Numerical Analysis and Scientific Computing Preprint Seria An algebraic solver for the Oseen problem with application to hemodynamics

The paper studies an iterative solver for algebraic problems arising in numerical simulation of blood flows. Here we focus on a numerical solver for the fluid part of otherwise coupled fluid-structure system of equations which models the hemodynamics in vessels. Application of the finite element method and semiimplicit time discretization leads to the discrete Oseen problem on every time step of the simulation. The problem challenges numerical methods by anisotropic geometry, open boundary conditions, small time steps and transient flow regimes. We review known theoretical results and study the performance of recently proposed preconditioners based on two-parameter threshold ILU factorization of nonsymmetric saddle point problems. The preconditioner is applied to the linearized Navier–Stokes equations discretized by the stabilized Petrov–Galerkin finite element (FE) method. Careful consideration is given to the dependence of the solver on the stabilization parameters of the FE method. We model the blood flow in the digitally reconstructed right coronary artery under realistic physiological regimes. The paper discusses what is special in such flows for the iterative algebraic solvers, and shows how the two-parameter ILU preconditioner is able to meet these specifics. Igor N. Konshin Institute of Numerical Mathematics, Institute of Nuclear Safety, Russian Academy of Sciences, Moscow; e-mail: [email protected] Maxim A. Olshanskii Department of Mathematics, University of Houston; e-mail: [email protected] Yuri V. Vassilevski Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow Institute of Physics and Technology, Moscow; e-mail: [email protected]