A Posteriori Error Estimation for the p-curl Problem
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 Nédélec 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2016
- DOI:
- 10.48550/arXiv.1605.06532
- arXiv:
- arXiv:1605.06532
- Bibcode:
- 2016arXiv160506532W
- Keywords:
-
- Mathematics - Numerical Analysis;
- 35K65;
- 65M60;
- 65M15;
- 78M10
- E-Print:
- 32 pages