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This paper is concerned with a blood flow problem coupled with a slow plaquegrowth at the artery wall. In the model, the micro (fast) system is theNavier-Stokes equation with a periodically applied force and the macro (slow)system is a fractional reaction equation, which is used to describe the plaquegrowth with memory effect. We construct an auxiliary temporal periodic problemand an effective time-average equation to approximate the original problem andanalyze the approximation error of the corresponding linearized PDE (Stokes)system, where the simple front-tracking technique is used to update the slowmoving boundary. An effective multiscale method is then designed based on theapproximate problem and the front tracking framework. We also present atemporal finite difference scheme with a spatial continuous finite elementmethod and analyze its temporal discrete error. Furthermore, a fast iterativeprocedure is designed to find the initial value of the temporal periodicproblem and its convergence is analyzed as well. Our designed front-trackingframework and the iterative procedure for solving the temporal periodic problemmake it easy to implement the multiscale method on existing PDE solvingsoftware. The numerical method is implemented by a combination of the finiteelement platform COMSOL Multiphysics and the mainstream software MATLAB, whichsignificantly reduce the programming effort and easily handle thefluid-structure interaction, especially moving boundaries with more complexgeometries. We present some numerical examples of ODEs and 2-D Navier-Stokessystem to demonstrate the effectiveness of the multiscale method. Finally, wehave a numerical experiment on the plaque growth problem and discuss thephysical implication of the fractional order parameter.

A fast front-tracking approach and its analysis for a temporal multiscale flow problem with a fractional-order boundary growth