Computational Modeling of the Kidney Hemodynamics With a Coupled Unsteady Stokes–Darcy Model

We consider the numerical simulation of blood flows in a patient-specific kidney including the renal artery, the renal vein, and the kidney tissue using a coupled system of unsteady Stokes-Darcy equations. The Stokes equations and the Darcy equations are implicitly coupled on the interfaces by enforcing three conditions, namely the conservation of mass, the balance of the normal force and the Beavers-Joseph-Saffman condition. To discretize the system we introduce a stabilized P1-P1-P1 finite element method for the spatial variables and an implicit backward Euler method for the temporal variable. A mathematical theory is developed to guarantee the stability and the convergence of the proposed discretization method. To efficiently solve the large, sparse and highly ill-conditioned algebraic systems, we further propose a Krylov subspace method preconditioned by a robust two-scale additive Schwarz method consisting of a mixed-dimensional coarse preconditioner with a 1D central-line preconditioner in the vascular region and a 3D preconditioner for the kidney tissue with some compatibility conditions imposed on the 1D and 3D interfaces. Some numerical experiments for a benchmark problem and a patient-specific kidney with physiologic parameters are presented to verify the accuracy, the robustness, and the effectiveness of the proposed method.