Discontinuous Petrov Galerkin Methods and Applications

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Computer simulation of many natural and technological processes relies on robust and efficient algorithms for solution of partial differential equations. In the continuing pursuit of such algorithms, a class of new Discontinuous Petrov-Galerkin (DPG) methods emerged as hybrid methods with a least-squares character. Their unusual stability and localization properties have the potential to expand the research frontiers in high performance computing. This project extends, improves, and identifies new applications for these DPG methods. These methods use a number of local operations, implementable on heterogenous computational clusters, to guarantee stability. Building on the method's known stability properties, the following five projects are proposed: (i) an explicit space-time extension of DPG methods providing a new way to simulate evolution of Friedrichs systems (ii) analysis and incorporation of adaptivity into DPG schemes thereby allowing computational resources to be allocated where they are needed most (iii) understanding DPG methods for harmonic wave phenomena in acoustics, elasticity, and electromagnetics, (iv) design of efficient preconditioners and other fast solution strategies for DPG methods, and (v) new tailor-made computational techniques to simulate biological pattern formation via chemotactic feedback.The proposed research is on a new method to solve partial differential equations, the Discoutinuous Petrov-Galerkin method. Impacts of development of the new method will apply to several areas, including propagation of acoustic, electromagnetic, and seismic waves in heterogenous media, and potential contributions in computational fluid dynamics applied to wind energy. The research component on biological patterns is inspired by questions of immediate relevance in health sciences, including a model for simulating tumor invasion, and simulation of chemotactic cancer cell movement, both intimately related to cells aggregating to form patterns. Finally, trained workforce additions will be accomplished by integrating graduate student involvement into the proposed research.