence on different surfaces.

Abstract

This paper addresses the analysis and numerical assessment of a computational method for solving the Cahn--Hilliard equation defined on a surface. The proposed approach combines the stabilized trace finite element method for spatial discretization with an implicit--explicit scheme for temporal discretization. The method belongs to a class of unfitted finite element methods that use a fixed background mesh and a level-set function for implicit surface representation. We establish the numerical stability of the discrete problem by showing a suitable energy dissipation law for it. We further derive optimal-order error estimates assuming simplicial background meshes and finite element spaces of order $m \geq 1$. The effectiveness of the method is demonstrated through numerical experiments on several two-dimensional closed surfaces, confirming the theoretical results and illustrating the robustness and convergence properties of the scheme.