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

TL;DR

This paper introduces discrete functional analysis methods for analyzing the convergence of numerical schemes solving diffusion equations with irregular data, emphasizing linear elliptic equations and extensions to nonlinear models.

Contribution

It presents a framework using discrete Sobolev norms to prove convergence without regularity assumptions, applicable to both stationary and transient diffusion problems.

Findings

Convergence of numerical schemes established without regularity assumptions.

Techniques based on discrete Sobolev norms are effective for irregular data.

Extensions to nonlinear models like Navier--Stokes are discussed.

Abstract

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 on the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques can be adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on discrete Sobolev norms and the translation to the discrete setting of functional analysis results.