Finite element approximation of stationary Fokker--Planck--Kolmogorov equations with application to periodic numerical homogenization

TL;DR

This paper develops and analyzes finite element methods for stationary Fokker-Planck-Kolmogorov equations with periodic boundary conditions, addressing both weakly differentiable and bounded measurable coefficients, with applications to homogenization.

Contribution

It introduces a rigorous finite element approach for stationary FPK equations under different coefficient regularities and proposes an approximation scheme for the effective diffusion matrix.

Findings

Proved convergence and stability of the finite element method.

Demonstrated the method's effectiveness through numerical experiments.

Provided a new approach for homogenization problems with large drifts.

Abstract

We propose and rigorously analyze a finite element method for the approximation of stationary Fokker--Planck--Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. In particular, the Cordes setting guarantees the existence and uniqueness of a square-integrable invariant measure. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. Finally, we demonstrate the performance of the methods through numerical experiments.