### 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) |
---|---|

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 |

### Fingerprint

### 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

### Cite this

*Journal of Computational and Applied Mathematics*,

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

**A memory-efficient finite element method for systems of reaction-diffusion equations with non-smooth forcing.** / Hanhart, Alexander L.; Gobbert, Matthias K.; Izu, Leighton T.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 169, no. 2, pp. 431-458. https://doi.org/10.1016/j.cam.2003.12.035

}

TY - JOUR

T1 - A memory-efficient finite element method for systems of reaction-diffusion equations with non-smooth forcing

AU - Hanhart, Alexander L.

AU - Gobbert, Matthias K.

AU - Izu, Leighton T

PY - 2004/8/15

Y1 - 2004/8/15

N2 - 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.

AB - 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.

KW - Cluster computing

KW - Galerkin method

KW - Matrix-free iterative method

KW - Non-smooth data

KW - Reaction-diffusion equation

UR - http://www.scopus.com/inward/record.url?scp=4444290686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4444290686&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2003.12.035

DO - 10.1016/j.cam.2003.12.035

M3 - Article

AN - SCOPUS:4444290686

VL - 169

SP - 431

EP - 458

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 2

ER -