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A Robust HHO Method for the Phase Field Model with Logarithmic Potential and Mass Source
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DOI：10.1016/j.cnsns.2026.110058.png)
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Abstract
En 中文
• This study develops a Hybrid High-Order (HHO) method for the Allen-Cahn equation with a logarithmic potential and a mass source term on general polygonal meshes, ensuring high accuracy in complex two-dimensional domains through polynomial recon-structions. • The HHO framework (Di Pietro & Droniou) is based on local reconstruction of differential operators using cell and face unknowns, along with stabilization terms to control discontinuities, leading to symmetric, coercive, and well-conditioned systems with optimal convergence. • A key computational advantage is static condensation, which eliminates local degrees of freedom at the element level, resulting in compact global systems and effcient implementation, making HHO methods highly effective across various applications. • The main contribution of this work is the rigorous convergence analysis of the Allen-Cahn equation with a singular logarithmic free-energy potential and a nonlinear source term, which introduces significant analytical and numerical challenges. • Two major diffculties are addressed: preserving the physical bounds of the solution within (-1; 1) and ensuring discrete energy stability, particularly on general polygonal meshes where standard techniques are not directly applicable. • The analysis is carried out for both semi-discrete and fully discrete schemes, where time discretization is performed using BDF1 and BDF2 methods, requiring careful handling of the nonlinear source and singular behavior near the phase boundaries. • To overcome these challenges, a Lyapunov functional-based framework is de-veloped, which allows the proof of monotonic decay of discrete energy and ensures the preservation of the dissipative structure with improved polynomial growth bounds for long-time stability. • Optimal convergence rates are established and validated numerically, showing errors of order O(hp+1+τ) and O(hp+1+τ2) , confirming that the proposed HHO–BDF approach is accurate, stable, and robust for nonlinear phase-separation problems.
Keywords:
Hybrid High-Order method
Allen-Cahn equation
logarithmic potential
mass source term
phase-field model
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