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

9973328本研究的主要目标是构建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、Stokes和Navier-Stokes方程、线性弹性方程，以及构建、分析和测试求解所得到的离散系统的快速方法。重点将放在发展新的准确、稳健和高效的计算技术上。包括标准伽辽金有限元方法、最小二乘法和混合有限元方法在内的近似策略将从其准确性、稳定性和有效实施的角度进行研究。将开发的解决技术将强调通过域分解或多网格/多层方法进行预处理。局部网格细化和误差控制与多级求解方法相结合将在地下水和石油应用中得到应用。技术的发展越来越依赖于大规模的科学计算。这种计算通常受到数值方法的准确性和求解所需的计算量的限制。本提案资助的研究将为线性和非线性方程组的有效解提供更好的数值方法和新技术。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，例如污染物在多组分地下水系统中的行为。

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.