Scalable space-time adaptive simulation tools for computational electrocardiology
This work is concerned with the development of computational tools for the solution of reaction-diffusion equations from the field of computational electrocardiology. We designed lightweight spatially and space-time adaptive schemes for large-scale parallel simulations. We propose two different adaptive schemes based on locally structured meshes, managed either via a conforming coarse tessellation or a forest of shallow trees. A crucial ingredient of our approach is a non-conforming mortar element discretization which is used to glue together individually structured meshes by means of constraints. For the solution of variational problems in the proposed trial spaces we investigate two diametrically opposite approaches. First, we discuss the implementation of a matrix-free scheme for the solution of the monodomain equation on patch-wise adaptive meshes. Second, an approach to the construction of standard linear algebra data structures on tree-based meshes is considered. In particular, we address the element-wise assembly of stiffness matrices on constrained spaces via an algebraic representation of the inclusion map. We evaluate the performance of our adaptive schemes for small- and large-scale problems and demonstrate their applicability to the design of realistic large-scale heart models. In order to enable local time stepping in the context of (semi-)implicit integration schemes, we present a space-time discretization based on the proposed lightweight adaptive mesh data structures. By means of a discontinuous Galerkin method in time, the solution of the linear or non-linear system of equations is reduced to a sequence of smaller systems of adjustable size. We discuss the stabilization of the arising discrete problems and present extensive numerical evaluations of the space-time adaptive solution of the (1+1)-, (2+1)- and (3+1)-dimensional heat equation as well as the monodomain reaction-diffusion equation. Our results show both feasibility and potential of adaptive space-time discretizations for the solution of reaction-diffusion equations in computational electrocardiology.