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

Erik Burman, Alexandre Ern, Romain Mottier

TL;DR

This work addresses elliptic interface problems on unfitted meshes by developing a high-order hybrid discretization (HHO) stabilized with polynomial extension to stabilize ill-cut cells. The method uses a gradient reconstruction defined on subcells, extended by a pairing operator that links ill-cut cells to neighboring well-cut cells, and employs a three-pronged stabilization (s_h^0, s_h^Γ, s_h^N) to guarantee stability and robustness to large contrasts in the diffusion coefficients. The authors prove stability, well-posedness, and optimal error estimates in the energy norm, with numerical results confirming convergence rates, robustness to coefficient contrasts, and favorable sparsity patterns. The approach enables high-order accuracy on fixed unfitted meshes without resorting to mesh modification, offering practical benefits for complex geometries and interfaces. Overall, the paper contributes a rigorous unfitted HHO framework with polynomial extension that delivers stable, high-order solutions for elliptic interface problems on general geometries.

Abstract

In this work, we study the design and 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.

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

TL;DR

This work addresses elliptic interface problems on unfitted meshes by developing a high-order hybrid discretization (HHO) stabilized with polynomial extension to stabilize ill-cut cells. The method uses a gradient reconstruction defined on subcells, extended by a pairing operator that links ill-cut cells to neighboring well-cut cells, and employs a three-pronged stabilization (s_h^0, s_h^Γ, s_h^N) to guarantee stability and robustness to large contrasts in the diffusion coefficients. The authors prove stability, well-posedness, and optimal error estimates in the energy norm, with numerical results confirming convergence rates, robustness to coefficient contrasts, and favorable sparsity patterns. The approach enables high-order accuracy on fixed unfitted meshes without resorting to mesh modification, offering practical benefits for complex geometries and interfaces. Overall, the paper contributes a rigorous unfitted HHO framework with polynomial extension that delivers stable, high-order solutions for elliptic interface problems on general geometries.

Abstract

In this work, we study the design and 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.

Paper Structure

This paper contains 20 sections, 7 theorems, 113 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Unfitted HHO methods
  4. Unfitted meshes
  5. Unfitted HHO spaces
  6. Local reconstruction operator
  7. Stabilization
  8. Discrete problem
  9. Stability, consistency, and error estimate
  10. Analysis tools
  11. Stability and well-posedness
  12. Interpolation operator
  13. Consistency
  14. Error estimate
  15. Numerical results
  16. ...and 5 more sections

Key Result

Lemma 3.1

Let ${\color{black}u^{\textup{ex}}}$ be the weak solution to eq:weak. Let $\hat{v}_h\in\widehat{\mathcal{U}}_{h0}$ be arbitrary and set Then, the following holds for all $\hat{w}_h \in \widehat{\mathcal{U}}_{h0}$,

Figures (12)

  • Figure 1: Model problem
  • Figure 2: Left: Illustration of the different types of cell in the unfitted mesh. Right: Zoom on the local degrees of freedom in a cut cell $T \in \mathcal{T}_h^{\textup{OK}}$; here, the approximation is affine in the sub-cells and constant on the sub-faces.
  • Figure 3: Pairing operators $\mathcal{N}_1$ and $\mathcal{N}_2$ acting on the ill-cut cells.
  • Figure 4: Local stencil for gradient reconstruction operator
  • Figure 5: Circular (top row) and flower-like (bottom row) interfaces. Left column: Coarsest mesh. Central column: Outcome of pairing procedure for polynomial extension stabilization, cells in $\mathcal{T}_h^{\textup{KO},1}$ are colored in green, and cells in $\mathcal{T}_h^{\textup{KO},2}$ are colored in blue, arrows indicate the pairing operator. Right column: Outcome of pairing procedure for cell-agglomeration stabilization, agglomerated cells are colored in dark.
  • ...and 7 more figures

Theorems & Definitions (14)

  • Lemma 3.1: Consistency: preparatory identity
  • proof
  • Lemma 4.1: Discrete inverse inequalities
  • Lemma 4.2: Approximation
  • Lemma 4.3: Stability and boundedness
  • proof
  • Lemma 4.4: Approximation
  • proof
  • Lemma 4.5: Consistency
  • proof
  • ...and 4 more