Flux recovery for Cut finite element method and its application in a posteriori error estimation

TL;DR

This paper develops a local flux recovery method for the Cut Finite Element Method applied to Poisson problems, enabling effective a posteriori error estimation and adaptive mesh refinement with proven reliability and efficiency.

Contribution

It introduces a local flux recovery technique in Raviart-Thomas space for Cut FEM, avoiding mixed problem solutions and providing reliable error estimates.

Findings

Proves global reliability and local efficiency of the error estimator.

Demonstrates optimal flux error convergence rates in numerical experiments.

Validates theoretical results through numerical verification.

Abstract

In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we are able to prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.