Robust a posteriori error estimation for finite element approximation to (curl) problem

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 several H(curl) interface problems.

A Posteriori Error-Estimators for Induction Heating Processes Modelling

“…Superconvergent points enable to recover the gradient of the solution with higher order approximation (errors of the order 𝑂(ℎ 𝑝+1 )). Applications in the electromagnetic fields to 𝐻(𝑐𝑢𝑟𝑙) problem, were reported in [31], [32].…”

A Posteriori Error-Estimators for Induction Heating Processes Modelling

“…Superconvergent points enable to recover the gradient of the solution with higher order approximation (errors of the order 𝑂(ℎ 𝑝+1 )). Applications in the electromagnetic fields to 𝐻(𝑐𝑢𝑟𝑙) problem, were reported in [31], [32].…”

An Equilibrated a Posteriori Error Estimator for Arbitrary-Order Nédélec Elements for Magnetostatic Problems

“…From Corollary 3.3, it follows that a constant-free upper bound on the error can be obtained from any fieldH Δ that satisfies (7).…”

“…A different equilibration-based error estimator for magnetostatics was introduced in [21] and, for an eddy current problem, in [10,11]. Constant-free upper bounds are also obtained by the functional estimate in [18], when selecting a proper function y in their estimator, and by the recovery-type error estimator in [7], in case the equations contain an additional term βu, with β > 0.…”