Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

Abstract

Abstract. This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization of the differential operator based on the Maxwellian M corresponding to U , which vanishes along ∂D, we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through M , in the principal part of the operator. The general class of admissible potentials considered includes the FENE (finitely extendible nonlinear elastic) model. We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discrete spectral Galerkin method for such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted H 1 norm on D. In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with the original operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions. Mathematics Subject Classification.

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

“…Note that the algorithm proposed in [15,3] involves an LU decomposition for dense matrices, so the computational cost is O(8M 3 N 3 ) for initial setup and then O(4M 3 N 2 ) for every time step. The algorithm proposed in [20] has a slightly better computational complexity: O(8M 3 N 3 ) for initial setup and O(4M 2 N 2 ) for every time step.…”

“…In [15], the case s = b/2 was considered. In this particular case, the weak formulation in [15] is equivalent to (2.13).…”

“…While for given K and U (1.7) is linear, the unbounded FENE potential term ∇ · (f ∇U ) introduces difficulties in both mathematical analysis and numerical approximation. Recently, there have been many mathematical and numerical studies of the homogeneous system (1.7); see, for instance, [20,7,8,19,29,15,14] and the references therein. In particular, some delicate numerical algorithms based on spectral methods were developed recently for the Fokker-Planck equation of the FENE model in [20,7,15,14,6].…”

“…Later, Fan [10] improved Warner's method and used the results of the Fokker-Planck equation to verify the approximation of the FENE-P model. Our work is inspired by the recent work of Knezevic and Süli [15,14] and aimed at developing optimal spectral algorithms with minimal assumptions on the initial condition. More precisely, our approach will be based on a new weighted weak formulation which allows initial conditions in a much larger functional space, and we construct unconditionally stable semi-implicit schemes in time and spectral-Galerkin algorithms in space with optimal computational complexity while properly handing the pole conditions and unbounded drift.…”

Finite element approximation of kinetic dilute polymer models with microscopic cut-off

“…For a review of numerical algorithms for the approximation of kinetic models of dilute polymers see, for example, Section 4 of the survey article of Li and Zhang [32]; for recent progress on deterministic algorithms for the approximation of Fokker-Planck and coupled Navier-Stokes-Fokker-Planck systems, see, for example, Lozinski et al [37,38], and Knezevic and Süli [26,27].…”