A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915082446831616 |
|---|---|
| author | Dong, Zhaonan Georgoulis, Emmanuil H. Herbert, Philip J. |
| author_facet | Dong, Zhaonan Georgoulis, Emmanuil H. Herbert, Philip J. |
| contents | We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a \emph{numerical hypocoercivity} property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting ``stronger-than-energy'' stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the ``time'' variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation Dong, Zhaonan Georgoulis, Emmanuil H. Herbert, Philip J. Numerical Analysis 65N30 We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a \emph{numerical hypocoercivity} property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting ``stronger-than-energy'' stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the ``time'' variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings. |
| title | A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation |
| topic | Numerical Analysis 65N30 |
| url | https://arxiv.org/abs/2401.12921 |