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HoSGFEM: High-order stable generalized finite element method for elliptic interface problem

Bingying Zhao, Yin Song, Quanling Deng, Xin Li

TL;DR

The paper addresses high-order unfitted finite element methods for elliptic interface problems, where material discontinuities across an interface Γ pose stability and accuracy challenges. It introduces HoSGFEM, a high-order stable generalized finite element method that uses Bernstein-based partition of unity functions on interface elements and element-centroid-centered local enrichment with Gram–Schmidt orthogonalization to create a enriched space with a large angle to the standard FEM space. The authors prove optimal convergence in the energy norm with the estimate |u − u_h|_{H^1(Ω)} ≤ C h^p ||u||_{H^{p+1}(Ω)} and demonstrate numerically that HoSGFEM achieves optimal rates for p=2…5 while maintaining a stiffness matrix conditioning growing like O(h^{-2}), comparable to FEM, and robust behavior as the interface nears element boundaries. The results on straight and curved interfaces, including a bi-material elastostatic test, show HoSGFEM outperforms existing GFEM variants in stability and accuracy, enabling high-order unfitted simulations without fitted meshes. Potential future work includes extending to three dimensions and developing scalable preconditioners for large-scale problems.

Abstract

The Generalized Finite Element Method (GFEM) is an effective unfitted numerical method for handling interface problems. By augmenting the standard FEM space with an appropriate enrichment space, GFEM can accurately capture C^0 solutions across the interfaces. While numerous GFEMs for interface problems have been studied, establishing a stable high-order GFEM with optimal convergence rates and robust system conditioning remains a challenge. The highest known order of two was established by Zhang and Babuška (SGFEM2, Comput. Methods Appl. Mech. Engrg. 363 (2020), 112889). In this paper, we propose a unified enrichment space construction and establish arbitrary high-order stable GFEMs (HoSGFEM) for elliptic interface problems. The main idea distinguishes itself from Zhang and Babuška's SGFEM2 substantially and it is twofold: a) we construct dimensionality-reduced auxiliary locally supported piecewise polynomials that satisfy the partition of unity property for elements containing interfaces; b) we construct the enrichment scheme based on d{1,(x-x_c^e),...,(y-y_c^e)^{p-1}} (d is the distance function; (x_c^e, y_c^e) is the center of the element containing interface, thus element-based) for arbitrary p-th order elements instead of d, d{1,x,y} or d{1,x,y,x^2,xy,y^2} (global functions) for p=1,2 in the literature. This idea results in an enrichment space that has a large angle with the standard FEM space, leading to the stability of the method with system condition number growing in order O(h^{-2}). We establish optimal convergence rates for HoSGFEM solutions under the proposed construction. Various numerical experiments with both straight and curved interfaces demonstrate the optimal convergence, FEM-comparable system condition number with O(h^{-2}) growth, and robustness as element boundaries approach interfaces.

HoSGFEM: High-order stable generalized finite element method for elliptic interface problem

TL;DR

The paper addresses high-order unfitted finite element methods for elliptic interface problems, where material discontinuities across an interface Γ pose stability and accuracy challenges. It introduces HoSGFEM, a high-order stable generalized finite element method that uses Bernstein-based partition of unity functions on interface elements and element-centroid-centered local enrichment with Gram–Schmidt orthogonalization to create a enriched space with a large angle to the standard FEM space. The authors prove optimal convergence in the energy norm with the estimate |u − u_h|_{H^1(Ω)} ≤ C h^p ||u||_{H^{p+1}(Ω)} and demonstrate numerically that HoSGFEM achieves optimal rates for p=2…5 while maintaining a stiffness matrix conditioning growing like O(h^{-2}), comparable to FEM, and robust behavior as the interface nears element boundaries. The results on straight and curved interfaces, including a bi-material elastostatic test, show HoSGFEM outperforms existing GFEM variants in stability and accuracy, enabling high-order unfitted simulations without fitted meshes. Potential future work includes extending to three dimensions and developing scalable preconditioners for large-scale problems.

Abstract

The Generalized Finite Element Method (GFEM) is an effective unfitted numerical method for handling interface problems. By augmenting the standard FEM space with an appropriate enrichment space, GFEM can accurately capture C^0 solutions across the interfaces. While numerous GFEMs for interface problems have been studied, establishing a stable high-order GFEM with optimal convergence rates and robust system conditioning remains a challenge. The highest known order of two was established by Zhang and Babuška (SGFEM2, Comput. Methods Appl. Mech. Engrg. 363 (2020), 112889). In this paper, we propose a unified enrichment space construction and establish arbitrary high-order stable GFEMs (HoSGFEM) for elliptic interface problems. The main idea distinguishes itself from Zhang and Babuška's SGFEM2 substantially and it is twofold: a) we construct dimensionality-reduced auxiliary locally supported piecewise polynomials that satisfy the partition of unity property for elements containing interfaces; b) we construct the enrichment scheme based on d{1,(x-x_c^e),...,(y-y_c^e)^{p-1}} (d is the distance function; (x_c^e, y_c^e) is the center of the element containing interface, thus element-based) for arbitrary p-th order elements instead of d, d{1,x,y} or d{1,x,y,x^2,xy,y^2} (global functions) for p=1,2 in the literature. This idea results in an enrichment space that has a large angle with the standard FEM space, leading to the stability of the method with system condition number growing in order O(h^{-2}). We establish optimal convergence rates for HoSGFEM solutions under the proposed construction. Various numerical experiments with both straight and curved interfaces demonstrate the optimal convergence, FEM-comparable system condition number with O(h^{-2}) growth, and robustness as element boundaries approach interfaces.
Paper Structure (13 sections, 3 theorems, 68 equations, 18 figures, 1 algorithm)

This paper contains 13 sections, 3 theorems, 68 equations, 18 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Elliptic interface problems
  3. HoSGFEM main idea
  4. GFEM framework
  5. Overview of GFEMs
  6. HoSGFEM
  7. Construction of PU functions for an interface element
  8. Construction of the enrichment space
  9. Error Analysis
  10. Numerical experiments
  11. Error convergence rates and scaled condition numbers
  12. Application: bi-material with a circular inclusion
  13. Conclusions

Key Result

Lemma 1

For an interface element $e_i$, there exists a polynomial $\eta_i$ of degree $(p - 1)$ such that where $\omega_i$ is the support of PU function $\phi_i$ corrosponding to interface element $e_i$.

Figures (18)

  • Figure 1: An example of a domain $\Omega$ with a curved interface $\Gamma$.
  • Figure 2: Illustration of different index sets.
  • Figure 3: An example of the 4-th order Bernstein polynomials
  • Figure 4: Construction of PU functions $\varphi_{k}(x, y)$.
  • Figure 5: The left plot highlights several interface elements in light blue, and the right plot shows the corresponding PU functions.
  • ...and 13 more figures

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof