Compatible and Nearly Compatible Finite Element Discretizations: Algorithms, Analysis and Applications

研究者正在研究传统有限元法与尖端不连续伽辽金法之间的联系。直到最近才发现，通过这些哲学上不同的方法离散的微分算子具有密切相关的特征谱。研究者正在综合这两个领域的分析和数值工具，以检查和提高后一种方法在使用高度非一致性离散化分辨率时的鲁棒性。此外，研究者和研究生正在创建一个面向对象的库，允许非专家通过简单直观的界面使用这些方法的组合。随着物理现象的计算模拟新技术的发展，确定它们在什么情况下以可预测的方式表现最佳是极其重要的。最近发展的不连续伽辽金模拟方法引起了人们的极大兴趣，因为它可以解决现有方法难以实现的大规模问题。例如，这些方法可以潜在地提高大型复杂飞机雷达散射建模的精度、效率和范围。研究人员正在研究如何预测这种新方法何时会给出物理上合理的解决方案，例如在计算下一代飞机产生的噪音时。这项调查的最终结果将是一套关于如何以及何时最好地使用这些方法的指导方针。此外，研究者正在开发一个软件库，通过简化物理模拟工具的新关键核心组件的快速原型和测试过程，该软件库将简化这些高分辨率方法的转移。

The investigator is studying connections between conventional finite elementmethods and the cutting edge discontinuous Galerkin method. Only recently hasit been discovered that the differential operators discretized via thesephilosophically distinct methods have closely related eigenspectra. Theinvestigator is synthesizing analytical and numerical tools from both fields toexamine and improve the robustness of the latter method when used with highlynon-conforming discretization resolution. In addition, the investigator and agraduate student are creating an object oriented library which allowsnon-experts to use combinations of these methods through a simple and intuitiveinterface.As new techniques for computational simulation of physical phenomena aredeveloped, it is extremely important to determine under what circumstances theyperform at their best and in a predictable way. There has been significantinterest in the recently developed discontinuous Galerkin simulation method,because it can solve large scale problems not readily attainable with existingmethods. For example, these methods can potentially increase the accuracy,efficiency, and scope of modeling radar scattering from large complex aircraft.The investigator is studying how to predict when this new method will givephysically reasonable solutions, for example in computing the noise generatedby next generation aircraft. The end product of this investigation will be aset of guidelines on how and when to best use the methods. Furthermore, theinvestigator is developing a software library which will ease transfer of thesehigh resolution methods by simplifying the process of rapid prototyping andtesting of new critical core components for physics simulation tools.