Collaborative Rsch: Adaptive Hybridized DG Methods for Acoustic and Electromagnetic Scattering

在这个项目中，研究人员和他的同事将开发，分析和实施准最佳自适应杂交间断Galerkin方法的声学和电磁散射问题所描述的亥姆霍兹方程和时间谐波麦克斯韦方程在二维和三维域使用一般的本地基地有界域。这些方法的特点是传统的分段多项式基地，以及特殊用途的波元素，和多级预处理迭代求解器的自适应细化单纯形，四边形和六面体网格。自适应网格细化是由残差型后验误差估计驱动的。特别是，该团队将证明自适应解过程的收敛性以及其在计算复杂性方面的准最优性，相对于一个适当指定的近似类。高质量的软件实现将用于测试算法并为分析提供信息。声学和电磁仿真数值方法的最新进展在多个应用中具有潜在的重大影响。例如，声波传播的更准确的模拟可以用于显著提高医学超声扫描的分辨率，从而扩大非侵入性诊断的范围。这个项目的重点是开发新的数值方法，计算物理声学和电磁学现象的可证明准确的近似。

In this project, the investigator and his colleagues will develop, analyze, and implement quasi-optimal adaptive hybridized Discontinuous Galerkin methods for acoustic and electromagnetic scattering problems as described by the Helmholtz equation and the time-harmonic Maxwell equation on bounded domains in two and three dimensional domains using general local bases. These methods feature traditional piecewise polynomial bases as well as special purpose wave elements, and multilevel preconditioned iterative solvers on adaptively refined simplicial, quadrilateral, and hexahedral meshes. The adaptive mesh refinement is driven by residual-type a posteriori error estimators. In particular, the team will prove convergence of the adaptive solution process as well as its quasi-optimality in terms of the computational complexity with respect to a properly specified approximation class. High quality software implementations will be used to test the algorithms and inform the analysis.Advances in the state of the art in numerical methods for performing acoustic and electromagnetic simulations have potentially high impact in several applications. As an example more accurate simulation of acoustic wave propagation can be used to considerably enhance resolution of medical ultra sound scans, and thus expand the reach of non-invasive diagnostics. The emphasis of this project is the development of novel numerical methods that compute provably accurate approximations of physical acoustics and electromagnetics phenomena.