A Posteriori Error Estimates for Non-Stationary Non-Linear Convection-Diffusion Equations

TL;DR

This paper develops a posteriori error estimates for non-stationary, non-linear convection-diffusion equations, addressing challenges posed by non-linearity and parameter interactions, with implications for stochastic convection-diffusion problems.

Contribution

It introduces novel a posteriori error estimates tailored for complex non-linear, non-stationary convection-diffusion equations with non-differentiable nonlinearities.

Findings

Error estimates account for parameter interplay

Framework applicable to stochastic convection-diffusion problems

Addresses non-linearity without standard differentiability assumptions

Abstract

Motivated by stochastic convection-diffusion problems we derive a posteriori error estimates for non-stationary non-linear convection-diffusion equations acting as a deterministic paradigm. The problem considered here neither fits into the standard linear framework due to its non-linearity nor into the standard non-linear framework due to the lacking differentiability of the non-linearity. Particular attention is paid to the interplay of the various parameters controlling the relative sizes of diffusion, convection, reaction, and non-linearity (noise).