Spectral and Spectral Element Methods

该项目是博伊德和舒尔茨持续开发和应用光谱方法的延续。 这项工作在过去两年中出版了一本书和三十三篇论文，范围从渐近切比雪夫系数的理论分析到地球物理和工程流的实际应用。 在拟议的工作中，我们将继续将数值导向与应用导向的研究结合起来。 谱域分解和奇点的特殊处理将是主要主题。 我们将应用多元素勒让德谱方法来解决两个问题：椭圆轴颈轴承的润滑和树维海洋环流模型。 在这些工程研究中，我们将比较显式和隐式时间推进方案、具有各种平滑器（特别是 GMRES）的多重网格，以及处理角奇点和前沿的不同策略。 此外，我们还将对几个主题进行理​​论研究：和加速伪谱方法及其与有限差分的关系、伪谱残差和误差的比较、用于解决强非线性边值问题的简化基和Delves-Freeman迭代，以及锋和冲击的谱方法。

This project is the continuation of Boyd and Schultz's ongoing development and application of spectral methods. This effort has produced one book and thirty-three papers over the past two years, ranging form theoretical analyses of asymptotic Chebyshev coefficients to practical applications to geophysical and engineering flows. In the proposed work, we will continue to mix numerically- oriented with application-oriented studies. Spectral domain decomposition and special handling of singularities will be major themes. We will apply multi- element Legendre spectral methods to two problems: lubrication in an elliptical journal bering, and a tree-dimensional ocean general circulation model. In these engineering studies, we will compare explicit and implicit time- marching schemes, multigrid with various smoothers (notably GMRES), and different strategies for coping with corner singularities and fronts. In addition, we will perform theoretical investigations on several topics: sum-accelerated pseudospectral methods and their relationship to finite differences, comparison of pseudospectral residuals and errors, reduced basis and Delves-Freeman iterations for solving strongly nonlinear boundary value problems, and spectral methods for fronts and shocks.