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

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.

28 pages

References in corpus (6)

• Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
• Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations
• Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton--Jacobi--Bellman--Isaacs problems with application to numerical homogenization
• Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form
• Computational multiscale methods for nondivergence-form elliptic partial differential equations
• Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem