  Computing the solution of linear systems of equations is invariably the most
time consuming task in the numerical solutions of PDEs in many fields of
computational science. In this study, we focus on the numerical simulation of
cardiovascular hemodynamics with rigid and deformable walls, discretized in
space and time through the variational multi-scale finite element method. We
focus on three approaches: the problem agnostic generalized minimum residual
(GMRES) and stabilized bi-conjugate gradient (BICGS) methods, and a recently
proposed, problem specific, bi-partitioned (BIPN) method. We also perform a
comparative analysis of several preconditioners, including diagonal,
block-diagonal, incomplete factorization, multi-grid, and resistance based
methods. Solver performance and matrix characteristics (diagonal dominance,
symmetry, sparsity, bandwidth and spectral properties) are first examined for
an idealized cylindrical geometry with physiologic boundary conditions and then
successively tested on several patient-specific anatomies representative of
realistic cardiovascular simulation problems. Incomplete factorization
pre-conditioners provide the best performance and results in terms of both
strong and weak scalability. The BIPN method was found to outperform other
methods in patient-specific models with rigid walls. In models with deformable
walls, BIPN was outperformed by BICG with diagonal and Incomplete LU
preconditioners.
