Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
The paper is concerned with the analysis and implementation of a spectral Galerkin method 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 nu...
| Główni autorzy: | , |
|---|---|
| Format: | Report |
| Wydane: |
Unspecified
2007
|
| _version_ | 1826306919379238912 |
|---|---|
| author | Knezevic, D Suli, E |
| author_facet | Knezevic, D Suli, E |
| author_sort | Knezevic, D |
| collection | OXFORD |
| description | The paper is concerned with the analysis and implementation of a spectral Galerkin method 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 class of admissible potentials 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 approximation of such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted H1 norm on D. The theoretical results are illustrated by numerical experiments for the FENE model in two space dimensions. |
| first_indexed | 2024-03-07T06:55:12Z |
| format | Report |
| id | oxford-uuid:fdeaa025-2b83-4d67-a1c1-e0dc7ce7126a |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:55:12Z |
| publishDate | 2007 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:fdeaa025-2b83-4d67-a1c1-e0dc7ce7126a2022-03-27T13:32:14ZSpectral Galerkin approximation of Fokker-Planck equations with unbounded driftReporthttp://purl.org/coar/resource_type/c_93fcuuid:fdeaa025-2b83-4d67-a1c1-e0dc7ce7126aMathematical Institute - ePrintsUnspecified2007Knezevic, DSuli, EThe paper is concerned with the analysis and implementation of a spectral Galerkin method 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 class of admissible potentials 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 approximation of such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted H1 norm on D. The theoretical results are illustrated by numerical experiments for the FENE model in two space dimensions. |
| spellingShingle | Knezevic, D Suli, E Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift |
| title | Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift |
| title_full | Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift |
| title_fullStr | Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift |
| title_full_unstemmed | Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift |
| title_short | Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift |
| title_sort | spectral galerkin approximation of fokker planck equations with unbounded drift |
| work_keys_str_mv | AT knezevicd spectralgalerkinapproximationoffokkerplanckequationswithunboundeddrift AT sulie spectralgalerkinapproximationoffokkerplanckequationswithunboundeddrift |