Matthias K. Gobbert, Bradford E. Peercy, and Michael Muscedere, Department of Mathematics and Statistics, UMBC

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 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 parallel implementation of the algorithm allows for the solution on meshes with yet finer resolution than possible in serial.

Publications:

1. Matthias K. Gobbert, Parallel Performance Studies for an Elliptic Test Problem, Technical Report number HPCF-2008-1, UMBC High Performance Computing Facility, 2008. (HPCF machines used: hpc and kali.) PDF
2. Michael Muscedere and Matthias K. Gobbert, Parallel Performance Studies for a Parabolic Test Problem, Technical Report number HPCF-2008-2, UMBC High Performance Computing Facility, University of Maryland, Baltimore County, 2008. (HPCF machines used: hpc and kali.) PDF