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Stable quadratic generalized IsoGeometric analysis for elliptic interface problem

Yin Song, Wenkai Hu, Xin Li

TL;DR

This work develops SGIGA2, a stable quadratic generalized isogeometric analysis method for two-dimensional elliptic interface problems, combining a modified quasi-interpolation that remains accurate across $C^0$ interfaces with a specially designed enrichment space to control stiffness conditioning. The authors prove an $H^1$-error bound of $|u-u_h|_{H^1(\Omega^p)}\le C h^2 \|u\|_{H^3(\Omega^p)}$ and demonstrate optimal convergence through rigorous analyses and numerical experiments on straight, circular, and arc interfaces. SGIGA2 avoids the numerical drawbacks of LPCA-based enrichments and does not require knot repetition, while maintaining CAD-geometric fidelity intrinsic to IGA. The results show robust, stable performance with scaled condition numbers that grow like $O(h^{-2})$, making SGIGA2 a practical unfitted-mesh approach for interface problems in two dimensions with potential extension to higher orders and dimensions.

Abstract

Unfitted mesh formulations for interface problems generally adopt two distinct methodologies: (i) penalty-based approaches and (ii) explicit enrichment space techniques. While Stable Generalized Finite Element Method (SGFEM) has been rigorously established for one-dimensional and linear-element cases, the construction of optimal enrichment spaces preserving approximation-theoretic properties within isogeometric analysis (IGA) frameworks remains an open challenge. In this paper, we introduce a stable quadratic generalized isogeometric analysis (SGIGA2) for two-dimensional elliptic interface problems. The method is achieved through two key ideas: a new quasi-interpolation for the function with C0 continuous along interface and a new enrichment space with controlled condition number for the stiffness matrix. We mathematically prove that the present method has optimal convergence rates for elliptic interface problems and demonstrate its stability and robustness through numerical verification.

Stable quadratic generalized IsoGeometric analysis for elliptic interface problem

TL;DR

This work develops SGIGA2, a stable quadratic generalized isogeometric analysis method for two-dimensional elliptic interface problems, combining a modified quasi-interpolation that remains accurate across interfaces with a specially designed enrichment space to control stiffness conditioning. The authors prove an -error bound of and demonstrate optimal convergence through rigorous analyses and numerical experiments on straight, circular, and arc interfaces. SGIGA2 avoids the numerical drawbacks of LPCA-based enrichments and does not require knot repetition, while maintaining CAD-geometric fidelity intrinsic to IGA. The results show robust, stable performance with scaled condition numbers that grow like , making SGIGA2 a practical unfitted-mesh approach for interface problems in two dimensions with potential extension to higher orders and dimensions.

Abstract

Unfitted mesh formulations for interface problems generally adopt two distinct methodologies: (i) penalty-based approaches and (ii) explicit enrichment space techniques. While Stable Generalized Finite Element Method (SGFEM) has been rigorously established for one-dimensional and linear-element cases, the construction of optimal enrichment spaces preserving approximation-theoretic properties within isogeometric analysis (IGA) frameworks remains an open challenge. In this paper, we introduce a stable quadratic generalized isogeometric analysis (SGIGA2) for two-dimensional elliptic interface problems. The method is achieved through two key ideas: a new quasi-interpolation for the function with C0 continuous along interface and a new enrichment space with controlled condition number for the stiffness matrix. We mathematically prove that the present method has optimal convergence rates for elliptic interface problems and demonstrate its stability and robustness through numerical verification.

Paper Structure

This paper contains 14 sections, 5 theorems, 58 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Non-uniform rational B-splines
  3. Interface problem and solving framework
  4. Interface problem using finite element method
  5. Interface problem using isogeometric analysis
  6. An example of a straight line interface
  7. New GIGA enrichment space construction
  8. Modified quasi-interpolation
  9. New enrichment space and basis function
  10. Proof for the optimal convergence rates
  11. Numerical experiment
  12. An example of a circular interface
  13. An example of an arc interface
  14. Conclusion and future work

Key Result

Theorem 1

There exists a constant $0<C<1$ such that for all $u \in W^{3, \infty}(\Omega)$ with step size $h$, we have

Figures (15)

  • Figure 1: Illustration of the quadratic basis functions in FEM (left) and IGA (right).
  • Figure 2: Illustration of the quasi-interpolation for quadratic B-splines.
  • Figure 3: Illustration of the notations for the construction of enrichment space.
  • Figure 4: Mapping between parameter domain and physical domain.
  • Figure 5: The $H^1$ errors and the SCNs with respect to $h$ (from $\frac{1}{5}$ to $\frac{1}{160}$) for a straight line interface ($a_0 = 20, a_1 = 1$).
  • ...and 10 more figures

Theorems & Definitions (8)

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