Spectral and Spectral Element Methods

该项目是 博伊德和舒尔茨正在 的开发与应用 谱方法 这一努力 出版了一本书，三十三本。 在过去的两年里， 从理论上分析 渐近切比雪夫 实用系数 应用于地球物理和 工程流程 在拟议的工作中，我们将 继续在数字上混合- 以应用为导向 问题研究 谱域 分解和特殊处理 奇点的数量将是主要的 主题. 我们将应用多- 勒让德元谱方法 解决两个问题： 椭圆轴颈轴承， 三维海洋概况 循环模型 在这些 工程研究，我们将 比较显性和隐性时间- 行进模式，多重网格， 各种平滑器（尤其是GMRES）， 以及不同的应对策略 有角点奇点， 战线 此外，我们将 进行理论研究 关于几个主题：加速求和 伪谱方法及其 与有限差分的关系， 伪谱比较 残差和误差，缩减基 Delves-Freeman迭代 求解强非线性边界 价值问题，以及光谱 前沿和冲击的方法。

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.