Mathematical Sciences: Algorithms and Numerical Analysis for Partial Differential and Integral Equations

小行星9311406 研究人员和他的同事研究新的技术和方法的分析行为在现有的近似方法，并介绍和分析新的算法的数值逼近的各种问题。 需要考虑的具体领域包括：i）快速求解所得代数方程的预处理迭代法。 ii）具有弯曲边界和界面的域中的有限元方法。 （3）奇异微分和积分方程问题的有限元方法的局部性态。 iv）微分和积分方程中的超收敛现象。 数值试验和算例将在各种数值方法的发现和阐明中发挥重要作用。 当然，实际应用是这里研究的问题的一个激励因素。 拟议研究的主要目的之一是加深对许多现有方法的理解，改进这些方法或开发处理新老问题的新方法。 这种理解对于设计用于当前和未来计算机环境的准确和有效的算法至关重要。 调查人员正在研究各种问题，这些问题涉及工程和科学中感兴趣的问题的实际解决方案。 例如，这些问题会产生非常大的方程组，研究人员研究与计算其解相关的数学问题。 一个目标是构建比现有方法更快、更准确的方法。 第二个目标是发明和研究解决迄今为止甚至超出最强大的计算机范围的问题的技术。 上述研究非常重要的一些应用领域是结构力学、流体动力学、地下水流和电磁理论。

9311406 Bramble The investigator and his colleague study new techniques and methods of analysis of behavior in existing approximation methods, and introduce and analyze new algorithms for the numerical approximation of various problems. Specific areas to be considered include: i) Preconditioned iterative methods for fast solution of resulting algebraic equations. ii) Finite element methods in domains with curved boundaries and interfaces. iii) Local behavior of finite element methods in differential and integral equation problems with singularities. iv) Superconvergence phenomena in differential and integral equations. Numerical experimentation and examples will play important roles in the discovery and elucidation of various numerical methods. Of course, practical application is a motivating factor for the problems studied here. One of the principal aims of the studies proposed is to develop a deeper understanding of many existing methods, improve them or develop new ways of treating old and new problems. This kind of understanding is central to the design of accurate and efficient algorithms for the use in present and future computer environments. The investigators are researching various questions concerning the practical solution of problems of interest in engineering and science. For example, such problems give rise to extremely large sets of equations and the investigators study mathematical questions related to computing their solutions. One aim is to construct methods that are faster and more accurate than existing ones. A second aim is to invent and study techniques for the solution of problems heretofore beyond the scope of even the most powerful computers. Some areas of application in which the above mentioned studies are very important are Structural Mechanics, Fluid Dynamics, Groundwater flow and Electromagnetic theory.