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Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator
We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
On convergences of the squareroot approximation scheme to the Fokker-Planck operator
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.