Collaborative Research: Tuning-Free Adaptive Multilevel Discontinuous Galerkin Methods for Maxwell's Equations

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The investigator and colleagues are formulating, analyzing, and implementing adaptive multilevel Discontinuous Galerkin methods for coupled interior/exterior domain problems associated with the time-harmonic Maxwell equations. These advanced finite element methods are being realized as multilevel techniques on the basis of an adaptively generated hierarchy of triangulations of the computational domain. The research team is focusing on three central issues related to the basic steps `SOLVE', `ESTIMATE', `MARK', and `REFINE' of the adaptive loop. First, the smoothing process within the multilevel solver is performed only on the newly refined part of the triangulation obtained by a residual type a posteriori error estimator. Second, the a posteriori error analysis, which additionally has to take into account the effect of such local smoothing, aims to provide conditions guaranteeing a reduction of the global discretization error at each refinement step.Third, the selection of elements, faces and edges of the triangulation for refinement are based on a bulk criterion with an automatic (`tuning free') choice of the parameters controlling the amount of refinement in order to achieve optimal performance of the overall algorithm. Finally, the team is developing criteria to choose the parameters of artificial radiation boundary conditions automatically, such that no tuning on behalf of the user is required there as well.Simulation of electromagnetic phenomenais a particularly challenging problem in computational mathematics.The investigator and colleagues areestablishing a profound theoretical foundation for adaptive multilevel discontinuous Galerkin methods in electromagnetic field computations. They are developing a reliable algorithmic tool, of optimal computational complexity, that can be used for the numerical solution of challenging real-life problems in electrical engineering applications. The methods developed in this project have numerous technical and scientific applications, for instance semiconductor simulation or particle accelerator design. The results will be disseminated through publication of algorithms and results and reference computer codes being developed during this project will be made available to practitioners.