Adaptive Methods for Systems of Reaction-Diffusion Equations in Three Space Dimensions

三维空间反应扩散方程组的自适应方法

• 批准号: 9973048
• 负责人: Peter Moore
• 金额: $ 7.39万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2001-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973048&HistoricalAwards=false
• 关键词: Adaptive; Methods; Systems; Reaction; Diffusion

9973048Systems of reaction-diffusion equations occur frequently in scientific and engineering applications. Some important examples include the bidomain model with the Beeler-Reuter ionic membrane kinetics in cardiac electrophysiology, the Brusselator model of the Belousov-Zhabotinsky reaction and Turing models of pattern formation Adaptive finite element methods are particularly well-suited to these problems and indeed have proved effective in solving a wide variety of partial differential equations without requiring user intervention. This proposal aims to develop software for solving reaction-diffusion systems in three dimensions efficiently and accurately. Adaptive codes depend on two building blocks, a posteriori error estimates and adaptive grid strategies. The finite element method will employ hp-adaptivity where h and p will be allowed to vary in different directions (anisotropic refinement). A new data structure for storing anisotropic grids will be implemented. The adaptive strategy will be coupled with new a posteriori error indicators based on successful indicators developed by the proposer for linear elements. The large system of linear equations arising from the temporal and spatial discretization process will be solved using preconditioned GMRES. An adaptive preconditioning selection strategy will be used to choose the appropriate preconditioner during program execution.Many physical and biological processes can be modeled by reaction-diffusion equations. Some important examples include pattern formation in biological systems and voltage propagation in the heart. Specifically models for voltage propagation in the heart aid in understanding the onset of cardiac arrhythmias and may lead to better therapies. These models involve a large number of equations over both simple and complex geometries. The goal of this proposal is to provide researchers, such as biomedical engineers, with the accurate and efficient computational tools necessary in order to analyze their models in the case of simple geometries. These tools can then be used in solving problems on more complex geometries. This work is being carried out in conjunction with the work of a biomedical engineer at Tulane University.

9973048反应扩散方程组在科学和工程应用中经常出现。 一些重要的例子包括心脏电生理学中 Beeler-Reuter 离子膜动力学的双域模型、Belousov-Zhabotinsky 反应的 Brusselator 模型和图案形成的图灵模型。自适应有限元方法特别适合这些问题，并且确实已经证明无需用户干预即可有效求解各种偏微分方程。 该提案旨在开发用于高效、准确地求解三维反应扩散系统的软件。 自适应代码依赖于两个构建块：后验误差估计和自适应网格策略。 有限元方法将采用 hp 自适应，其中允许 h 和 p 在不同方向上变化（各向异性细化）。 将实施用于存储各向异性网格的新数据结构。 自适应策略将与基于线性元素提案者开发的成功指标的新后验误差指标相结合。 由时间和空间离散化过程产生的大型线性方程组将使用预处理的 GMRES 进行求解。 自适应预处理选择策略将用于在程序执行期间选择合适的预处理器。许多物理和生物过程可以通过反应扩散方程来建模。 一些重要的例子包括生物系统中的模式形成和心脏中的电压传播。 心脏中电压传播的具体模型有助于了解心律失常的发生，并可能带来更好的治疗方法。 这些模型涉及大量简单和复杂几何形状的方程。 该提案的目标是为生物医学工程师等研究人员提供必要的准确有效的计算工具，以便在简单几何形状的情况下分析他们的模型。 然后可以使用这些工具来解决更复杂的几何形状的问题。 这项工作是与杜兰大学生物医学工程师的工作结合进行的。