Spectral and Spectral Element Methods

这个项目是博伊德和舒尔茨正在进行的光谱方法开发和应用的延续。这一努力在过去两年里出版了一本书和33篇论文，从渐近切比雪夫系数的理论分析到地球物理和工程流程的实际应用。​在建议的工作中，我们将继续混合以数字为导向和以应用为导向的研究。谱域分解和奇点的特殊处理将是主要的主题。我们将把多元勒让德谱方法应用于两个问题：椭圆轴颈轴承的润滑和三维海洋环流模型。在这些工程研究中，我们将比较显式和隐式时间推进方案，具有各种平滑器的多网格（特别是GMRES），以及处理拐角奇点和锋面的不同策略。此外，我们将对几个主题进行理论调查：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减少基和Delves-Freeman迭代，以及锋面和冲击的谱方法。

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.