Robust A Posteriori Error Estimation for Finite Element Approximation to H(curl) Problem

TL;DR

This paper proposes a new a posteriori error estimator for finite element solutions to H(curl) problems, improving error assessment accuracy without requiring quasi-monotonicity, and demonstrates its effectiveness through numerical tests.

Contribution

It introduces a recovery-based a posteriori error estimator for H(curl) finite element problems that does not rely on quasi-monotonicity assumptions.

Findings

Estimator accurately approximates true error in energy norm

Effective for inhomogeneous media and L^2 right-hand side

Validated through numerical experiments on interface problems

Abstract

In this paper, we introduce a novel a posteriori error estimator for the conforming finite element approximation to the H(curl) problem with inhomogeneous media and with the right-hand side only in L^2. The estimator is of the recovery type. Independent with the current approximation to the primary variable (the electric field), an auxiliary variable (the magnetizing field) is recovered in parallel by solving a similar H(curl) problem. An alternate way of recovery is presented as well by localizing the error flux. The estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable. It is proved that the estimator is approximately equal to the true error in the energy norm without the quasi-monotonicity assumption. Finally, we present numerical results for two H(curl) interface problems.