RIA: An Adaptive Spectral Element Method for the Solution ofTransient Partial Differential Equations in Complex Geometries

本研究的主要目标是利用元算法求解时变偏微分方程，以模拟复杂几何中的物理现象。自适应方法通过自动细化，保证计算机资源在存储和cpu时间方面的最优分配。粗化或重构几何网格，使方程在整个计算过程中在空间上离散化。该方法将以一般的方式发展，因为创新将独立于要模拟的几何和方程。这种方法将有利于应用高阶谱方法模拟复杂物理现象的许多领域。主要任务是构建基于规则的系统进行细化，针对不断变化的网格连通性信息制定动态空间分配方案，提出了网格优化准则，保证了网格变化的时效性和保守性，改进了现有的误差估计器和预校正器。这项研究需要理论和计算工作，并将包括一些测试和新颖的模拟。谱元法的自适应表述旨在提高一般高阶方法的灵活性和能力范围。高阶方法提供高度精确的模拟，具有非常快的收敛速度，通常用于基本物理现象的直接模拟。拟议的研究将允许高阶方法来处理“真正的”工程问题，而不是理想化的研究问题，通过将所需的计算机资源减少一到两个数量级

The main objective of this research initiation project (RIA) element algorithm for the solution of time dependent partial differential equations to simulate physical phenomena in complex geometries. The adaptive method will ensure optimal allocation of computer resources in terms of storage and cpu time by automatically refining, coarsening or reconstructing the geometric mesh used to discretize the equations in space throughout the calculation. The method wil be developed ina general way, in that the innovations will be independent of the geometry and equations to be simulated. This approach will benefit the many fields in which the application of high order spectral methods can be used for simulation of complex physical phenomena. THe main tasks are the construction of a rule-based system for refinement, the development of a scheme for dynamic allocation of space for the changing mesh connectivity information, the development of a mesh optimization criterion, ensuring that the mesh alteration is time-accurate and conservative and finally improving the current error estimators and preconditioners. The research entails theoretical and computational work and will include a number of test and novel simulations. The adaptive formulation of the spectral element method is aimed at increasing the flexibility and range of capabilities of high order methods in general. High order methods provide highly accurate simulations with very fast rates of convergence and are typically used in the direct simulation of basic physical phenomena. The proposed research will allow high order methods to deal with "real" engineering problems, rather than idealized research problems, by reducing the needed computer resources by one to two orders of magnitude.//