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

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biddomain模型，Belousov-Zhabotinsky反应的Brusselator模型和图灵模式形成模型。自适应有限元方法特别适合于这些问题，并且确实被证明在不需要用户干预的情况下有效地求解各种偏微分方程。本课题旨在开发一种能够高效、准确地求解三维反应扩散系统的软件。自适应编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h和p将允许在不同方向上变化（各向异性细化）。将实现一种新的存储各向异性网格的数据结构。自适应策略将与新的后验误差指标相结合，该指标基于提出的线性元素的成功指标。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GMRES求解。自适应预处理选择策略将用于在程序执行期间选择适当的预处理。许多物理和生物过程可以用反应扩散方程来模拟。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。具体来说，心脏电压传播模型有助于理解心律失常的发生，并可能导致更好的治疗方法。这些模型涉及大量简单和复杂几何形状的方程。这项提议的目标是为研究人员，如生物医学工程师，提供准确和有效的计算工具，以便在简单几何形状的情况下分析他们的模型。这些工具可以用来解决更复杂的几何问题。这项工作是与杜兰大学一位生物医学工程师的工作结合进行的。

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.