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Preprints, Working Papers, ... Year : 2024

A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

Abstract

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.
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hal-04413333 , version 1 (23-01-2024)

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Zhaonan Dong, Emmanuil H Georgoulis, Philip J. Herbert. A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation. 2024. ⟨hal-04413333⟩
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