Modeling 3-D compliant blood flow with FOSLS

Blood flow in large vessels is typically modeled using the Navier-Stokes equations for the fluid domain and elasticity equations for the vessel wall. As the wall deforms, additional complications are introduced because the shape of the fluid domain changes, necessitating the use of a re-mapping or re-griding process for the fluid region. Typically, this system (fluid, solid, mapping) is solved using an iterative approach in which the fluid, elastic, and mapping equations are solved in series until the iterations converge. We present a new approach based on multilevel minimization of the finite element approximation error using a least-squares (LS) norm. This approach allows for minimization of the error for the entire system or in selected parts. The multilevel LS approach overcomes many shortcomings of standard techniques. Most notably, the computational cost of solving the problem increases linearly with the degrees of freedom and the associated least-squares functional provides an a posteriori error measure. This paper compares the LS finite element approach to other popular numerical methods, specifically, the commercial package CFD-ACE. The focus of the comparison is on accuracy, computational cost, scalability (both parallel and serial), and flexibility. We show that the multilevel LS finite element approach scales optimally (i.e., linearly in serial environments), while the other methods degrade substantially as the problem size increases.