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

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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.