This work focuses on developing computational tools for solving reaction-diffusion equations in computational electrocardiology. We designed lightweight, adaptive schemes for large-scale parallel simulations, proposing two adaptive methods based on locally structured meshes, managed through either conforming coarse tessellations or shallow tree forests. A key aspect of our approach is a non-conforming mortar element discretization that connects individually structured meshes using constraints. We explore two contrasting methods for solving variational problems in our trial spaces. The first involves implementing a matrix-free scheme for the monodomain equation on patch-wise adaptive meshes. The second considers constructing standard linear algebra data structures on tree-based meshes, specifically addressing element-wise assembly of stiffness matrices on constrained spaces through an algebraic representation of the inclusion map. We assess the performance of our adaptive schemes and their application in designing realistic large-scale heart models. To facilitate local time stepping in (semi-)implicit integration schemes, we introduce a space-time discretization based on our lightweight adaptive mesh structures. Using a discontinuous Galerkin method in time, we reduce the solution of linear or non-linear systems to smaller, adjustable systems. We discuss the stabilization of these discrete problems and present extensive numeric
