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Construction of Basis Functions for the Geometry Conforming Immersed Finite Element Method

Slimane Adjerid, Tao Lin, Haroun Meghaichi

TL;DR

This work develops high-order geometry-conforming immersed finite element (GC-IFE) bases for elliptic interface problems by leveraging a Frenet transformation to local coordinates, enabling exact enforcement of interface jump conditions on each interface element. It introduces a two-stage construction: an initial basis that satisfies jumps, followed by a reconstruction that yields an $L^2$-orthonormal basis using SVD-based orthogonalization of a generalized Vandermonde or mass matrix, with careful preconditioning to manage ill-conditioning. The approach is implemented in MATLAB with explicit routines for the Frenet maps, basis assembly, and quadrature, and includes numerical demonstrations of optimal $L^2$ projection accuracy and improved conditioning. The framework supports high-order approximations and can extend to more complex PDE systems, providing a practical and extensible toolkit for interface problems.

Abstract

The Frenet apparatus is a new framework for constructing high order geometry-conforming immersed finite element functions for interface problems. In this report, we present a procedure for constructing the local IFE bases in some detail as well as a new approach for constructing orthonormal bases using the singular value decomposition of the local generalized Vandermonde matrix. A sample implementation in MATLAB is provided to showcase the simplicity and extensionability of the framework.

Construction of Basis Functions for the Geometry Conforming Immersed Finite Element Method

TL;DR

This work develops high-order geometry-conforming immersed finite element (GC-IFE) bases for elliptic interface problems by leveraging a Frenet transformation to local coordinates, enabling exact enforcement of interface jump conditions on each interface element. It introduces a two-stage construction: an initial basis that satisfies jumps, followed by a reconstruction that yields an -orthonormal basis using SVD-based orthogonalization of a generalized Vandermonde or mass matrix, with careful preconditioning to manage ill-conditioning. The approach is implemented in MATLAB with explicit routines for the Frenet maps, basis assembly, and quadrature, and includes numerical demonstrations of optimal projection accuracy and improved conditioning. The framework supports high-order approximations and can extend to more complex PDE systems, providing a practical and extensible toolkit for interface problems.

Abstract

The Frenet apparatus is a new framework for constructing high order geometry-conforming immersed finite element functions for interface problems. In this report, we present a procedure for constructing the local IFE bases in some detail as well as a new approach for constructing orthonormal bases using the singular value decomposition of the local generalized Vandermonde matrix. A sample implementation in MATLAB is provided to showcase the simplicity and extensionability of the framework.

Paper Structure

This paper contains 11 sections, 1 theorem, 73 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Calculations in local variables
  3. Calculus in terms of the local coordinates
  4. The Frenet transformations between the physical and local coordinates
  5. Construction of a basis for the GC-IFE space
  6. An initial construction of a GC-IFE basis
  7. A generalized initial construction of a GC-IFE basis
  8. Reconstruction of a GC-IFE basis
  9. A numerical example for illustrating key computations
  10. Reproducibility
  11. Conclusion

Key Result

Lemma 1

If $\hat{\mathcal{B}}(C)=\hat{\mathcal{B}}((C^-,C^+))$ is a basis for $\hat{\mathcal{V}}^m_{\beta}(\hat{K}_F)$ and $Q$ is an invertible $(m+1)^2\times (m+1)^2$ matrix, then $\hat{\mathcal{B}}((C^- Q,C^+ Q))$ is also a basis for $\hat{\mathcal{V}}^m_{\beta}(\hat{K}_F)$.

Figures (5)

  • Figure 1: An interface element $K$ and its associated sets $\hat{K}_F, K_F$ and $\hat{\Gamma}_{K_F}$.
  • Figure 2: The sample points $\mathbf{g}(\xi^{(i)})$ (black) on the interface $\Gamma$ (red)
  • Figure 3: The condition number of $\mathbf{A}$ before and after the preconditioning procedure under degree refinement (left, in semi-log plot), and under mesh refinement (right, in log-log plot)
  • Figure 4: The condition number of $\tilde{\mathbf{A}}$ before and after the preconditioning procedure.
  • Figure 5: A quadrature rule constructed on an interface element of type I (left) and type II (right) with n_qp=3.

Theorems & Definitions (2)

  • Lemma 1
  • proof