Partitioned schemes for the blood solute dynamics model by the variational multiscale method

In this paper, we consider a heterogeneous model of solute absorption processes by the arterial wall. This model is based on an advection-diffusion equation describing the solute dynamics in the vascular lumen, the convective field being provided by the blood velocity. A pure diffusive model coupled with this equation is considered for the solute dynamics inside the arterial wall. The two subdomains are physically separated by the endothelial layer, which acts as a selectively permeable membrane and the interface condition matching the two subproblems follow from the nature of this membrane. To compute the approximate solutions, we propose two partitioned schemes for this model by the variational multiscale method. Stability and convergence results are proved for both schemes. We derive error bounds of the fully discrete solution which are first order in time. The optimal error estimates in space could be achieved for the velocity and concentration in the H¹-norm, and pressure in the L²-norm with the proper choosing of stabilized parameters. Theoretical results are supported by numerical examples, and two model problems from the physiological interest are also considered.