A Posteriori Error Estimation for the p-curl Problem

TL;DR

This paper develops reliable a posteriori error estimates for a nonlinear p-curl finite element problem in superconductivity, enabling improved error control without linearization and applicable to practical computational settings.

Contribution

It introduces a novel residual-based a posteriori error estimator for the nonlinear p-curl problem, handling non-conformity and nonlinearity without linearization.

Findings

Error estimates are reliable for non-conforming Nédélec elements.

The method effectively estimates errors for the AC loss quantity.

Numerical results confirm the theoretical reliability of the error estimator.

Abstract

We derive a posteriori error estimates for a semi-discrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the $p$-curl problem. In particular, we show the reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual type argument and a Helmholtz-Weyl decomposition of $W^p_0(\text{curl};\Omega)$. As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the $p$-Laplacian. It is handled without linearizing around the approximate solution. The non-conformity is dealt by adapting error decomposition techniques of Carstensen, Hu and Orlando. Geometric non-conformities also appear because the continuous problem is defined over a bounded $C^{1,1}$ domain while the discrete problem is formulated over a weaker polyhedral domain. The semi-discrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.