Journal article

A domain decomposition method for advection-diffusion processes with application to blood solutes

We consider a heterogeneous model for the dynamics of a blood solute both in the vascular lumen and inside the arterial wall. In the lumen, we consider an advection-diffusion equation, where the convective field is provided by the velocity of blood, which is in turn obtained by solving the Navier-Stokes equations. Inside the arterial wall we consider a pure diffusive dynamics. Since the endothelial layer at the interface between the lumen and the wall acts as a permeable membrane, whose permeability depends on the shear rate exerted by the blood, the solute concentration is discontinuous across this membrane. A possible approach for the numerical study of this kind of problem is inspired by domain decomposition techniques. In particular, we introduce a splitting in the computation and alternate the solution of the advection diffusion equation in the lumen with that of the diffusion equation in the wall. We set up an efficient iterative method, based on a suitable reformulation of the problem in terms of a Steklov-Poincare interface equation. This formulation is a nonstandard one because of the concentration discontinuity at the lumen-wall interface and plays a key role in the proof of convergence of our method. In particular, we prove that the convergence rate performed by the proposed method is independent of the finite element space discretization and provides a criterion for the selection of an acceleration parameter. Several numerical results, referred to as biomedical applications, support our theoretical conclusions and illustrate the efficiency of this algorithm

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