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A Posteriori Error Estimates with Boundary Correction for a Cut Finite Element Method

Erik Burman, Cuiyu He, Mats G. Larson

TL;DR

This work introduces, analyzes and implements a residual-based a posteriori error estimation for the CutFEM fictitious domain method applied to an elliptic model problem and can prove that the error estimation is both reliable and efficient.

Abstract

In this work we study a residual based a posteriori error estimation for the CutFEM method applied to an elliptic model problem. We consider the problem with non-polygonal boundary and the analysis takes into account the geometry and data approximation on the boundary. The reliability and efficiency are theoretically proved. Moreover, constants are robust with respect to how the domain boundary cuts the mesh.

A Posteriori Error Estimates with Boundary Correction for a Cut Finite Element Method

TL;DR

This work introduces, analyzes and implements a residual-based a posteriori error estimation for the CutFEM fictitious domain method applied to an elliptic model problem and can prove that the error estimation is both reliable and efficient.

Abstract

In this work we study a residual based a posteriori error estimation for the CutFEM method applied to an elliptic model problem. We consider the problem with non-polygonal boundary and the analysis takes into account the geometry and data approximation on the boundary. The reliability and efficiency are theoretically proved. Moreover, constants are robust with respect to how the domain boundary cuts the mesh.

Paper Structure

This paper contains 14 sections, 8 theorems, 82 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Problem and the Cut Finite Element Method
  3. The Continuous Problem
  4. The Mesh, Discrete Domains, and Finite Element Spaces
  5. The Cut Finite Element Method
  6. Some Important Inequalities
  7. Global Reliability
  8. Efficiency
  9. Numerical Results
  10. Computation of $\| \nabla \tilde{e}\|_{{\Omega}}$
  11. Numerical Examples
  12. Example 3
  13. Example 4
  14. Acknowledgement

Key Result

Lemma 2.1

Let $v \in H_0^1({\Omega})$. Then for any $K$ such that $K \in {\mathcal{T}}_h^b$ or $K \cap ({\Omega} \setminus {\Omega}_h) \neq \emptyset$ there exists a local convex neighborhood $\mathcal{S}_K$ of $K$ such that $v$ vanishes on a nonzero subset of $\partial S_K$ and where we defined $v$ outside ${\Omega}$ using the trivial extension $v\vert_{{\Omega}^c} = 0$.

Figures (14)

  • Figure 1: An irregular element $K$ and $D_K$ (the shaded area)
  • Figure 1: Partition of the intersection cell based on the cut of level set.
  • Figure 2: An example of the boundary correction mesh for Example 1
  • Figure 3: Final meshes generated without and with $\| \nabla \tilde{e}\|_K$.
  • Figure 4: Estimator convergence rate without and with $\| \nabla \tilde{e}\|$
  • ...and 9 more figures

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • Proof 1
  • Theorem 3.1
  • Proof 2
  • Remark 3.1
  • Lemma 3.1
  • Proof 3
  • Remark 3.1
  • ...and 15 more