Mathematical Sciences: Algorithms and Numerical Analysis forDifferential Equations

该研究将在定常和非定常问题的偏微分方程数值解的算法的制定和它们的分析方法的发展。具体主题包括构建用于并行计算机架构的迭代算法的前提条件、域分解技术、奇异摄动结构问题、角点奇点的处理、现有有限元的超收敛逼近、具有时间相关系数的抛物方程、对流主导问题的流线扩散法，包括拐点、自适应网格细化和污染效应等。今天，科学和工程中的技术问题是如此复杂，以至于大多数现实问题都需要借助计算机和超级计算机进行数值求解。这里的研究是在当前和未来的计算机环境中使用的准确和有效的算法设计的核心。

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.