On the a posteriori error analysis for linear Fokker-Planck models in convection-dominated diffusion problems

TL;DR

This paper develops reliable a posteriori error estimates for convection-dominated diffusion problems modeled by linear Fokker-Planck equations, with numerical validation demonstrating their effectiveness in adaptive mesh refinement.

Contribution

It introduces computable functional-type error bounds for static and time-dependent Fokker-Planck models under various boundary conditions, enhancing error control in numerical simulations.

Findings

Error estimates are reliable and efficient.

Numerical examples validate the error bounds.

Adaptive mesh refinement improves solution accuracy.

Abstract

This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection-dominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain computable error bounds of the functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). Finally, we present a set of various numerical examples including discussions on mesh adaptivity and space-time discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived.