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

• Published in: Journal of computational and applied mathematics 2004-08, Vol.169 (2), p.431-458
• Main Authors: Hanhart, Alexander L., Gobbert, Matthias K., Izu, Leighton T.
• Format: Article
• Language: English
• Subjects: Cluster computing; Exact sciences and technology; Galerkin method; Mathematical analysis; Mathematics; Matrix-free iterative method; Methods of scientific computing (including symbolic computation, algebraic computation); Non-smooth data; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Reaction–diffusion equation; Sciences and techniques of general use
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2003.12.035

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.