Adaptive Mortar Edge Element Methods in Electromagnetic Field Computation

We consider adaptive mortar edge element discretizations in the numerical solution of the quasi-stationary limit of Maxwell’s equations, also known as the eddy currents equations, in three space dimensions. The approach is based on the macro-hybrid variational formulation with respect to a geometrically conforming, non overlapping decomposition of the computational domain. Using adaptively generated hierarchies of locally quasi-uniform and shape regular simplicial triangulations of the subdomains, the edge element discretizations involve the lowest order curl-conforming edge elements of Nédélec’s first family. Due to the occurrence of nonmatching grids at the interfaces of adjacent subdomains, weak continuity of the tangential traces across the skeleton of the decomposition is realized by appropriately chosen Lagrange multipliers. The mortar edge element discretized problems give rise to the numerical solution of algebraic saddle point problems which is taken care of by a multilevel iterative solver featuring a hybrid smoother that involves an additional defect correction on the subspace of irrotational vector fields. Particular emphasis is on mesh adaptivity which is provided on the basis of an efficient and reliable residual-type a posteriori error estimator that relies on a Helmholtz decomposition of the error into an irrotational and weakly solenoidal part. As applications, we consider the computation of eddy currents in converter modules which are used as electric drives for high power electro motors. A further example addresses the numerical simulation of LWD (Logging While Drilling) tools used in oil exploration for in situ measurements of geological formations and thus illustrates that the developed algorithm can as well be used for the time-harmonic Maxwell equations.