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

Bramble 9626567 该项目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、斯托克斯和纳维斯托克斯方程、线弹性方程、板弯曲问题，以及用于求解所得代数系统的快速方法的构造、分析和测试。 重点是开发新的准确、稳健和高效的计算技术。 特别从精度和稳定性的角度研究了涉及标准伽辽金有限元法、新型最小二乘法和混合有限元法的逼近策略。 开发的解决方案技术强调通过域分解或多网格/多级方法进行预处理。 该计划的总体目标是充分利用严格数学的力量来分析当前科学计算中使用的数值方法的行为，并开发新的算法，以提高准确性、稳定性和性能。 更高效和可扩展的算法可以对有效模拟器的开发产生重大影响，以了解具有国家重要性的物理过程。 例如，地下水过程的计算机建模涉及大型复杂非线性偏微分方程系统的求解。 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模拟、传热传质以及高性能计算等问题。

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.