We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection–diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Péclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Péclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

The numerical modeling of solutes absorption processes by the arterial wall is of paramount interest for the understanding of the relationships between the local features of blood flow, the nourishing of the inner arterial wall by the blood solutes, and the pathologies that can appear when this process is for some reason perturbed. In the present work, two models for the solutes dynamics are investigated. In the first model, which is essentially based on the one introduced by Rappitsch and Perktold [J. Biomech. Engrg., 118 (1996), pp. 511--519] and Rappitsch, Perktold, and Pernkopf [Internat. J. Numer. Methods Fluids, 25 (1997), pp. 847--857], the Navier--Stokes equations for an incompressible fluid, describing the blood velocity and pressure fields, are coupled with an advection-diffusion equation for the solute concentration. The wellposedness of this model is discussed. The second model considers also the solutes dynamics "inside" the arterial wall, described by a pure diffusion equation. Actually, this is a heterogeneous model, coupling different equations in different parts of the domain at hand. Its wellposedness is proven. Moreover, in view of the numerical study, an iterative finite element method by subdomains is proposed and its convergence properties are analyzed. Finally, several numerical results comparing the different models in situations of physiologic interest are illustrated.

We propose a domain decomposition method for advection‐diffusion‐reaction equations based on Nitsche’s transmission conditions. The advection‐dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries. We prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method. The convergence of the resulting domain decomposition method is proved, and this result holds true uniformly with respect to the diffusion parameter. The numerical scheme that we propose here can thus be applied straightforwardly to diffusion‐dominated, advection‐dominated, and hyperbolic problems. Some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method, and we compare our method with some other domain decomposition techniques for advection‐diffusion equations.