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A Memory-Efficient Finite Element Method for Systems of Reaction-Diffusion Equations with Non-Smooth Forcing (2003)
| Citations: | 17 - 12 self |
BibTeX
@MISC{Hanhart03amemory-efficient,
author = {Alexander L. Hanhart and Matthias K. Gobbert and Leighton T. Izu},
title = {A Memory-Efficient Finite Element Method for Systems of Reaction-Diffusion Equations with Non-Smooth Forcing},
year = {2003}
}
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Abstract
The release of calcium ions in a human heart cell is modeled by a system of reactiondi #usion equations, which describe the interaction of the chemical species and the e#ects of various cell processes on them. The release is modeled by a forcing term in the calcium equation that involves a superposition of many Dirac delta functions in space; such a non-smooth right-hand side leads to divergence for many numerical methods. The calcium ions enter the cell at a large number of regularly spaced points throughout the cell; to resolve those points adequately for a cell with realistic three-dimensional dimensions, an extremely fine spatial mesh is needed. A finite element method is developed that addresses the two crucial issues for this and similar applications: Convergence of the method is demonstrated in extension of the classical theory that does not apply to non-smooth forcing functions like the Dirac delta function; and the memory usage of the method is optimal and thus allows for extremely fine three-dimensional meshes with many millions of degrees of freedom, already on a serial computer. Additionally, a coarse-grained parallel implementation of the algorithm allows for the solution on meshes with yet finer resolution than possible in serial.
Keyphrases
memory-efficient finite element method reaction-diffusion equation calcium ion realistic three-dimensional dimension many numerical method many million similar application chemical specie finite element method many dirac delta function serial computer forcing term dirac delta function large number human heart cell coarse-grained parallel implementation crucial issue calcium equation non-smooth forcing function fine three-dimensional mesh memory usage non-smooth right-hand side various cell classical theory reactiondi usion equation fine spatial mesh
