四阶微分方程的谱和谱元方法

项目名称： 四阶微分方程的谱和谱元方法

项目编号： No.11426155

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 余旭洪

作者单位： 上海理工大学

项目金额： 3万元

中文摘要： 谱方法的主要特点是计算的高精度，并已广泛应用于流体力学、数值天气预报、统计物理和量子力学等有关问题的数值模拟。在以往的谱方法研究中更多的考虑二阶微分方程，并且已经取得许多研究成果。但是数学物理中大量问题可归结为四阶微分方程，至今这类问题的谱和谱元方法研究并不多见。在本项目中将针对二维四阶微分方程问题展开谱和谱元方法研究。本项目中首先考虑四边形区域上四阶问题的谱方法并建立相应的Legendre拟正交逼近理论。其次考虑复杂区域上四阶问题的谱元法并建立组合Legendre拟正交逼近理论。该项目的研究成果将拓展谱方法的应用范围，发展和丰富四阶微分方程的数值解法，并为科学和工程中有关问题的数值模拟提供一些原创性算法。

中文关键词： 谱方法；谱元方法；四阶偏微分方程；齐次化技巧；

英文摘要： The main advantage of spectral method is its high accuracy. It has been applied successfully to numerical simulations in many fields, such as fluid dynamics, numerical weather prediction, statistical physics and quantum mechanics. In the past, we mostly considered the second order differential equations, and have made a lot of results. But a lot of problems in the mathematical physics boil down to fourth-order differential equations. Whereas, so far, there is few results on the spectral method for such problems. The spectral method and spectral element method for the two-dimensional fourth-order differential equations will be researched in this project. We first consider the spectral method for fourth-order problems defined on quadrilaterals; establish the corresponding Legendre irrational orthogonal approximations. We then consider the spectral element method for fourth-order problems on complex domains, establish the corresponding composite Legendre quasi-orthogonal approximations. The research results of this project will expand the application of spectral method; develop and enrich the numerical methods for fourth-order problems. It will provide some original algorithm for the numerical simulation in the related problems of science and engineering.

英文关键词： spectral method；spectral element method；fourth-order partial differential equation；lifting technique；