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

9311406 Bramble研究者和他的同事研究了现有近似方法中行为分析的新技术和方法，并介绍和分析了各种问题的数值近似的新算法。需要考虑的具体领域包括：i)快速求解所得到的代数方程的预条件迭代方法。ii)曲面边界和界面域的有限元方法。具有奇异点的微分方程和积分方程问题的有限元方法的局部行为。微分方程和积分方程中的超收敛现象。数值实验和算例对于发现和阐明各种数值方法具有重要作用。当然，实际应用是这里研究问题的一个激励因素。提出的研究的主要目的之一是加深对许多现有方法的理解，改进它们或开发处理新旧问题的新方法。这种理解对于在当前和未来的计算机环境中使用准确有效的算法的设计是至关重要的。研究人员正在研究有关工程和科学中感兴趣的问题的实际解决方案的各种问题。例如，这样的问题会产生非常大的方程组，研究者研究与计算它们的解有关的数学问题。其中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即使是最强大的计算机也无法解决的问题的技术。上述研究在结构力学、流体力学、地下水流动和电磁理论等应用领域具有重要意义。

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.