The Fully Discrete Finite Element Algorithms For The Blood Solute Dynamics Model

In this paper we consider a heterogeneous model of solutes 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.This equation is coupled with a pure diffusive model accounting for the solute dynamics inside the arterial wall,where convection is negligible.The two subdomains(namely the lumen and the wall)are physically separated by the endothelial layer,which acts as a permeable membrane and the interface condition matching the two subproblems.The solute concentration is discontinuous across this membrane.We propose two decoupled backward Euler schemes based on different expressions of the interface.By domain decomposition methods,solving the different problems in the different subdomains.The convection dominated problem is solved by defining the stabilization terms via two local Gauss integrations at the element level.Stability and convergence results are derived for both schemes.The optimal error estimates in space could be achieved for velocity and concentration in the H~1norm with the proper choosing of stabilized parameters.The derived theoretical results are supported by numerical examples and a model problem from the physiological interest about transfusion is also considered.