Multiscale models for coupled reaction-diffusion equations
We consider reaction-diffusion equations in fractured domains, with a high dimensional gap between the dimension of the domain and the dimension of the fractures. An example is provided by fluid flow problems in three-dimensional porous media with a network of thin fractures. If the fractures are reduced to one-dimensional manifolds, the 3D equation of the asymptotic model features a measure term to account for mass conservation, which makes the considered problem non-standard. The main advantage of such reduced models is that the computational grid does not need to be adapted or conformal to the fractures. The 3D and 1D mesh are independent; moreover, for time dependent problems, the characteristic time/length scales of the 3D domain and of the fracture network can also be resolved independently. These features make this class of 3D-1D multiscale models attractive for applications to bioengineering problems, such as the analysis of tissue perfusion and of drug release from thin implantable devices, or for applications in geophysics. In this talk, we consider 3D-1D coupled reaction-diffusion problems and describe the main difficulties related to non-standard measure terms that govern the coupling conditions. We introduce special analytical and numerical tools and present some useful results about the analysis of the model and the numerical techniques for the approximate solution. We also describe the accuracy of the reduced model, and present a numerical convergence analysis. Finally, we show some of the computational results that have been obtained using the 3D-1D approach, with applications to blood flow simulation and to the analysis of drug release by biomedical devices.
