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Convergences of the squareroot approximation scheme to the Fokker-Planck operator. (English) Zbl 1411.65121

Summary: We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker-Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by H. C. Lie et al. [SIAM J. Matrix Anal. Appl. 34, No. 2, 738–756 (2013; Zbl 1274.60232)] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. 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.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35Q84 Fokker-Planck equations
49M25 Discrete approximations in optimal control
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
60H25 Random operators and equations (aspects of stochastic analysis)
80M40 Homogenization for problems in thermodynamics and heat transfer

Citations:

Zbl 1274.60232
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