WIAS Preprint No. 2399, (2017)

On convergences of the squareroot approximation scheme to the Fokker--Planck operator



Authors

  • Heida, Martin

2010 Mathematics Subject Classification

  • 35B27 35Q84 49M25 60H25 60K37 80M40 35R60 47B80

Keywords

  • Finite volumes, Voronoi, discretization, Fokker-Planck, Smoluchowski, Langevin dynamics, square root approximation, stochastic homogenization, G-convergence

DOI

10.20347/WIAS.PREPRINT.2399

Abstract

We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

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