Construction of Accurate, Robust and Efficient Numerical Techniques for Partial Differential Equations

9973328本研究的主要目标是构造和研究力学和工程的稳态和瞬态问题的有限元近似，包括对流扩散反应问题，斯托克斯和Navier-Stokes方程，线性弹性方程，以及构造，分析和测试求解离散系统的快速方法。 重点将放在发展新的准确，强大和有效的计算技术。 近似策略涉及标准的伽辽金有限元法，最小二乘法，混合有限元法将研究，特别是从他们的精度，稳定性和有效的实施的立场。 将开发的解决方案技术将强调通过区域分解或多重网格/多级方法的预处理。 局部网格加密和误差控制与多级求解方法相结合将用于地下水和石油应用。技术发展越来越依赖于大规模科学计算。 这种计算通常受到数值方法的精度和求解所需计算量的限制。 这项建议资助的研究将提供更好的数值方法和新技术，有效地解决由此产生的线性和非线性方程组。 这些进步将提高我们准确模拟更复杂过程（设备）的能力，以及我们理解复杂系统的能力，例如多组分地下水系统中污染物的行为。

9973328The main objectives of this research are the construction and study of finite element approximations to steady-state and transient problems of mechanics and engineering, including convection-diffusion-reaction problems, Stokes and Navier-Stokes equations, equations of linear elasticity, and the construction, analysis, and testing of fast methods for solving the resulting discrete systems. The emphasis will be on the development of new accurate, robust and efficient computational techniques. Approximation strategies involving standard Galerkin finite element methods, least-squares methods, and mixed finite element methods will be studied especially from the standpoint of their accuracy, stability, and efficient implementation. The solution techniques which will be developed will emphasize preconditioning via domain decomposition or multigrid/multilevel methods. Local grid refinement and error control coupled with multilevel solution methods will be used in ground water and petroleum applications.Technological development is becoming increasingly dependent on large scale scientific computation. Such computations are often limited by accuracy of the numerical method and the amount of computational effort required for its solution. The research funded by this proposal will provide better numerical methods and new techniques for the efficient solution of the resulting system of linear and non-linear equations. Such advances will improve our ability to accurately model more complex processes (devices) as well as our ability to understand complex systems such as the behavior of a contaminant in a multi-component groundwater system.