  We are interested in a reduced order method for the efficient simulation of
blood flow in arteries. The blood dynamics is modeled by means of the
incompressible Navier-Stokes equations. Our algorithm is based on an
approximated domain-decomposition of the target geometry into a number of
subdomains obtained from the parametrized deformation of geometrical building
blocks (e.g. straight tubes and model bifurcations). On each of these building
blocks, we build a set of spectral functions by proper orthogonal decomposition
of a large number of snapshots of finite element solutions (offline phase). The
global solution of the Navier-Stokes equations on a target geometry is then
found by coupling linear combinations of these local basis functions by means
of spectral Lagrange multipliers (online phase). Being that the number of
reduced degrees of freedom is considerably smaller than their finite element
counterpart, this approach allows us to significantly decrease the size of the
linear system to be solved in each iteration of the Newton-Raphson algorithm.
We achieve large speedups with respect to the full order simulation (in our
numerical experiments, the gain is at least of one order of magnitude and grows
inversely with respect to the reduced basis size), whilst still retaining
satisfactory accuracy for most cardiovascular simulations.
