Applications of Adjoint Solutions for Predicting and Analyzing Numerical Error of Forward Solutions Based on Higher Order Finite Element Modeling

摘要

Summary form only given. The fi nite element method (FEM) for discretizing partial differential equations in electromagnetics is an extremely powerful and versatile general numerical methodology for electromagnetic (EM) fi eld modeling and computation in microwave engineering. This paper addresses numerical error estimation, model sensitivity prediction, and adaptive mesh refinement in the context of FEM solutions to EM scattering problems. The paper explains the concept of the adjoint operator and describes applications of adjoint solutions for predicting and analyzing numerical error of forward solutions based on higher order FEM modeling, with a perfectly matched layer (PML) boundary conditions for domain truncation. We present examples of application of an adjoint operator to quantify sensitivity of a quantity of interest (QoI) to perturbations in an input parameter in three-dimensional (3-D) higher order FEM-PML EM scattering computation, where we use the sensitivity information to predict the QoI over the parameter domain. In a large variety of illustrative one-dimensional (1-D) higher order FEM-PML EM scattering problems, we compute the error estimate from the numerical forward and adjoint solutions on an element -by element basis and analyze element -wise contribution to the total error in a given QoI from the forward solution. We then discuss the usefulness of this information for adaptive mesh refinement, considering both h- and p -refinements. We demonstrate that developing efficient strategies for adaptive mesh refinement requires accounting for local cancellation of the element contributions to the error. We show that adjoint methods present a useful technique toward a posteriori error estimation and adaptive mesh refinement for the FEM computation of EM scattering.