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Accurate transcranial electrical stimulation (TES), electroconvulsive therapy (ECT), and electroencephalography (EEG) forward modeling requires resolving numerical singularities in the charge density near electrodes and tissue interfaces. We present an adaptive mesh refinement (AMR) strategy for the charge based boundary element method (BEM) accelerated by the fast multiple method (BEM-FMM) including electrode and interface singularities. We derive a new error estimator which considers both local and nonlocal contributions of the single-layer potential operator and construct a refinement criterion based on the difference in charge solution across AMR iterations. We evaluate this approach on a 5-layer sphere model and on multiple subject-specific head models derived from the 7-tissue SimNIBS (headreco) and 40-tissue Sim4Life (head40) segmentations, using both voltage-controlled and current-controlled electrode formulations. Through convergence analysis on the white matter and deep hippocampal targets, we find electric fields with relative residual errors below 0.1% and 1% for SimNIBS and Sim4Life models, respectively. Our results indicate that the residual based AMR applied to BEM-FMM leads to numerically stable TES and EEG forward solutions in realistic head models.
Drumm et al. (Sat,) studied this question.