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An Unfitted Interface Penalty DG--FE Method for Elliptic Interface Problems

Juan Han, Haijun Wu, Yuanming Xiao

TL;DR

The paper addresses elliptic interface problems with discontinuous coefficients on unfitted meshes by introducing the UIPDG-FEM, a hybrid discretization that confines DG treatment to macro-elements near the interface and uses standard FEM away from the interface. It combines harmonic weighting in the Nitsche-based interface conditions with a robust 2D element-merging algorithm to stabilize small-cut elements, and provides optimal error estimates (H^1, L^2, and flux) along with a condition-number bound comparable to fitted FEM and independent of interface position. The key contributions include a curvature-informed merging criterion with feasibility and reliability guarantees, and a comprehensive analysis showing coefficient-robust flux error and mesh-independence of conditioning. Numerical experiments validate the theoretical results, demonstrating optimal convergence and robustness to coefficient contrasts and complex interface geometry. This work enables accurate, stable unfitted discretizations for elliptic interface problems on simple background meshes, reducing meshing complexity in practical applications.

Abstract

We propose an unfitted interface penalty Discontinuous Galerkin-Finite Element Method (UIPDG-FEM) for elliptic interface problems. This hybrid method combines the interior penalty discontinuous Galerkin (IPDG) terms near the interface-enforcing jump conditions via Nitsche method-with standard finite elements away from the interface. The UIPDG-FEM retains the flexibilities of IPDG, particularly simplifying mesh generation around complex interfaces, while avoiding its drawback of excessive number of global degrees of freedom. We derive optimal convergence rates independent of interface location and establish uniform flux error estimates robust to discontinuous coefficients. To deal with conditioning issues caused by small cut elements, we develop a robust two-dimensional merging algorithm that eliminates such elements entirely, ensuring the condition number of the discretized system remains independent of interface position. A key feature of the algorithm is a novel quantification criterion linking the threshold for small cuts to the product of the maximum interface curvature and the local mesh size. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed method.

An Unfitted Interface Penalty DG--FE Method for Elliptic Interface Problems

TL;DR

The paper addresses elliptic interface problems with discontinuous coefficients on unfitted meshes by introducing the UIPDG-FEM, a hybrid discretization that confines DG treatment to macro-elements near the interface and uses standard FEM away from the interface. It combines harmonic weighting in the Nitsche-based interface conditions with a robust 2D element-merging algorithm to stabilize small-cut elements, and provides optimal error estimates (H^1, L^2, and flux) along with a condition-number bound comparable to fitted FEM and independent of interface position. The key contributions include a curvature-informed merging criterion with feasibility and reliability guarantees, and a comprehensive analysis showing coefficient-robust flux error and mesh-independence of conditioning. Numerical experiments validate the theoretical results, demonstrating optimal convergence and robustness to coefficient contrasts and complex interface geometry. This work enables accurate, stable unfitted discretizations for elliptic interface problems on simple background meshes, reducing meshing complexity in practical applications.

Abstract

We propose an unfitted interface penalty Discontinuous Galerkin-Finite Element Method (UIPDG-FEM) for elliptic interface problems. This hybrid method combines the interior penalty discontinuous Galerkin (IPDG) terms near the interface-enforcing jump conditions via Nitsche method-with standard finite elements away from the interface. The UIPDG-FEM retains the flexibilities of IPDG, particularly simplifying mesh generation around complex interfaces, while avoiding its drawback of excessive number of global degrees of freedom. We derive optimal convergence rates independent of interface location and establish uniform flux error estimates robust to discontinuous coefficients. To deal with conditioning issues caused by small cut elements, we develop a robust two-dimensional merging algorithm that eliminates such elements entirely, ensuring the condition number of the discretized system remains independent of interface position. A key feature of the algorithm is a novel quantification criterion linking the threshold for small cuts to the product of the maximum interface curvature and the local mesh size. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed method.
Paper Structure (8 sections, 15 theorems, 83 equations, 11 figures, 1 algorithm)

This paper contains 8 sections, 15 theorems, 83 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The unfitted interface penalty DG--FE Method
  3. Error estimates of SUIPDG-FEM
  4. $H^1$ and $L^2$ error estimate
  5. Flux error estimate
  6. Estimate of the condition number
  7. A merging algorithm
  8. Numerical tests

Key Result

Lemma 3.1

There exists a constant $h_0>0$, depending only on the interface $\Gamma$ and the shape regularity of the meshes, such that for all $h\in(0,h_0)$, and every interface-cut element $K\in{\cal M}_{h,i}$, $i=1, 2$, the following estimates hold:

Figures (11)

  • Figure 2.1: Merged elements for $h=1/16$.
  • Figure 3.1: Transfer the trace from short edge to long edge.
  • Figure 5.1: Merged elements do not overlap under Assumption \ref{['connection']}.
  • Figure 5.2: Two types of small elements.
  • Figure 5.3: Local deviation of $\mathcal{C}$ from its tangent line.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Theorem 3.6
  • ...and 21 more