A vast amount of natural processes can be modelled by
partial differential equations involving diffusion operators.
The Navier-Stokes equations of fluid dynamics is one of the most
popular of such models, but many other equations describing
flows involve diffusion processes.
These equations are often non-linear and coupled, and theoretical
analysis can only provided limited information on the qualitative
behaviours of their solutions. Numerical analysis is then used to
obtain a prediction of the fluid's behaviour.
In many circumstances, the numerical methods used to approximate the
models must satisfy engineering or computational constraints. For
examples, in underground flows in porous media (involved in oil
recovery, carbon storage or hydrogeology), the diffusions properties
of the medium vary a lot between geological layers, and can be
strongly skewed in one direction. Moreover, the available meshes
used to discretise the equations may be quite irregular. The sheer
size of the domain of study (a few kilometres wide) also calls for
methods that can be easily parallelised and give good and stable
results on relatively large grids. These constraints make the construction
and study of numerical methods for diffusion models very challenging.
In the first part of this talk, I will present some numerical schemes,
developed in the last 10 years and designed to discretise diffusion
equations as encountered in reservoir engineering, with all the
associated constraints. In the second part, I will focus
on mathematical tools and techniques constructed to analyse the
convergence of numerical schemes under realistic hypotheses (i.e.
without assuming non-physical smoothness on the data or the solutions).
These techniques are based on the adaptation to the discrete setting
of functional analysis results used to study the continuous equations.