Mathematical Sciences: Algorithms and Numerical Analysis forDifferential Equations

这项研究将是关于稳态和非稳态问题的偏微分方程数值解的算法的制定和分析方法的发展。具体主题包括构造用于并行计算机体系结构的迭代算法的前提条件，区域分解技术，具有奇异摄动结构的问题，角点奇点的处理，现有有限元的超收敛近似，具有时间相关系数的抛物型方程，具有转折点的对流占优问题的流线扩散法，自适应网格细化和污染效应等。当今科学和工程中的技术问题是如此复杂，以至于最现实的问题需要计算机和超级计算机的帮助下的数值解。这里的研究对于设计在现在和未来的计算机环境中使用的准确和高效的算法是至关重要的。

The research will be in the formulation of algorithms for the numerical solutions of partial differential equations for stationary and non-stationary problems and the development of methods for their analysis. Specific topics include construction of preconditions for use in iterative algorithms for parallel computer architectures, domain decomposition techniques, problems with singular perturbation structures, treatment of corner singularities, superconvergent approximations for existing finite elements, parabolic equations with time dependent coefficients, streamline diffusion method for convection dominated problems with turning points, adaptive mesh refinements and pollution effects etc. Technical problems in science and engineering are so complex today that most realistic ones will require numerical solutions with the help of computers and supercomputer. The research here is central to the design of accurate and efficient algorithms for use in present and future computer environments.