Discontinuous Petrov Galerkin Methods and Applications

间断 Petrov Galerkin 方法及应用

• 批准号: 1318916
• 负责人: Jay Gopalakrishnan
• 金额: $ 30.2万
• 依托单位: Portland State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318916&HistoricalAwards=false
• 关键词: Discontinuous; Petrov; Galerkin; Methods; Applications

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.

许多天然和技术过程的计算机模拟依赖于偏微分方程解决方案的鲁棒和有效算法。在继续追求此类算法时，一类新的不连续的彼得 - 盖尔金（DPG）方法出现为具有最小二乘特征的混合方法。它们的异常稳定性和本地化属性有可能在高性能计算中扩展研究前沿。该项目扩展，改进和确定这些DPG方法的新应用程序。这些方法使用许多可在异质计算簇上实现的本地操作来确保稳定性。 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电磁技术的设计有效的预处理和其他快速解决方案策略的DPG方法以及（v）新的量身定制的计算技术，以通过趋化反馈模拟生物模式形成。拟议的研究是对一种新方法，用于求解部分微分方程的新方法。新方法开发的影响将适用于多个领域，包括在异源培养基中传播声学，电磁波和地震波的传播，以及用于风能的计算流体动力学的潜在贡献。生物学模式的研究成分灵感来自健康科学中直接相关性的问题，包括模拟肿瘤侵袭的模型，以及模拟趋化性癌细胞运动的模型，这两者都与汇总到形成模式的细胞密切相关。最后，通过将研究生的参与纳入拟议的研究来实现训练有素的劳动力增加。