DOI: 10.1007/s10092-003-0077-y CALCOLO 40, 249 – 271 (2003) CALCOLO © Springer-Verlag 2003 A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes S. Nicaise, E. Creusé Université deValenciennes et du Hainaut Cambrésis MACS, Institut des Sciences et Tech- niques deValenciennes, Valencienne, France e-mail: Serge.Nicaise; Emmanuel.Creuse@univ-valenciennes.fr Received: May 2003 /Accepted: October 2003 – © Springer-Verlag 2003 Abstract. We consider residual-based a posteriori error estimators for the heterogeneous Maxwell equations using isotropic as well as anisotropic meshes. The continuous problem is approximated by using conforming ap- proximated spaces with minimal assumptions. Lower and upper bounds are obtained under standard assumptions on the meshes. The lower bound holds unconditionally, while the upper bound depends on alignment properties of the meshes with respect to the solution. In particular for isotropic meshes the upper bound also holds unconditionally. A numerical test is presented which confirms our theoretical results. 1 Introduction The classical eddy current electric formulation in a three-dimensional bounded polyhedral domain  is given by the parabolic initial boundary value problem [7, 5] ∂ t (σ E) + curl (χ curl E) =-∂ t j in , E × n = 0 on Ŵ = ∂, E(· ,t = 0) = E 0 in , where E is the unknown electric field, χ denotes the inverse of the magnetic permeability, σ is the conductivity of the body occupying  and j(· ,t) is