Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

For the adaptive solution of the Maxwell equations on three-dimensional domains with N´ed´elec edge finite element methods, we consider an implicit a posteriori error estimation technique. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. We show that the discrete bilinear form of the local problems satisfies an inf-sup condition ensuring the well posedness of the error equations. An adaptive algorithm is developed based on the estimated error. We show that the method accurately predicts the regions in the domain with a larger error. The performance of the method is tested on various problems on non-convex domains with non-smooth boundaries. The numerical results show an accurate approximation of the true error. On the meshes generated adaptively with the help of the implicit a posteriori error estimation technique an error is obtained which is smaller than on globally refined meshes. Moreover, the convergence of the error on the locally adapted meshes is faster than that on the globally refined mesh.