A parallel solver for reaction-diffusion systems in computational electrocardiology

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain %(AB) cardiac model, consisting of a system of two degenerate parabolic reaction-diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. %In case of equal anisotropy ratio, this system reduces to The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction-diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh-Nagumo (FHN) model to the more complex phase-I Luo-Rudy model (LR1). The solver employs structured isoparametric $Q_1$ finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to $O(10^7)$ unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains

Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the latter remains bounded and the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied

Comparison of single- and multistage strategies during fenestrated-branched endovascular aortic repair of thoracoabdominal aortic aneurysms

Objective: The aim of this study was to compare outcomes of single or multistage approach during fenestrated-branched endovascular aortic repair (FB-EVAR) of extensive thoracoabdominal aortic aneurysms (TAAAs). Methods: We reviewed the clinical data of consecutive patients treated by FB-EVAR for extent I to III TAAAs in 24 centers (2006-2021). All patients received a single brand manufactured patient-specific or off-the-shelf fenestrated-branched stent grafts. Staging strategies included proximal thoracic aortic repair, minimally invasive segmental artery coil embolization, temporary aneurysm sac perfusion and combinations of these techniques. Endpoints were analyzed for elective repair in patients who had a single- or multistage approach before and after propensity score adjustment for baseline differences, including the composite 30-day/in-hospital mortality and/or permanent paraplegia, major adverse event, patient survival, and freedom from aortic-related mortality. Results: A total of 1947 patients (65% male; mean age, 71 ± 8 years) underwent FB-EVAR of 155 extent I (10%), 729 extent II (46%), and 713 extent III TAAAs (44%). A single-stage approach was used in 939 patients (48%) and a multistage approach in 1008 patients (52%). A multistage approach was more frequently used in patients undergoing elective compared with non-elective repair (55% vs 35%; P < .001). Staging strategies were proximal thoracic aortic repair in 743 patients (74%), temporary aneurysm sac perfusion in 128 (13%), minimally invasive segmental artery coil embolization in 10 (1%), and combinations in 127 (12%). Among patients undergoing elective repair (n = 1597), the composite endpoint of 30-day/in-hospital mortality and/or permanent paraplegia rate occurred in 14% of single-stage and 6% of multistage approach patients (P < .001). After adjustment with a propensity score, multistage approach was associated with lower rates of 30-day/in-hospital mortality and/or permanent paraplegia (odds ratio, 0.466; 95% confidence interval, 0.271-0.801; P = .006) and higher patient survival at 1 year (86.9±1.3% vs 79.6±1.7%) and 3 years (72.7±2.1% vs 64.2±2.3%; adjusted hazard ratio, 0.714; 95% confidence interval, 0.528-0.966; P = .029), compared with a single stage approach. Conclusions: Staging elective FB-EVAR of extent I to III TAAAs was associated with decreased risk of mortality and/or permanent paraplegia at 30 days or within hospital stay, and with higher patient survival at 1 and 3 years

BDDC and FETI-DP preconditioners for spectral element discretizations

Two of the most recent and important nonoverlapping domain decomposition methods, the BDDC method (Balancing Domain Decomposition by Constraints) and the FETI-DP method (Dual–Primal Finite Element Tearing and Interconnecting) are here extended to spectral element discretizations of second-order elliptic problems. In spite of the more severe ill-conditioning of the spectral element discrete systems, compared with low-order finite elements and finite differences, these methods retain their good properties of scalability, quasi-optimality and independence on the discontinuities of the elliptic operator coefficients across subdomain interfaces