A number of pathologies related to the electrical activity in the heart can be studied using computer simulations of reaction-diffusion models. With the recent extracellular-membrane-intracellular (EMI) model, the geometry of each cell is resolved in the mesh, allowing for a more accurate representation of cardiac tissue on the cell scale. However, the EMI model requires a very fine mesh, and the linear systems arising from the diffusion process in the extracellular and the intracellular domains are ill-conditioned.

In this talk, we present an improved operator-splitting method that decouples the intracellular and extracellular domains, such that each sub-problem becomes a classical elliptic partial differential equation. Using this operator-splitting method, the computing time scales linearly with the problem size. This operator-splitting method enables us to solve the linear systems efficiently on shared-memory parallel computers, and we demonstrate that we are able to solve a system with 512 x 256 cardiac cells, solving linear systems with approximately 250 million degrees of freedom.