Discontinuous Petrov Galerkin Methods and Applications

许多自然和技术过程的计算机模拟依赖于求解偏微分方程的鲁棒和高效算法。在对这类算法的不断追求中，出现了一类新的具有最小二乘特征的不连续彼得罗夫-伽辽金（DPG）方法。它们不寻常的稳定性和局域性有可能扩展高性能计算的研究前沿。该项目扩展、改进并确定了这些DPG方法的新应用。这些方法使用许多可在异构计算集群上实现的本地操作来保证稳定性。基于该方法已知的稳定性特性，提出了以下五个项目：(i) DPG方法的明确时空扩展提供了一种模拟Friedrichs系统演化的新方法；（ii）分析并将自适应纳入DPG方案，从而允许将计算资源分配到最需要的地方；（iii）理解声学、弹性和电磁学中谐波现象的DPG方法；（iv）设计有效的预处理器和DPG方法的其他快速解决策略。(v)新的定制计算技术，通过趋化反馈来模拟生物模式的形成。提出了一种求解偏微分方程的新方法——非连续Petrov-Galerkin法。新方法的发展影响将应用于几个领域，包括声波、电磁波和地震波在非均质介质中的传播，以及应用于风能的计算流体动力学的潜在贡献。生物模式的研究部分受到健康科学中直接相关问题的启发，包括模拟肿瘤侵袭的模型，以及模拟趋化癌细胞运动，两者都与细胞聚集形成模式密切相关。最后，通过将研究生参与到拟议的研究中来完成训练有素的劳动力增加。

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.