Convergence and Error Estimates for the Conservative Spectral Method for Fokker-Planck-Landau Equations

TL;DR

This paper rigorously derives error estimates for a conservative spectral method solving the space-homogeneous Fokker-Planck-Landau equation, establishing convergence and boundedness of solutions for the first time.

Contribution

It provides the first known error estimates for numerical methods approximating FPL equations across all potential ranges, demonstrating convergence and solution boundedness.

Findings

Unique solution with bounded moments

Derivatives remain bounded in $L^2$ spaces over time

Convergence to equilibrium states

Abstract

Error estimates are rigorously derived for a semi-discrete version of a conservative spectral method for approximating the space-homogeneous Fokker-Planck-Landau (FPL) equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in $L^2$ spaces globally time, under certain conditions. These estimates, combined with control of the spectral projection, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates FPL equations associated to any range of potentials.