**Series in Computational Science
edited by Illia Horenka, Rolf
Krause, Olaf Schenk
Volume 4**

Dorian Krause

**Scalable
Space-Time Adaptive Simulation Tools for **

**Computational Electrocardiology **

First Edition 2014. 184
pages, EUR 64,00

ISBN 978-3-86628-494-4

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 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 equation. Our
results show both feasibility and potential of adaptive space-time discretizations for the solution of reaction-diffusion
equations in computational electrocardiology.

**Keywords: **Computational Science;
Computational electrocardiology; High performance computing;
Parallel computing; Space-time adaptivity;
Lightweight adaptive mesh data structures; Non-conforming discretizations;
Fast solution techniques for reaction-diffusion equations; Large-scale heart models

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