### Abstract

The release of calcium ions in a human heart cell is modeled by a system of reaction-diffusion equations, which describe the interaction of the chemical species and the effects 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 nonsmooth 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 nonsmooth 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.

Original language | English (US) |
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Pages (from-to) | 431-458 |

Number of pages | 28 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 169 |

Issue number | 2 |

DOIs | |

State | Published - Aug 15 2004 |

Externally published | Yes |

### Keywords

- Cluster computing
- Galerkin method
- Matrix-free iterative method
- Non-smooth data
- Reaction-diffusion equation

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

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## Cite this

*Journal of Computational and Applied Mathematics*,

*169*(2), 431-458. https://doi.org/10.1016/j.cam.2003.12.035