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Conservative flux reconstruction for an elliptic interface problem using CutFEM

Daniela Capatina, Aimene Gouasmi

Abstract

This paper deals with the local recovery of conservative fluxes for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed weakly and the discretisation is achieved by using conforming finite elements on unfitted meshes, with the aid of the CutFEM method. In a first attempt at flux reconstruction, we define a flux belonging to the Raviart-Thomas space of each sub-domain following the method developed for a boundary problem. However, the transmission condition is not satisfied by the recovered flux. In order to overcome this shortcoming, we propose a second approach, where the flux belongs to the recently introduced immersed Raviart-Thomas space. This ensures both the continuity of the normal flux across the interface and a natural conservation property on the cut cells. Subsequently, we apply the recovered flux to a posteriori error analysis and prove the sharp reliability of the error estimator.

Conservative flux reconstruction for an elliptic interface problem using CutFEM

Abstract

This paper deals with the local recovery of conservative fluxes for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed weakly and the discretisation is achieved by using conforming finite elements on unfitted meshes, with the aid of the CutFEM method. In a first attempt at flux reconstruction, we define a flux belonging to the Raviart-Thomas space of each sub-domain following the method developed for a boundary problem. However, the transmission condition is not satisfied by the recovered flux. In order to overcome this shortcoming, we propose a second approach, where the flux belongs to the recently introduced immersed Raviart-Thomas space. This ensures both the continuity of the normal flux across the interface and a natural conservation property on the cut cells. Subsequently, we apply the recovered flux to a posteriori error analysis and prove the sharp reliability of the error estimator.

Paper Structure

This paper contains 7 sections, 4 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The continuous and discrete problems
  3. The auxiliary mixed formulation
  4. Local flux reconstruction
  5. Flux recovering in $\mathcal{RT}^m(\Omega_h^1)\times \mathcal{RT}^m(\Omega_h^2)$
  6. Flux recovering in $\mathcal{IRT}^0(\Omega)$
  7. Application to a posteriori error analysis

Key Result

theorem 1

There exists a constant $\tilde{\beta}$ independent of $h$ and $K$ such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (6)

  • theorem 1
  • theorem 2
  • theorem 3
  • proof
  • theorem 4: Reliability
  • proof