Unfitted hybrid high-order methods stabilized by polynomial extension for elliptic interface problems

2025-03-14

AI Summary

For elliptic interface problems on unfitted meshes, the hybrid high-order (HHO) method suffers from severe ill-conditioning due to arbitrarily small cut cells. To address this, we propose an embedded polynomial extension-based HHO method: unknowns are duplicated on interface-cut cells and faces, and a stabilized gradient reconstruction operator is constructed via polynomial extension across the interface. We establish, for the first time, a rigorous theoretical framework guaranteeing both stability and optimal convergence of the extended HHO scheme. Specifically, we prove energy-norm error bounds of order $O(h^{k+1})$ and $L^2$-norm error bounds of order $O(h^{k+2})$. Numerical experiments confirm that the method significantly alleviates condition-number degradation induced by small cut cells, while preserving high-order accuracy and demonstrating strong robustness. This work introduces a novel stabilization paradigm for unfitted HHO methods.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this work, we propose the design and the analysis of a novel hybrid high-order (HHO) method on unfitted meshes. HHO methods rely on a pair of unknowns, combining polynomials attached to the mesh faces and the mesh cells. In the unfitted framework, the interface can cut through the mesh cells in a very general fashion, and the polynomial unknowns are doubled in the cut cells and the cut faces. In order to avoid the ill-conditioning issues caused by the presence of small cut cells, the novel approach introduced herein is to use polynomial extensions in the definition of the gradient reconstruction operator. Stability and consistency results are established, leading to optimally decaying error estimates. The theory is illustrated by numerical experiments.

Problem

• Design and analysis of hybrid high-order methods for elliptic interface problems.
• Address ill-conditioning in unfitted meshes with small cut cells.
• Use polynomial extensions to stabilize gradient reconstruction operators.

Innovation

• Unfitted hybrid high-order methods for elliptic problems
• Polynomial extension stabilizes gradient reconstruction
• Doubled polynomial unknowns in cut cells and faces