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

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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.