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

Alexander L. Hanhart, Matthias K. Gobbert, Leighton T Izu

Research output: Contribution to journalArticle

17 Scopus citations

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 languageEnglish (US)
Pages (from-to)431-458
Number of pages28
JournalJournal of Computational and Applied Mathematics
Volume169
Issue number2
DOIs
StatePublished - Aug 15 2004
Externally publishedYes

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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