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A High Order Geometry Conforming Immersed Finite Element for Elliptic Interface Problems

Slimane Adjerid, Tao Lin, Haroun Meghaichi

TL;DR

The paper develops a high-order immersed finite element method for elliptic interface problems on interface-independent Cartesian meshes by leveraging a Frenet-coordinate transformation to map curved interface segments within interface elements to straight lines in a Frenet fictitious element. The resultant IFE space, built from $Q^m$ polynomials and transferred back to the physical element, satisfies the interface jump conditions exactly and integrates with a symmetric interior-penalty DG scheme without extra penalties. The authors prove optimal approximation properties that are robust to coefficient contrasts and interface geometry, and demonstrate $h$- and $p$-convergence through numerical experiments up to degree $m=8$, including challenging nonlinear curves and small-cut configurations. The approach offers robust, high-order accuracy on Cartesian meshes, with potential for extension to 3D and rigorous error analysis.

Abstract

We present a high order immersed finite element (IFE) method for solving the elliptic interface problem with interface-independent meshes. The IFE functions developed here satisfy the interface conditions exactly and they have optimal approximation capabilities. The construction of this novel IFE space relies on a nonlinear transformation based on the Frenet-Serret frame of the interface to locally map it into a line segment, and this feature makes the process of constructing the IFE functions cost-effective and robust for any degree. This new class of immersed finite element functions is locally conforming with the usual weak form of the interface problem so that they can be employed in the standard interior penalty discontinuous Galerkin scheme without additional penalties on the interface. Numerical examples are provided to showcase the convergence properties of the method under $h$ and $p$ refinements.

A High Order Geometry Conforming Immersed Finite Element for Elliptic Interface Problems

TL;DR

The paper develops a high-order immersed finite element method for elliptic interface problems on interface-independent Cartesian meshes by leveraging a Frenet-coordinate transformation to map curved interface segments within interface elements to straight lines in a Frenet fictitious element. The resultant IFE space, built from polynomials and transferred back to the physical element, satisfies the interface jump conditions exactly and integrates with a symmetric interior-penalty DG scheme without extra penalties. The authors prove optimal approximation properties that are robust to coefficient contrasts and interface geometry, and demonstrate - and -convergence through numerical experiments up to degree , including challenging nonlinear curves and small-cut configurations. The approach offers robust, high-order accuracy on Cartesian meshes, with potential for extension to 3D and rigorous error analysis.

Abstract

We present a high order immersed finite element (IFE) method for solving the elliptic interface problem with interface-independent meshes. The IFE functions developed here satisfy the interface conditions exactly and they have optimal approximation capabilities. The construction of this novel IFE space relies on a nonlinear transformation based on the Frenet-Serret frame of the interface to locally map it into a line segment, and this feature makes the process of constructing the IFE functions cost-effective and robust for any degree. This new class of immersed finite element functions is locally conforming with the usual weak form of the interface problem so that they can be employed in the standard interior penalty discontinuous Galerkin scheme without additional penalties on the interface. Numerical examples are provided to showcase the convergence properties of the method under and refinements.
Paper Structure (13 sections, 15 theorems, 121 equations, 9 figures, 1 algorithm)

This paper contains 13 sections, 15 theorems, 121 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Frenet coordinates for interface elements
  4. An IFE Space by Frenet Coordinates
  5. A local IFE space on every interface element
  6. A basis for the local IFE space
  7. Computational aspects of the construction of an IFE basis
  8. The approximation capabilities of the IFE space
  9. An immersed discontinuous Galerkin scheme
  10. The implementation of the Frenet transformation
  11. The construction of the local matrices
  12. Numerical Examples
  13. Conclusion

Key Result

Lemma 1

Given a polynomial $\hat{\phi}^{\pm} \in Q^m(\hat{K}_F^\pm)$, there is a unique polynomial $\hat{\phi}^{\mp} \in Q^m(\hat{K}_F^\mp)$ such that the piecewise $Q^m$ polynomial $\hat{\phi} = \hat{\phi}^- \chi_{\hat{K}_F^-} + \hat{\phi}^+ \chi_{\hat{K}_F^+}$ satisfies the jump conditions eqn:extended_

Figures (9)

  • Figure 1: Curves parallel to $\Gamma$ and lines normal to $\Gamma$ form a curvilinear coordinate system in the vicinity of the interface $\Gamma$. The innermost curve, parallel to $\Gamma$, and the outermost curve, also parallel to $\Gamma$, enclose a region known as a tubular neighborhood of $\Gamma$.
  • Figure 2: The two green parallel curves form an $\epsilon$ tubular neighborhood $N_\Gamma(\epsilon)$ of the interface curve $\Gamma$ in red color, the two blue parallel curves form the $h$ tubular neighborhood $N_\Gamma(h)$.
  • Figure 3: The plot on the right is for a square interface element $K$ and the associated fictitious element $K_F$ which is the black trapezoid with two bases parallel to the interface curve $\Gamma$ in red color. The black rectangle in the plot on the left is the related fictitious element $\hat{K}_F$ in the $\eta$-$\xi$ plane such that $K_F = P_\Gamma(\hat{K}_F)$.
  • Figure 4: Condition numbers of $\tilde{\mathbf{A}}(m,\varepsilon)$ for different degrees $m$ and different values for $\varepsilon$.
  • Figure 5: The relative error of the $L^2$ projection method applied to Example 1 with $\beta^+=10$ ( left), $\beta^+=100$ (middle), $\beta^+=1000$ (right).
  • ...and 4 more figures

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 1
  • Lemma 5
  • ...and 17 more