Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Elliptic Interface Problem approximated by CutFEM: II. A posteriori error analysis based on equilibrated fluxes

Daniela Capatina, Aimene Gouasmi

Abstract

This paper investigates an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes, employing the CutFEM method. The main contribution is the a posteriori error analysis based on equilibrated fluxes belonging to the immersed Raviart-Thomas space. We establish sharp reliability and local efficiency of a new error estimator, which includes both volume and interface terms, carefully tracking the dependence of the efficiency constant on the diffusion coefficients and the mesh/interface configuration. Numerical results highlight the robustness of the proposed approach.

Elliptic Interface Problem approximated by CutFEM: II. A posteriori error analysis based on equilibrated fluxes

Abstract

This paper investigates an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes, employing the CutFEM method. The main contribution is the a posteriori error analysis based on equilibrated fluxes belonging to the immersed Raviart-Thomas space. We establish sharp reliability and local efficiency of a new error estimator, which includes both volume and interface terms, carefully tracking the dependence of the efficiency constant on the diffusion coefficients and the mesh/interface configuration. Numerical results highlight the robustness of the proposed approach.

Paper Structure

This paper contains 17 sections, 157 equations, 9 figures, 3 tables.

Table of Contents

  1. Continuous and discrete problems
  2. Definition of the flux and the a posteriori error estimators
  3. Reliability
  4. Sharp upper bound of the error
  5. A continuous approximation of the CutFEM solution
  6. Estimate of the interpolation error
  7. Properties of a piecewise Raviart-Thomas space on the cut cells
  8. Auxiliary results
  9. Construction of specific shape functions
  10. Bounds of the specific shape functions
  11. Decomposition of the flux correction in the cut cells
  12. Efficiency
  13. Bound of $\eta_T$ by the residuals
  14. Bound of the residuals by the local error
  15. Final bound of $\eta_T$
  16. ...and 2 more sections

Figures (9)

  • Figure 1: Notation on a cut element in view of the interpolation
  • Figure 2: \ref{['example2']} Final adapted meshes for $\mu= 10$
  • Figure 3: \ref{['example2']} Convergence rate of the error, $\eta$, $\eta+\hat{\eta}$ and $\eta+\eta_{\Gamma}$ for $\mu=10$
  • Figure 4: \ref{['example2']} AMR using $\eta_T$: estimator and error convergence (left) and final mesh (right) for $\mu=10^6$
  • Figure 5: \ref{['example3']} Initial and final meshes
  • ...and 4 more figures

Theorems & Definitions (28)

  • proof
  • proof
  • proof
  • proof
  • proof
  • Remark 1
  • proof
  • Remark 2
  • proof
  • proof
  • ...and 18 more