Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and discrete forms of functional analysis results. The discrete functional analysis tools presented here are versatile and applicable to a number of numerical methods and models.