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

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.