Numerical Solution of Partial Differential Equations and Applications

研究者将设计、改进和分析由偏微分方程模拟的复杂物理现象的数值模拟方法，强调三个主要领域：弹性方程的混合方法、不连续伽辽金方法和计算广义相对论。在弹性方面，研究者将以最近的突破为基础，建立第一个具有多项式试函数的位移-应力公式的稳定混合有限元方法，并致力于开发更简单的非一致性混合有限元方法和扩展到三维弹性问题。关于不连续伽辽金方法——近似分段多项式函数不连续的有限元方法，并将修改纳入变分公式以达到一致性-研究者将与他的合作者一起建立在这些方法的统一和分类方面的最新工作，以开发一种统一的方法来分析和区分椭圆方程的大量不连续伽辽金方法。应用于含剪切弹性板的数值模拟。第三个也是最大的努力涉及爱因斯坦场方程的数值解，该方程与质量和时空曲率有关。这里的重点是理解爱因斯坦方程的基本性质，这些性质与它们的数值解最相关，以及困扰它们的先前数值模拟尝试的基本困难。这项工作的指导目标是模拟黑洞激励对的合并以及由此产生的引力辐射，这是引力物理学的一个基本问题，也是因为这种模拟对于实现基于引力波探测器的新一代天文台至关重要。计算机仿真是复杂工程结构设计和测试的重要工具。近几十年来，计算机模拟也加入了实验和理论，成为科学研究的主要范式之一。在这两个领域，许多最复杂的系统首先是由偏微分方程系统建模的——在偏微分方程系统中，微积分语言被用来表示相关物理量在空间和时间上的变化——然后这些微分方程系统必须用数值算法来近似，它利用现代计算机的力量，每秒执行数十亿次算术运算，以提取方程的解，达到所需的精度。近几十年来，这种算法的设计和验证原则已经为科学技术中遇到的许多基本微分方程系统开发出来，但许多更复杂的系统迄今为止还无法进行有效的计算，这就是本研究的重点。一个特别的重点将是在数值算法准确确定内部应力弹性结构，这是必不可少的建设安全和经济的工程结构。第二个重点将是开发模拟爱因斯坦广义相对论方程的方法，特别是用于预测引力辐射的输出——在时空曲面上传播的微小涟漪——来自黑洞碰撞等大规模宇宙事件。要实现目前正在建设的基于引力辐射的新型天文台的有效性，需要能够进行此类预测的计算机代码，这将为人类提供第一个了解占宇宙90%的暗物质的窗口。

The investigator will devise, improve, and analyze methods for the numerical simulation of complex physical phenomena modeled by partial differential equations, emphasizing three main areas: mixed methods for elasticity equations, discontinuous Galerkin methods, and computational general relativity. For elasticity, the investigator will build on a recent breakthrough that enabled the construction of the first stable mixed finite element methods for the displacement-stress formulations with polynomial trial functions, and also work towards the development of simpler nonconforming mixed finite element methods and extensions to three dimensional elasticity problems. Concerning discontinuous Galerkin methods--finite element methods in which the approximating piecewise polynomial functions are discontinuous, with modifications incorporated into the variational formulation to achieve consistency--the investigator will work with his collaborators to build on recent work on the unification and classification of such methods to develop a unified approach to the analysis of and discrimination among a wide class of discontinuous Galerkin methods for elliptic equations. Application will also be made to the numerical simulation of elastic plates incorporating shear. The third and largest effort concerns the numerical solution of Einstein's field equations relating mass and the curvature of space-time. The emphasis here will be on understanding the fundamental properties of the Einstein equations most relevant to their numerical solution and the basic difficulties that have beset previous attempts at numerical simulations of them. The work will be guided by the goal of simulating the coalescence of inspiralling pairs of black holes and the resulting emission of gravitational radiation, which is a problem of fundamental importance to gravitational physics and also because such simulations will be essential to realization of a new generation of observatories based on gravitational wave detectors.Computer simulation is a key tool for the design and testing of complex engineering structures. In recent decades computer simulation has also joined experiment and theory as one of the main paradigms of scientific investigation. In both areas, many of the most complex systems are first modeled by systems of partial differential equations--in which the language of calculus is used to express the variations of the relevant physical quantities in space and time--and then these systems of differential equations must be approximated by numerical algorithms, which harness the power of modern computers to perform billions of arithmetical operations a seconds to extract the solutions to the equations to the required degree of accuracy. In recent decades the principles for the design and validation of such algorithms have been developed for many of the basic systems of differential equations encountered in science in technology, but many more complex systems have so far resisted effective computation, and that is the thrust of this research. A particular emphasis will be on numerical algorithms for accurate determination of the stresses internal to elastic structures, which is essential to building safe and economic engineering structures. A second emphasis will be on developing methods to simulate Einstein's equations of general relativity, especially for predicting the output of gravitational radiation--minute ripples that propagate on the curved surface of space-time--from massive cosmological events such as black hole collisions. Computer codes capable of making such predictions are needed to realize the effectiveness of a new type of observatory based on gravitational radiation currently being constructed, which will provide mankind with its first window on the dark matter that makes up 90% of the universe.