Collaborative Research: Tuning-free adaptive multilevel Discontinuous Galerkin methods for Maxwell's equations

研究者及其同事正在制定、分析和实现与时谐麦克斯韦方程组相关的耦合内外域问题的自适应多层不连续Galerkin方法。这些先进的有限元方法正在被实现为基于自适应生成的计算域三角剖分层次的多层技术。研究小组专注于与自适应回路的基本步骤“SOLVE”、“ESTIMATE”、“MARK”和“REFINE”相关的三个核心问题。首先，多层解算器内的平滑处理仅对残差型后验误差估计器获得的三角剖分新细化部分进行。其次，后验误差分析还必须考虑这种局部平滑的影响，其目的是提供条件，保证在每个细化步骤中减小全局离散化误差。第三，三角剖分的元素、面和边的选择是基于一个批量标准的，该标准具有自动（“无调优”）选择参数来控制改进的数量，以实现整体算法的最佳性能。最后，该团队正在开发自动选择人工辐射边界条件参数的标准，这样就不需要代表用户进行调整。电磁现象的模拟是计算数学中一个特别具有挑战性的问题。研究者和同事们正在为电磁场计算中的自适应多层不连续伽辽金方法建立一个深刻的理论基础。他们正在开发一种可靠的算法工具，具有最佳的计算复杂性，可用于电子工程应用中具有挑战性的现实问题的数值解决方案。在这个项目中开发的方法有许多技术和科学应用，例如半导体模拟或粒子加速器设计。结果将通过出版算法来传播，而在本项目期间开发的参考计算机代码将提供给从业人员。

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.