Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements

Abstract

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted $L^2$ projection onto $H(\mathrm{div})$ conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps.