On Objective-Aware Adaptive Methods in Computational Electromagnetics: Comparing Error Estimation and Indication Approaches for Problems with Singular Solutions

We summarize the application of goal-oriented adaptive methods in computational electromagnetics (CEM), that, as opposed to methods based purely on indicators, tailor discretizations in accordance with a collection of objectives, namely the accurate computation of certain quantities of interest (QoIs). When the accuracy of the goal separates from the properties of an underlying indicator (e.g., the jump in some functional of the solution), numerical results (and the predicted accuracy of those results) may subsequently diverge, rendering undetectably unreliable simulations. In the presence of singular, or otherwise burdensome solution behavior, which requires many degrees of freedom (DoFs) to model with acceptable accuracy, neglecting the objective in the refinement procedure may yield particularly poor results in addition to refinement inefficiencies. To elucidate these connections, we compare a goaloriented error estimation approach with a pure indicator methodology for the Maxwell eigenvalue problem, with a focus on an eigenpair with a singular eigenfunction. Driving ineffectual refinements, refinement based on indicators disconnected from the accuracy of the goal quantity can deliver poor convergence behavior in comparison to equivalent goal-oriented adaption.