Local Error Estimators for Integral Equations: A Progress Report

Adaptive refinement procedures for numerical solutions of differential equations are in widespread use, but analogous approaches are not widely used with integral equations due to their perceived cost. During the past decade, a number of inexpensive local error estimators have been developed and assessed for perfectly conducting (PEC) targets modeled with flat triangular patches and surface integral equations (Kim and Peterson, “Evaluation of local error estimators for the RWG based EFIE,” IEEE Trans. Antennas Propagat., vol. 66, pp. 819–826, 2018; Braddock and Peterson, “Comparison of error estimators for the method of moments,” ACES Journal, vol. 35, pp. 1406–1407, Nov. 2020; Steines, Baxley, and Peterson, “Performance of inexpensive local error estimation techniques for integral equation numerical solutions,” IEEE J. Multiscale and Multiphysics Comp. Tech., vol. 8, pp. 158–164, 2023). These estimators are an essential part of an adaptive refinement algorithm for optimizing the performance of numerical solution schemes. Adaptive refinement involves adjusting the cell sizes or representation order in sub-regions of the target where the current density exhibits excessive error, to achieve necessary accuracy in a more efficient way than by uniform mesh refinement.