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A unified immersed finite element error analysis for one-dimensional interface problems

Slimane Adjerid, Tao Lin, Haroun Meghaichi

Abstract

It has been noted that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.

A unified immersed finite element error analysis for one-dimensional interface problems

Abstract

It has been noted that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.
Paper Structure (13 sections, 33 theorems, 182 equations, 1 figure)

This paper contains 13 sections, 33 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of the RIFE space VI
  4. An immersed Bramble-Hilbert lemma and the approximation capabilities of the LIFE space
  5. Analysis of an IFE-DG method for a class of hyperbolic systems with discontinuous coefficients
  6. Problem statement and preliminary results
  7. The immersed Radau projection and the convergence analysis
  8. Novel proofs for results already established in the literature
  9. The $m$-th degree IFE space for an elliptic interface problem
  10. Euler-Bernoulli Beam interface problem
  11. Conclusion
  12. Proof of \ref{['lem:norm_on_Q']}
  13. Proof of \ref{['lem:bounds_on_Hermite']}

Key Result

Lemma 1

Let ${\tilde{m}} \geq m\ge 0, \{r_k\}_{k=0}^{\tilde{m}} \subset\mathbb{R}_+, \check{\alpha}\in (0,1)$ and $s\in\{+,-\}$. The following statements hold

Figures (1)

  • Figure 1: The relative position of the interface (on the right) changes as the we refine the mesh (on the left).

Theorems & Definitions (65)

  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 55 more