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

树莓9626567 该项目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，斯托克斯和Navier-Stokes方程，线性弹性方程，板弯曲问题，以及构建，分析和测试快速方法来解决由此产生的代数系统。 重点是发展新的准确，强大和高效的计算技术。 近似策略涉及标准的Galerkin有限元方法，新颖的最小二乘法，混合有限元方法进行了研究，特别是从他们的精度和稳定性的观点。 开发的解决方案的技术，强调通过区域分解或多重网格/多级方法的预处理。 该计划的广泛目标是利用严格数学的全部力量来分析目前在科学计算中使用的数值方法的行为，并开发具有更高精度，稳定性和性能的新算法。 更有效和可扩展的算法可以对有效的模拟器的发展产生重大影响，以了解国家重要的物理过程。 例如，地下水过程的计算机建模涉及非线性偏微分方程的大型复杂系统的解。 这些模拟的结果提供了重要的预测信息，可以影响地下水修复策略和国家保护计划的决策。 由于几何和物理过程的复杂性，有意义的模拟压倒了当今最强大的并行计算机，因此需要更有效的数值技术。 所开发的技术也适用于智能材料、传热传质和高性能计算等问题的模拟。

Bramble 9626567 The main objectives of this project 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, plate bending problems, and the construction, analysis, and testing of fast methods for solving the resulting algebraic systems. The emphasis is on the development of new accurate, robust and efficient computational techniques. Approximation strategies involving standard Galerkin finite element methods, novel least-squares methods, and mixed finite element methods are studied especially from the standpoint of their accuracy and stability. The solution techniques that are developed emphasize preconditioning via domain decomposition or multigrid/multilevel methods. The broad objective of this program is to use the full power of rigorous mathematics both to analyze the behavior of numerical methods currently used in scientific computation and to develop new algorithms exhibiting improved accuracy, stability and performance. More efficient and scalable algorithms can have a significant impact on the development of effective simulators for understanding physical processes of national importance. For example, computer modeling of ground water processes involve the solution of large complex systems of nonlinear partial differential equations. The results of these simulations provide important predictive information that can influence decisions with regard to ground water remediation strategies and national conservation programs. Due to the complexity in the geometry and the physical processes, meaningful simulations overwhelm today's most powerful parallel computers and hence the need for more effective numerical techniques. The techniques developed are applicable as well to problems such as simulation of smart materials, heat and mass transfer, and high performance computing.