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Flux-equilibrated based a posteriori error analysis for an interface problem with CutFEM

Daniela capatina, Aimene Gouasmi

Abstract

This paper addresses the local recovery of conservative fluxes and the a posteriori error analysis for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed by means of Nitsche's method and the discretization is carried out using conforming finite elements on unfitted meshes via the CutFEM method. A flux is subsequently defined in the global Raviart-Thomas space, ensuring that it satisfies the natural conservation property on the cut cells, and is then employed in the a posteriori error analysis. We prove here the sharp reliability of the error estimator and show a numerical experiment which illustrates the approach.

Flux-equilibrated based a posteriori error analysis for an interface problem with CutFEM

Abstract

This paper addresses the local recovery of conservative fluxes and the a posteriori error analysis for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed by means of Nitsche's method and the discretization is carried out using conforming finite elements on unfitted meshes via the CutFEM method. A flux is subsequently defined in the global Raviart-Thomas space, ensuring that it satisfies the natural conservation property on the cut cells, and is then employed in the a posteriori error analysis. We prove here the sharp reliability of the error estimator and show a numerical experiment which illustrates the approach.

Paper Structure

This paper contains 7 sections, 6 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The continuous and discrete problems
  3. The auxiliary mixed formulation
  4. Recovery of a conservative flux in the global Raviart-Thomas space
  5. Application to a posteriori error analysis
  6. Reliability
  7. Numerical experiment

Key Result

Lemma 3.1

The space $\mathrm{Ker}\, b_h = \{v_h \in \mathcal{D}_h ;\, b_h(\mu_h, v_h) = 0, \, \forall \mu_h \in \mathcal{M}_h \}$ is equal to $\mathcal{C}_h$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Notation on a cut element for the interpolation
  • Figure 2: Initial and adaptive final meshes. Error slope

Theorems & Definitions (8)

  • Lemma 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3: Reliability