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.
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
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