Collaborative Research: 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、分析和实现准最优自适应杂交间断Galerkin方法，用于求解二维和三维区域中有界域上的Helmholtz方程和时间调和Maxwell方程所描述的声波和电磁散射问题，使用一般的局部基。这些方法具有传统的分段多项式基和特殊用途的波元，以及自适应细化的单纯形、四边形和六面体网格上的多级预条件迭代求解器。自适应网格加密由残差型后验误差估值器驱动。特别是，该团队将证明自适应解过程的收敛以及其相对于适当指定的近似类的计算复杂性的准最优性。将使用高质量的软件实现来测试算法并为分析提供信息。执行声学和电磁模拟的数值方法的最新进展在几个应用中具有潜在的高影响。例如，可以使用更准确的声波传播模拟来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，计算出物理声学和电磁现象的可证明的准确近似。

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.