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 phenomena is a particularly challenging problem in computational mathematics. The investigator and colleagues are establishing 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.
This Collaborative Research Project has focused on the development, analysis and implementation of adaptive multilevel Discontinuous Galerkin (DG) and Hybridized DG (H-DG) methods for efficient and accurate numerical solution of the frequency domain Maxwell’s equations. Methods and algorithms that produce high quality solutions to the ubiquitous Maxwell's equations describing electromagnetic phenomena are broadly applicable in science and engineering. Paramount applications include improving the numerical capabilities of design tools for modeling antennae, electrical circuits, transmission wires and similar applications, for instance, the design of biomedical micro-electro-mechanical devices. The main results that have been obtained are in the area of local multigrid methods on adaptively refined meshes both for H(curl)-elliptic problems and the time-harmonic Maxwell equations. The characteristic feature of this combination of local mesh refinement, and local multigrid solvers is that using localized residual-type a posteriori error estimators and local solution iteration directs computational effort to the part of the domain where the solution needs most improvement and this has been shown to reduce substantially the overall time to obtain a sufficiently accurate solution. Interior Penalty DG (IPDG) as well as hybridized IPDG (IPDG-H) methods have been considered for both problem classes as well as for acoustic problems as described by Helmholtz type equations. Mathematical convergence results have been derived that prove the algorithms generate a sequence of approximate solutions that converge to physically realistic and accurate solutions.
