Hybrid element-based approximation for the Navier-Stokes equations in pipe-like domains

Motivated by its applicability to model blood flow in the cardiovascular system, in this work we propose a numerical approximation of the Navier-Stokes equations maintaining comparable accuracy with traditional finite element methods while performing a substantial reduction in the problem size. The method is envisaged for domains in which flow phenomena exhibits a dominant axial direction. The strategy consists of combining a finite element approximation for the description of fields (velocity components and pressure) in the primary (axial) direction, while taking a polynomial spectral approximation for the description of fields across the secondary (transversal) direction. In essence, this method can be understood as an extension of the so-called hierarchical modeling (HiMod) approach. Special attention is given to domains featuring sudden constrictions/expansions and bifurcations, for which a modified spectral approximation is considered. The capabilities and potentialities of the proposed approach to simulate fluid flow phenomena in pipe-like domains are presented through several examples.