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An unfitted finite element method for elliptic interface problem with low regularity estimates

Fanyi Yang

Abstract

In this paper, we present and analyze an unfitted finite element method for the elliptic interface problem. We consider the case that the interface is $C^2$-smooth or polygonal, and the exact solution $u \in H^{1+s}(Ω_0 \cup Ω_1)$ for any $s > 0$. The stability near the interface is guaranteed by a local polynomial extension technique combined with ghost penalty bilinear forms, from which the robust condition number estimates and the error estimates are derived. Furthermore, the jump penalty term for weakly enforcing the jump condition in our method is also defined based on the local polynomial extension, which enables us to establish the error estimation particularly for solutions with low regularity. We perform a series of numerical tests in two and three dimensions to illustrate the accuracy of the proposed method.

An unfitted finite element method for elliptic interface problem with low regularity estimates

Abstract

In this paper, we present and analyze an unfitted finite element method for the elliptic interface problem. We consider the case that the interface is -smooth or polygonal, and the exact solution for any . The stability near the interface is guaranteed by a local polynomial extension technique combined with ghost penalty bilinear forms, from which the robust condition number estimates and the error estimates are derived. Furthermore, the jump penalty term for weakly enforcing the jump condition in our method is also defined based on the local polynomial extension, which enables us to establish the error estimation particularly for solutions with low regularity. We perform a series of numerical tests in two and three dimensions to illustrate the accuracy of the proposed method.
Paper Structure (10 sections, 9 theorems, 89 equations, 5 figures, 7 tables)

This paper contains 10 sections, 9 theorems, 89 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem setting and preliminaries
  3. Numerical Scheme
  4. Numerical Results
  5. Example 1.
  6. Example 2.
  7. Example 3.
  8. Example 4.
  9. Example 5.
  10. Conclusions

Key Result

Lemma 1

For the $C^2$ interface $\Gamma$, there exists a constant $h_0$ such that for any $h < h_0$, there holds

Figures (5)

  • Figure 1: The interface intersects $B_{\tau(K)}$ in two dimensions.
  • Figure 2: The interface intersects $B_{\tau(K)}$ in three dimensions.
  • Figure 3: The unfitted mesh and the interface in two dimensions.
  • Figure 4: The unfitted mesh and the interface in three dimensions.
  • Figure 5: The simplex with vertices $A_0,\ldots,A_d$ in two and three dimensions.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 11 more