Many models of flows in porous media, involved for example in oil recovery, carbon sequestration or underground water engineering, involve Partial Differential Equations with diffusion terms. Real-world data (rock properties, meshes available in reservoir simulation, etc.) encountered in these applications are hardly adapted to classical numerical methods. Moreover, convergence analysis of numerical schemes for diffusion equations is usually done by establishing error estimates, which requires non-physical regularity assumptions on the data or the solution and is only feasible on the most simple models (linear or semi/quasi-linear).
In this talk, I will present one of the numerical methods developed in the last 10-15 years for discretising diffusion terms on the kind of unstructured and generic grids used in reservoir simulations. I will also introduce a set of techniques, called Discrete Functional Analysis and built on the Functional Analysis developed for the study of the continuous PDEs, that allow us to perform convergence analysis of these methods for numerous complex models and under regularity assumptions fully compatible with real-world data.