Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

David J. Knezevic; Endre Süli

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 43, Issue: 3, page 445-485
  • ISSN: 0764-583X

Abstract

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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 H1 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.

How to cite

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Knezevic, David J., and Süli, Endre. "Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift." ESAIM: Mathematical Modelling and Numerical Analysis 43.3 (2008): 445-485. <http://eudml.org/doc/194458>.

@article{Knezevic2008,
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 H1 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. },
author = {Knezevic, David J., Süli, Endre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Spectral methods; Fokker-Planck equations; transport-diffusion problems; FENE.; spectral methods; finitely extendible nonlinear elastic model (FENE); unbounded coefficients; polymer chains},
language = {eng},
month = {12},
number = {3},
pages = {445-485},
publisher = {EDP Sciences},
title = {Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift},
url = {http://eudml.org/doc/194458},
volume = {43},
year = {2008},
}

TY - JOUR
AU - Knezevic, David J.
AU - Süli, Endre
TI - Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/12//
PB - EDP Sciences
VL - 43
IS - 3
SP - 445
EP - 485
AB - 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 H1 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.
LA - eng
KW - Spectral methods; Fokker-Planck equations; transport-diffusion problems; FENE.; spectral methods; finitely extendible nonlinear elastic model (FENE); unbounded coefficients; polymer chains
UR - http://eudml.org/doc/194458
ER -

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Citations in EuDML Documents

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  1. John W. Barrett, Endre Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off
  2. John W. Barrett, Endre Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off
  3. John W. Barrett, Endre Süli, Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers
  4. John W. Barrett, Endre Süli, Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers
  5. David J. Knezevic, Endre Süli, A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

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