A multilevel adaptive approach for computational cardiology

A space-time adaptive mesh refinement (AMR) algorithm is proposed for cardiac problems, arising from numerical modeling of electrical wave propagation in the heart. A cardiac problem, such as the bidomain or monodomain model, consists of a singularly perturbed reaction-diffusion system coupled with a set of nonlinear stiff ordinary differential equations. The algorithm first uses an operator splitting technique to separate linear diffusion from nonlinear stiff reactions in the model problem to be solved. The reactions are integrated adaptively with a second-order singly diagonally implicit Runge-Kutta method. The decoupled linear diffusion is implicitly discretized with a conforming finite element approximation on adaptively refined grids, which are dynamically created by the AMR algorithm. The resulting composite grid equations are solved by a standard multilevel/multigrid iteration algorithm. The AMR algorithm uses quadrilateral or hexahedral elements to construct a hierarchy of properly nested level grids. The grid on each level is generated through regular bisection or refinement from a subgrid of that on the previous coarser level. We represent a grid by lists of elements and the lower dimensional facets, including nodes, edges and sides. With this grid representation, the connectivity of grid entities is well established, which makes grid-based operations easy to implement. Since the composite grids in the hierarchy do not match along coarse-fine grid interfaces, the Steklov-Poincare continuity conditions are weakly enforced, which results in a conforming finite element discretization on composite grids. The AMR algorithm follows Berger-Oliger's approach in time stepping. The hierarchical system is recursively integrated. Starting from the root level, a coarse level is first integrated with a large time step, followed by integrating its next finer level with a few small time steps until both levels are synchronized. At the moment of synchronization, the algorithm invokes some of the typical routines such as mesh regridding and data up-/downscaling. An error estimation technique of Richardson extrapolation type is proposed for tagging of cells. Numerical experiments with the AMR algorithm are presented at the end as part of the thesis work. It is demonstrated that the convergence rate of our AMR algorithm is second-order. Numerical results also show the numerical accuracy, algorithm efficiency, geometry flexibility and the promising application to realistic simulations of our AMR algorithm.