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Accelerated adaptive numerical methods for computational electromagnetics: enhancing goal-oriented approaches to error estimation, refinement, and uncertainty quantification




Harmon, Jake J., author
Notaroš, Branislav M., advisor
Estep, Don, committee member
Ilić, Milan, committee member
Oprea, Iuliana, committee member

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This dissertation develops strategies to enhance adaptive numerical methods for partial differential equation (PDE) and integral equation (IE) problems in computational electromagnetics (CEM). Through a goal-oriented emphasis, with a particular focus on scattered field and radar cross-section (RCS) quantities of interest (QoIs), we study automated acceleration techniques for the analysis of scattering targets. A primary contribution of this work, we propose an error prediction refinement strategy, which, in addition to providing rigorous global error estimates (as opposed to just error indicators), promotes equilibration of local error contribution estimates, a key requirement of efficient discretizations. Furthermore, we pursue consistent exponential convergence of the QoIs with respect to the number of degrees of freedom without prior knowledge of the solution behavior (whether smooth or otherwise) or the sensitivity of the QoIs to the discretization quality. These developments, in addition to supporting significant reductions in computation time for high accuracy, offer enhanced confidence in simulation results, promoting, therefore, higher quality decision making and design. Moreover, aside from the need for rigorous error estimation and fully automated discretization error control, practical simulations necessitate a study of uncertain effects arising, for example, from manufacturing tolerances. Therefore, by repeating the emphasis on the QoI, we leverage the computational efforts expended in error estimation and adaptive refinement to relate perturbations in the model to perturbations of the QoI in the context of applications in CEM. This combined approach permits simultaneous control of deterministic discretization error and its effect on the QoI as well as a study of the QoI behavior in a statistical sense. A substantial implementation infrastructure undergirds the developments pursued in this dissertation. In particular, we develop an approach to conducting flexible refinements capable of tuning both local spatial resolution ($h$-refinements) and enriching function spaces ($p$-refinements) for vector finite elements. Based on a superposition of refinements (as opposed to traditional refinement-by-replacement), the presented $hp$-refinement paradigm drastically reduces implementation overhead, permits straightforward representation of meshes of arbitrary irregularity, and retains the potential for theoretically optimal rates of convergence even in the presence of singularities. These developments amplify the utility of high-quality error estimation and adaptive refinement mechanisms by facilitating the insertion of new degrees of freedom with surgical precision in CEM applications. We apply the proposed methodologies to a strong set of canonical targets and benchmarks in electromagnetic scattering and the Maxwell eigenvalue problem. While directed at time-harmonic excitations, the proposed methods readily apply to other problems and applications in applied mathematics.


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adaptive refinement
error estimation
uncertainty quantification
boundary element methods
adaptive methods
finite element methods


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