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具有Beeler-Reuters离子膜动力学的Bidomain模型、Belousov-Zhabotinsky反应的布鲁塞尔模型和模式形成的图灵模型。自适应有限元方法特别适合于这些问题，并且确实被证明在无需用户干预的情况下求解各种偏微分方程组是有效的。该方案旨在开发高效、准确地求解三维反应扩散方程组的软件。自适应编码依赖于两个构建块，即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将被允许在不同方向上变化(各向异性细化)。将实现一种用于存储各向异性网格的新数据结构。自适应战略将与新的后验误差指标相结合，新的后验误差指标是根据提出者为线性要素制定的成功指标制定的。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条件GMRES求解。在程序执行过程中采用一种自适应的预条件选择策略来选择合适的预条件因子，许多物理和生物过程都可以用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。具体地说，心脏中电压传播的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。这些模型涉及简单和复杂几何结构上的大量方程。这项提议的目标是为研究人员，如生物医学工程师，提供必要的准确和高效的计算工具，以便在简单几何情况下分析他们的模型。然后，这些工具可以用于解决更复杂几何图形上的问题。这项工作是与杜兰大学一名生物医学工程师的工作一起进行的。

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.