Design and Analysis of a Lightweight Parallel Adaptive Scheme for the Solution of the Monodomain Equation

Numerical simulation of the non-linear reaction-diffusion equations in computational electrocardiology requires locally high spatial resolution to capture the multiscale effects related to the electrical activation of the heart accurately, namely the strongly varying action potential. Here, we propose a novel lightweight adaptive algorithm which aims at combining the plainness of structured meshes with the resolving capabilities of unstructered adaptive meshes. Our “patch-wise adaptive” approach is based on locally structured mesh hierarchies which are glued along their interfaces by a non-conforming mortar element discretization. To further increase the overall efficiency, we keep the spatial meshes constant over suitable time windows in which error indicators are accumulated. This approach facilitates strongly varying mesh sizes in neighboring patches as well as in consecutive time steps. For the transfer of the dynamic variables between different spatial approximation spaces we compare the L2-projection and a local approximation. Finally, since an implicit-explicit time discretization is employed for stability reasons, we derive a spatial preconditioner which is tailored to the special structure of the patch-wise adaptive meshes. We analyze the (parallel) performance and scalability of the resulting method by several realistic examples from computational electrocardiology of different sizes. Additionally, we compare our method to a standard adaptive refinement strategy using unstructured meshes. As it turns out, our novel adaptive scheme provides a very good balance between reduction in degrees of freedom and overall (parallel) efficiency.